The Practice of Social Research by Earl R. Babbie (z-lib.org)

426 ■ Chapter 14: Analyzing Quantitative Data In analyzing a discrete variable—a nominal or Table 14-5 ordinal variable, for example—some of the tech- Marijuana Legalization by Age of Respondents, 2006 niques discussed previously do not apply. Strictly speaking, modes should be calculated for nominal 55 and data, medians for interval data, and means for ratio Under 21 21–35 36–54 Older data, not for nominal data (see Chapter 6). If the variable in question is sex, for example, raw numbers Should be legalized 33% 37% 38% 29% (23 of the cross-dressing outlaw bikers in our sample are women) or percentages (7 percent are women) Should not be 66 63 62 71 can be appropriate and useful analyses, but neither legalized a median nor a mean would make any sense. Cal- culating the mode would be legitimate, though not 100% = (57) (574) (704) (513) very revealing, because it would only tell us “most were men.” However, the mode for data on religious Source: General Social Survey, 2006, National Opinion Research Center. affiliation might be more interesting, as in “most people in the United States are Protestant.” Table 14-6 Marijuana Legalization by Political Orientation, 2006 Detail versus Manageability Should Should Not 100% = In presenting univariate and other data, you’ll be Legalize Legalize constrained by two goals. On the one hand, you should attempt to provide your reader with the Extremely liberal 50% 50 (59) fullest degree of detail regarding those data. On the other hand, the data should be presented in a Liberal 52% 48 (197) manageable form. As these two goals often directly counter each other, you’ll find yourself continually Slightly liberal 48% 52 (217) seeking the best compromise between them. One useful solution is to report a given set of data in Moderate 36% 64 (669) more than one form. In the case of age, for exam- ple, you might report the distribution of ungrouped Slightly conservative 34% 66 (292) ages plus the mean age and standard deviation. Conservative 17% 83 (294) As you can see from this introductory discussion of univariate analysis, this seemingly simple matter Extremely conservative 17% 83 (73) can be rather complex. In any event, the lessons of this section pave the way for a consideration of sub- Source: General Social Survey, 2006, National Opinion Research Center. group comparisons and bivariate analyses. Often it’s appropriate to describe subsets of Subgroup Comparisons cases, subjects, or respondents. Here’s a simple example from the General Social Survey (GSS). Univariate analyses describe the units of analysis of In 2006, respondents were asked, “Should mari- a study and, if they are a sample drawn from some juana be made legal?” In response, 34.9 percent larger population, allow us to make descriptive in- said it should and 65.1 percent said it shouldn’t. ferences about the larger population. Bivariate and Table 14-5 presents the responses given to multivariate analyses are aimed primarily at expla- this question by respondents in different age nation. Before turning to explanation, however, we categories. should consider the case of subgroup description. Notice that the subgroup comparisons tell us how different groups in the population responded to this question. You can undoubtedly see a pattern in the results, though possibly not exactly what you expected; we’ll return to that in a moment. First, let’s see how another set of subgroups an- swered this question. Table 14-6 presents different political subgroups’ attitudes toward legalizing marijuana, based on whether respondents characterized themselves as

Subgroup Comparisons ■ 427 Table 14-7 Attitudes toward the United Nations:“How is the UN doing in solving the problems it has had to face?” West Germany Britain France Japan United States Very good job    2%    7%    2%    1%    5% Good job 46 39 45 11 46 Poor job 21 28 22 43 27 Very poor job 6 9 3 5 13 Don’t know 26 17 28 41 10 Source:“5-Nation Survey Finds Hope for U.N.,”New York Times, June 26, 1985, p. 6. conservative or liberal. Before looking at the table, selecting the two extreme response categories: you might try your hand at hypothesizing what The UN is doing a very good or a very poor job. the results are likely to be and why. Notice that I’ve Furthermore, although it might be tempting to changed the direction of percentaging this table, to read only the second line of the table (those saying make it easier to read. To compare the subgroups in “good job”), that would be improper. Looking at this case, you would read down the columns, not only the second row, we would conclude that West across them. ­Germany and the United States were the most pos- itive (46 percent) about the UN’s performance, fol- Before examining the logic of causal analysis, lowed closely by France (45 percent), with Britain let’s consider another example of subgroup com- (39 percent) less positive than any of those three parisons: one that will let us address some table- and Japan (11 percent) the least positive of all. formatting issues. This procedure is inappropriate in that it ig- “Collapsing” Response Categories nores all those respondents who gave the most positive answer of all: “very good job.” In a situa- “Textbook examples” of tables are often simpler tion like this, you should combine or “collapse” the than you’ll typically find in published research two ends of the range of variation. In this instance, reports or in your own analyses of data, so this combine “very good” with “good” and “very poor” s­ection and the next one address two common with “poor.” If you were to do this in the analysis problems and suggest solutions. of your own data, it would be wise to add the raw frequencies together and recompute percentages Let’s begin by turning to Table 14-7, which for the combined categories, but in analyzing a reports data collected in a multinational poll con- published table such as this one, you can simply ducted by the New York Times, CBS News, and add the percentages as illustrated by the results the Herald Tribune in 1985, concerning attitudes shown in Table 14-8. about the United Nations. The question reported in Table 14-7 deals with general attitudes about the With the collapsed categories illustrated in way the UN was handling its job. Table 14-8, we can now rather easily read across the several national percentages of people who Here’s the question: How do people in the five said the UN was doing at least a good job. Now the nations reported in Table 14-7 compare in their United States appears the most positive; Germany, support for the kind of job the UN was doing? Britain, and France are only slightly less positive As you review the table, you may find there are and are nearly indistinguishable from one another; simply so many numbers that it’s hard to see any and Japan stands alone in its quite low assessment meaningful pattern. of the UN’s performance. Although the conclusions to be drawn now do not differ radically from what Part of the problem with Table 14-7 lies in the relatively small percentages of respondents

428 ■ Chapter 14: Analyzing Quantitative Data Table 14-8 Collapsing Extreme Categories Good job or better West Germany Britain France Japan United States Poor job or worse Don’t know   48%   46%   47%   12%   51% 27 37 25 48 40 26 17 28 41 10 Table 14-9 Omitting the“Don’t Knows” West Germany Britain France Japan United States Good job or better 65% 55% 65% 20% 57% Poor job or worse 35% 45% 35% 81% 44% we might have concluded from simply reading Notice there is a good deal of variation in the the second line of Table 14-7, we should note that national percentages saying “don’t know” in this B­ ritain now appears relatively more supportive. instance, ranging from only 10 percent in the United States to 41 percent in Japan. The presence Here’s the risk I’d like to spare you. Suppose of substantial percentages saying they don’t know you had hastily read the second row of Table 14-7 can confuse the results of tables like these. For and noted that the British had a somewhat lower example, was it simply because so many ­Japanese assessment of the job the UN was doing than was didn’t express any opinion that they seemed true of people in the United States, West Germany, so much less likely to say the UN was doing a and France. You might feel obliged to think up an good job? explanation for why that was so—possibly creat- ing an ingenious psychohistorical theory about the Here’s an easy way to recalculate percentages, painful decline of the once powerful and dignified with the “don’t knows” excluded. Look at the British Empire. Then, once you had touted your first column of percentages in Table 14-8: West “theory” about, someone else might point out that G­ ermany’s answers to the question about the UN’s a proper reading of the data would show the Brit- performance. Notice that 26 percent of the respon- ish were actually not really less positive than the dents said they didn’t know. This means that those other three nations. This is not a hypothetical risk. who said “good” or “bad” job—taken together— Errors like these happen frequently, but they can represent only 74 percent (100 minus 26) of the be avoided by collapsing answer categories where whole. If we divide the 48 percent saying “good appropriate. job or better” by 0.74 (the proportion giving any opinion), we can say that 65 percent “of those with Handling “Don’t Knows” an opinion” said the UN was doing a good or very good job (48% 4 0.74 5 65%). Tables 14-7 and 14-8 illustrate another common problem in the analysis of survey data. It’s usually a Table 14-9 presents the whole table with the good idea to give people the option of saying “don’t “don’t knows” excluded. Notice that these new know” or “no opinion” when asking for their opin- data offer a somewhat different interpretation ions on issues. But what do you do with those an- than the previous tables do. Specifically, it would swers when you analyze the data? now appear that France and West Germany were the most positive in their assessments of the UN,

Subgroup Comparisons ■ 429 with the United States and Britain a bit lower. patients had said as well as offering a brief eth- A­ lthough Japan still stands out as lowest in nography of the setting and of certain behav- this regard, it has moved from 12 percent to ioural data. In addition, however, I constructed 20 percent positive. a coding form which enabled me to collate a number of crude measures of doctor and At this point, having seen three versions of p­ atient interactions. the data, you may be asking yourself, Which is the right one? The answer depends on your purpose in (1993: 163) analyzing and interpreting the data. For example, if it’s not essential for you to distinguish “very good” Not only did the numerical data fine-tune from “good,” it makes sense to combine them, be- S­ ilverman’s impressions based on his qualitative cause it’s easier to read the table. observations, but his in-depth understanding of the situation allowed him to craft an evermore Whether to include or exclude the “don’t appropriate quantitative analysis. Listen to the knows” is harder to decide in the abstract. It may ­interaction between qualitative and quantitative be a very important finding that such a large per- approaches in this lengthy discussion: centage of the Japanese had no opinion—if you wanted to find out whether people were familiar My overall impression was that private with the work of the UN, for example. On the ­consultations lasted considerably longer other hand, if you wanted to know how people than those held in the NHS clinics. When might vote on an issue, it might be more appropri- examined, the data indeed did show that ate to exclude the “don’t knows” on the assump- the ­former were almost twice as long as tion that they wouldn’t vote or that ultimately they the latter (20 minutes as against 11 minutes) would be likely to divide their votes between the and that the difference was statistically two sides of the issue. highly significant. However, I recalled that, for special reasons, one of the NHS clinics In any event, the truth contained within your had abnormally short consultations. I felt a data is that a certain percentage said they didn’t fairer comparison of consultations in the know and the remainder divided their opinions in two s­ ectors should exclude this clinic and whatever manner they did. Often, it’s appropriate should only compare consultations taken to report your data in both forms—with and with- by a single doctor in both sectors. This out the “don’t knows”—so your readers can draw s­ ubsample of cases revealed that the differ- their own conclusions. ence in length between NHS and private consultations was now ­reduced to an average Numerical Descriptions of under 3 minutes. This was still statisti- in Qualitative Research cally significant, although the significance was ­reduced. Finally, however, if I compared Although this chapter deals primarily with only new patients seen by the same doctor, quantitative research, the discussions also apply NHS ­patients got 4 minutes more on the to qualitative studies. Numerical testing can ­average—34 minutes as against 30 ­minutes often verify the findings of in-depth, qualitative in the private clinic. studies. Thus, for example, when David Silver- man wanted to compare the cancer treatments (1993: 163–64) received by patients in private clinics with the cancer treatments in Britain’s National Health This example further demonstrates the special Service, he primarily chose in-depth analyses of power that can be gained from a combination of the interactions between doctors and patients: approaches in social research. The combination of qualitative and quantitative analyses can be My method of analysis was largely qualitative e­ specially potent. and . . . I used extracts of what doctors and

430 ■ Chapter 14: Analyzing Quantitative Data Bivariate Analysis Table 14-10 In contrast to univariate analysis, subgroup Religious Attendance Reported by Men and Women in ­comparisons involve two variables. In this re- 2006 spect subgroup comparisons constitute a kind of ­bivariate analysis—that is, the analysis of two Men Women variables simultaneously. However, as with uni- variate analysis, the purpose of subgroup compari- Weekly 26% 35% sons is largely descriptive. Most bivariate analysis in Less often 74 65 social research adds another element: determining 100% = (2,049) (2,443) relationships between the variables themselves. Thus, univariate analysis and subgroup compari- Source: General Social Survey, 2006, National Opinion Research Center. sons focus on describing the people (or other units of analysis) under study, whereas bivariate analy- 2. People denied status gratification in the secular sis focuses on the variables and their empirical society may turn to religion as an alternative relationships. source of status. Table 14-10 could be regarded as an instance 3. Hence, women should be more religious than of subgroup comparison: It independently de- men. scribes the religious services attendance of men and women, as reported in the 2006 General ­Social Sur- The data presented in Table 14-10 confirm this vey. It shows—comparatively and d­ escriptively— reasoning. Thirty-five percent of the women attend that the women under study a­ ttended church religious services weekly, as compared with 26 per- more often than the men did. However, the same cent of the men. table, seen as an explanatory bivariate analysis, tells a somewhat different story. It suggests that the Using the logic of causal relationships among variable sex has an effect on the variable church at- variables has an important implication for the tendance. That is, we can view the behavior as a de- construction and reading of percentage tables. pendent variable that is partially determined by the One of the chief bugaboos for new-data analysts independent variable, sex. is deciding on the appropriate “direction of per- centaging” for any given table. In Table 14-10, for Explanatory bivariate analyses, then, involve example, I’ve divided the group of subjects into the “variable language” introduced in Chapter 1. two subgroups—men and women—and then In a subtle shift of focus, we’re no longer talking described the behavior of each subgroup. That about men and women as different subgroups but is the correct method for constructing this table. about sex as a variable: one that has an influence Notice, however, that we could—however inap- on other variables. The theoretical interpreta- propriately—construct the table differently. We tion of Table 14-10 might be taken from Charles could first divide the subjects into different degrees Glock’s Comfort Hypothesis as discussed in of religious services attendance and then describe Chapter 3: each of those subgroups in terms of the percentage of men and women in each. This method would 1. Women are still treated as second-class citizens make no sense in terms of explanation, however. in U.S. society. Table 14-10 suggests that your sex will affect your frequency of religious services attendance. Had bivariate analysis  The analysis of two variables we used the other method of construction, the simultaneously, for the purpose of determining the table would suggest that your religious services empirical relationship between them. The construc- attendance affects whether you’re a man or a tion of a simple percentage table or the computation woman—which makes no sense. Your behavior of a simple correlation coefficient are examples of can’t determine your sex. bivariate analyses.

Bivariate Analysis ■ 431 A related problem complicates the lives of total 100 percent each, it has been percentaged new-data analysts. How do you read a percent- across. The rule, then, is as follows: age table? There is a temptation to read Table 14-10 as follows: “Of the women, only 35 ­percent 1. If the table is percentaged down, read across. a­ ttended religious services weekly, and 65 percent said they attended less often; therefore, being a 2. If the table is percentaged across, read down. woman makes you less likely to attend religious services frequently.” This is, of course, an incor- Percentaging a Table rect reading of the table. Any conclusion that sex—as a v­ ariable—has an effect on religious Figure 14-7 reviews the logic by which we create service a­ ttendance must hinge on a comparison percentage tables from two variables. I’ve used as between men and women. Specifically, we com- variables sex and attitudes toward equality for men and pare the 35 p­ ercent with the 26 percent and note women. that women are more likely than men to attend religious services weekly. The comparison of sub- Here’s another example. Suppose we’re in- groups, then, is essential in reading an explana- terested in learning something about newspaper tory bivariate table. editorial positions regarding the legalization of marijuana. We undertake a content analysis of edi- In constructing and presenting Table 14-10, I’ve torials on this subject that have appeared during a used a convention called percentage down. This term given year in a sample of daily newspapers across means that you can add the percentages down the nation. Each editorial has been classified as each column to total 100 percent (with the pos- favorable, neutral, or unfavorable toward the legal- sibility of a rounding error). You read this form of ization of marijuana. Perhaps we wish to examine table across a row. For the row labeled “weekly,” the relationship between editorial policies and the what percentage of the men attend weekly? What types of communities in which the newspapers are percentage of the women ­attend weekly? published, thinking that rural newspapers might be more conservative in this regard than urban ones. The direction of percentaging in tables is ar- Thus, each newspaper (hence, each editorial) has bitrary, and some researchers prefer to percent- been classified in terms of the population of the age across. They would organize Table 14-10 so community in which it is published. that “men” and “women” were shown on the left side of the table, identifying the two rows, and Table 14-11 presents some hypothetical data “weekly” and “less often” would appear at the top describing the editorial policies of rural and urban to identify the columns. The actual numbers in the newspapers. Note that the unit of analysis in this table would be moved around accordingly, and example is the individual editorial. Table 14-11 tells each row of percentages would total 100 percent. us that there were 127 editorials about marijuana In that case, you would read the table down a in our sample of newspapers published in com- column, still asking what percentage of men and munities with populations under 100,000. (Note women attended frequently. The logic and the con- that this cutting point is chosen for simplicity of clusion would be the same in either case; only the illustration and does not mean that rural refers to form would differ. a community of less than 100,000 in any absolute sense.) Of these, 11 percent (14 editorials divided In reading a table that someone else has con- by the base of 127) were favorable toward legaliza- structed, therefore, you need to find out in which tion of marijuana, 29 percent were neutral, and direction it has been percentaged. Usually this will 60 ­percent were unfavorable. Of the 438 editori- be labeled or be clear from the logic of the vari- als that appeared in our sample of newspapers ables being analyzed. As a last resort, however, you published in communities of more than 100,000 should add the percentages in each column and residents, 32 percent (140 editorials) were favor- each row. If each of the columns totals 100 percent, able toward legalizing marijuana, 40 percent were the table has been percentaged down. If the rows neutral, and 28 percent were unfavorable.

432 ■ Chapter 14: Analyzing Quantitative Data Figure 14-7

Bivariate Analysis ■ 433 Table 14-11 the dependent variable. Thus, we proceed as follows: Hypothetical Data Regarding Newspaper Editorials on the Legalization of Marijuana 1. The cases are divided into men and women. Editorial Policy Community Size 2. Each sex subgrouping is described in terms of toward Legalizing approval or disapproval of sexual equality. Marijuana Under 100,000   Over 100,000 3. Men and women are compared in terms of the Favorable 11% 32% percentages approving of sexual equality. Neutral 29 40 Unfavorable 60 28 In the example of editorial policies regarding 100% = (127) (438) the legalization of marijuana, size of community is the independent variable, and a newspaper’s editorial When we compare the editorial policies of rural policy the dependent variable. The table would be and urban newspapers in our imaginary study, we constructed as follows: find—as expected—that rural newspapers are less favorable toward the legalization of marijuana than 1. Divide the editorials into subgroups according urban newspapers are. We determine this by noting to the sizes of the communities in which the that a larger percentage (32 percent) of the urban newspapers are published. editorials were favorable than the percentage of rural ones (11 percent). We might note as well that 2. Describe each subgroup of editorials in terms of more rural than urban editorials were unfavorable the percentages favorable, neutral, or unfavor- (60 percent compared with 28 percent). Note that able toward the legalization of marijuana. this table assumes that the size of a community might affect its newspapers’ editorial policies on this 3. Compare the two subgroups in terms of the issue, rather than that editorial policy might affect percentages favorable toward the legalization of the size of communities. marijuana. Constructing and Reading Bivariate analyses typically have an explana- Bivariate Tables tory causal purpose. These two hypothetical exam- ples have hinted at the nature of causation as social Let’s now review the steps involved in the con- scientists use it. struction of explanatory bivariate tables: Tables such as the ones we’ve been examin- 1. The cases are divided into groups according to ing are commonly called contingency tables: the attributes of the independent variable. Values of the dependent variable are contingent on (depend on) values of the independent variable. 2. Each of these subgroups is then described in Although contingency tables are common in social terms of attributes of the dependent variable. science, their format has never been standardized. As a result, you’ll find a variety of formats in re- 3. Finally, the table is read by comparing the inde- search literature. As long as a table is easy to read pendent variable subgroups with one another and interpret, there’s probably no reason to strive in terms of a given attribute of the dependent for standardization. However, there are several variable. guidelines that you should follow in the presenta- tion of most tabular data. Following these steps, let’s repeat the analysis of sex and attitude on sexual equality. For the rea- 1. A table should have a heading or a title that sons outlined previously, sex is the independent succinctly describes what is contained in the variable; attitude toward sexual equality constitutes table. contingency table  A format for presenting the relationships among variables as percentage distributions.

434 ■ Chapter 14: Analyzing Quantitative Data 2. The original content of the variables should Introduction to Multivariate be clearly presented—in the table itself if at Analysis all possible or in the text with a paraphrase in the table. This information is especially critical The logic of multivariate analysis, or the analy- when a variable is derived from responses to sis of more than two variables simultaneously, an attitudinal question, because the meaning of can be seen as an extension of bivariate analysis. the responses will depend largely on the word- Specifically, we can construct multivariate tables ing of the question. on the basis of a more complicated subgroup de- scription by following essentially the same steps 3. The attributes of each variable should be clearly outlined for bivariate tables. Instead of one in- indicated. Though complex categories will dependent variable and one dependent variable, have to be abbreviated, their meaning should however, we’ll have more than one independent be clear in the table and, of course, the full de- variable. Instead of explaining the dependent vari- scription should be reported in the text. able on the basis of a single independent variable, we’ll seek an explanation through the use of more 4. When percentages are reported in the table, than one independent variable. the base on which they are computed should be indicated. It’s redundant to present all Let’s return to the example of religious ser- the raw numbers for each category, because vices attendance. Suppose we believe that age these could be reconstructed from the per- would also affect such behavior (Glock’s Comfort centages and the bases. Moreover, the pre- H­ ypothesis suggests that older people are more sentation of both numbers and percentages religious than younger people). As the first step often confuses a table and makes it more in table construction, we would divide the total difficult to read. sample into subgroups based on the attributes of both independent variables simultaneously: 5. If any cases are omitted from the table because younger men, older men, younger women, and of missing data (“no answer,” for example), older women. Then the several subgroups would their numbers should be indicated in the be described in terms of the dependent v­ ariable, table. religious services attendance, and comparisons would be made. Table 14-12, from an analysis While I have introduced the logic of causal, bi- of the 2006 General Social Survey data, is the variate analysis in terms of percentage tables, there result. are many other formats appropriate to this topic. Scatterplot graphs are one possibility, providing a Table 14-12 has been percentaged down and visual display of the relationship between two vari- therefore should be read across. The interpretation ables. For an engaging example of this, you might of this table warrants several conclusions: check out the GapMinder software available on the web (see the link on your Sociology CourseMate 1. Among both men and women, older people at- at www.cengagebrain.com). Using countries as the tend religious services more often than younger unit of analysis, you can examine the relationship people do. Among women, 27 percent of those between birthrate and infant mortality, for exam- under 40 years of age, and 41 percent of those ple. In fact, you can watch the relationship develop 40 and older attend religious services weekly. over time. Among men, the respective figures are 19 and 31 percent. multivariate analysis  The analysis of the simul­tan­ eous relationships among several variables. ­Examining 2. Within each age group, women attend slightly simultaneously the effects of age, sex, and social class more frequently than men. Among those re- on religiosity would be an example of multivariate spondents under 40 years old, 27 percent of analysis. the women attend weekly, compared with 19 percent of the men. Among those 40 and over,

Sociological Diagnostics ■ 435 Table 14-12 Table 14-13 A Simplification of Table 14-12 Multivariate Relationship: Religious Service Attendance, Sex, and Age in 2006 Percent Who Attend about Weekly “How often do you attend religious services?” Men Women Under 40 40 and Older Under 40 19 27 40 and Older (832) (958) Men Women Men Women 31 41 About weekly* 19% 27% 31% 41% (1,211) (1,477) Less often 81 73 69 59 100% = (832) (958) (1,211) (1,477) Source: General Social Survey, 2006, National Opinion Research Center. *About weekly =“More than once a week,”“Weekly,”and“Nearly every week.” reported in the cells representing the intersections Source: General Social Survey, 2006, National Opinion Research Center. of the two independent variables. The numbers pre- sented in parentheses below each percentage repre- 41 percent of the women and 31 percent of the sent the number of cases on which the percentages men attend weekly. are based. Thus, for example, the reader knows there are 958 women under 40 years of age in the 3. As measured in the table, age appears to have a sample, and 27 percent of them attend religious ser- greater effect on attendance at religious services vices weekly. We can calculate from this that 262 of than does sex. those 958 women attend weekly and that the other 696 younger women (or 73 percent) attend less 4. Age and sex have independent effects on re- frequently. This new table is easier to read than the ligious service attendance. Within a given at- former one, and it does not sacrifice any detail. tribute of one independent variable, different attributes of the second still affect behaviors. Sociological Diagnostics 5. Similarly, the two independent variables The multivariate techniques we’re now exploring have a cumulative effect on behaviors. Older can serve as powerful tools for diagnosing social women attend the most often (41 percent), problems. They can be used to replace opinions and younger men attend the least often (19 with facts and to settle ideological debates with percent). data analysis. Before I conclude this section, it will be useful For an example, let’s return to the issue of to note an alternative format for presenting such sex and income. Many explanations have been data. Several of the tables presented in this chapter advanced to account for the long-standing pat- are somewhat inefficient. When the dependent tern of women in the labor force earning less than variable, religious attendance, is dichotomous (hav- men. One explanation is that, because of tradi- ing exactly two attributes), knowing one attribute tional family patterns, women as a group have permits the reader to reconstruct the other easily. participated less in the labor force and many only Thus, if we know that 27 percent of the women begin working outside the home after c­ ompleting under 40 attend religious services weekly, then certain child-rearing tasks. Thus, women as a we know automatically that 73 percent attend less group probably have less seniority at work than often. So reporting the percentages who attend less men do, and income increases with seniority. A often is unnecessary. 1984 study by the Census Bureau showed this reasoning to be partly true, as Table 14-14 shows. On the basis of this recognition, Table 14-12 could be presented in the alternative format of Table 14-13. In Table 14-13, the percentages of people say- ing they attend religious services “about weekly” are

436 ■ Chapter 14: Analyzing Quantitative Data Table 14-14 • Whether covered by a union contract Sex, Job Tenure, and Income, 1984* • Type of occupation • Number of employees in the firm Years Working with Average Hourly Income Women/Men • Whether private or public employer Current Employer Men Women Ratio • Whether they left previous job involuntarily • Time spent between current and previous job Less than 2 years $8.46 $6.03 0.71 • Race 2–4 years $9.38 $6.78 0.72 • Whether they have a disability 5–9 years $10.42 $7.56 0.73 • Health status 10 years or more $12.38 $7.91 0.64 • Age of children • Whether they took an academic curriculum in *Full-time workers 21–64 years of age high school Source: U.S. Bureau of the Census, Current Population Reports, Series P-70, No. 10, Male–Female Differences in Work Experience, Occupation, and Earning, • Number of math, science, and foreign language 1984 (Washington, DC: U.S. Government Printing Office, 1987), 4. classes in high school Table 14-14 indicates, first of all, that job tenure does indeed affect income. Among both men and • Whether they attended private or public high women, those with more years on the job earned more. This is seen by reading down the first two school columns of the table. • Educational level achieved The table also indicates that women earn less • Percentage of women in the occupation than men, regardless of job seniority. This can be • College major seen by comparing average wages across the rows of the table, and the ratio of women-to-men wages Each of the variables listed here might rea- is shown in the third column. Thus, years on the sonably affect earnings and, if women and men job is an important determinant of earnings, but differ in these regards, could help to account for seniority does not adequately explain the pattern of male/­female income differences. When all these women earning less than men. In fact, we see that variables were taken into account, the researchers women with ten or more years on the job earn could account for 60 percent of the discrepancy substantially less ($7.91/hour) than do men with between the incomes of men and women. The less than two years ($8.46/hour). remaining 40 percent, then, is a function of other “reasonable” variables and/or prejudice. This kind Although years on the job does not fully ex- of conclusion can be reached only by examining plain the difference between men’s and women’s the effects of several variables at the same time— pay, there are other possible explanations: level of that is, through multivariate analysis. education, child care responsibilities, and so forth. The researchers who calculated Table 14-14 also I hope this example shows how the logic examined some of the other variables that might implicit in day-to-day conversations can be reasonably explain the differences in pay without r­epresented and tested in a quantitative data representing gender discrimination, including these: a­ nalysis like this. Along those lines, you might be asking yourself, These data point to salary discrimi- • Number of years in the current occupation nation against women in 1984, but hasn’t that • Total years of work experience (any been remedied? Not really, as indicated by more- recent data. occupation) In 2008 the average full-time, year-round male • Whether they have usually worked full time worker earned $61,783. The average full-time, • Marital status year-round female worker earned $43,305, or • Size of city or town they live in about 70 percent as much as her male counter- part (U.S. Bureau of the Census 2011: Table 702,

Ethics and Quantitative Data Analysis ■ 437 Table 14-15 the same job in their employing organization. Fol- lowing their sex change, female-to-male transsexu- Average Earnings of Year-Round, Full-Time Workers by als were likely to enjoy pay raises and increased Educational Attainment, 2008 authority. In other studies, male-to-female trans- sexuals reported just the opposite experiences. Ratio of Women/ Personal accounts such as these flesh out statistical Men Women Men Earnings studies that consistently show women earning less than men, even when they do the same work. All workers $61,783 $43,305 0.70 Less than 9th grade 28,375 21,376 0.75 As another example of multivariate data analy- 9th–12th grades 33,457 22,246 0.66 sis in real life, consider the common observation H.S. graduates 43,493 31,666 0.73 that minority group members are more likely to Some college 50,433 36,019 0.71 be denied bank loans than white applicants are. A Associate degree 54,830 39,935 0.73 counterexplanation might be that the minority ap- Bachelor’s or more 94,206 60,293 0.64 plicants in question were more likely to have had a prior bankruptcy or that they had less collateral Note: These data point to a persistent difference between the incomes of men to guarantee the requested loan—both reasonable and women, even when both groups have achieved the same levels of education. bases for granting or denying loans. However, the kind of multivariate analysis we’ve just examined Source: U.S. Bureau of the Census, Statistical Abstract of the United States could easily resolve the disagreement. ­(Washington, DC: U.S. Government Printing Office, 2011), Table 702, p. 459. You can also access this table online at the link on your Sociology CourseMate at Let’s say we look only at those who have www.cengagebrain.com. not had a prior bankruptcy and who have a cer- tain level of collateral. Are whites and minorities p. 459). But does that difference represent sexual equally likely to get the requested loan? We could discrimination or does it reflect legitimate factors? conduct the same analysis in subgroups determined by level of collateral. If whites and minorities were Some argue that education, for example, af- equally likely to get their loans in each of the sub- fects income and that in the past, women have groups, we would need to conclude that there was gotten less education than men. We might start, no ethnic discrimination. If minorities were still less therefore, by checking whether educational dif- likely to get their loans, however, that would indi- ferences explain why women today earn less, on cate that bankruptcy and collateral differences were average, than men. Table 14-15 offers data to test not the explanation—strengthening the case that this hypothesis. discrimination was at work. As the table shows, at each level of comparable All this should make it clear that social research education, women earn substantially less than can play a powerful role in serving the human men do. Clearly, education does not explain the community. It can help us determine the cur- discrepancy. rent state of affairs and can often point the way to where we want to go. Sex and gender are not a simple matter of men and women for social researchers. For example, Welcome to the world of sociological transsexuals are individuals who choose to change diagnostics! their biological sex permanently through surgery and hormones. Clearly, such a radical change Ethics and Quantitative brings many adjustments and challenges that Data Analysis would make for interesting studies, but Kristen Schilt has taken an unusual tack. In Chapter 13, I pointed out that the subjectiv- ity present in qualitative data analysis increases While many kinds of research point to the the risk of biased analyses, which experienced disadvantaged status of women in the workplace, Schilt’s research on transsexuals reveals the impact of gender on a personal level. In many of the cases, the subjects changed their sex while maintaining

438 ■ Chapter 14: Analyzing Quantitative Data researchers learn to avoid. Some people believe • Researchers may use existing coding schemes, that quantitative analyses, however, are not sus- ceptible to subjective biases. Unfortunately, this such as the Census Bureau’s categorization of oc- isn’t exactly so. Even in the most mathematically cupations, or develop their own coding categories. explicit analysis, we can discover ample room for In either case, the coding scheme must be appro- defining and measuring variables in ways that priate to the nature and objectives of the study. encourage one finding over another. Quantitative analysts need to guard against this. Sometimes, the • A codebook is the document that describes careful specification of hypotheses in advance can offer protection, although this can also constitute a (1) the identifiers assigned to different variables straitjacket, hampering a full exploration of what and (2) the codes assigned to the attributes of data can tell us. those variables. The quantitative analyst has an obligation to Univariate Analysis report formal hypotheses and less-formal expecta- tions that didn’t pan out. Let’s suppose you think • Univariate analysis is the analysis of a single that a particular variable will prove a powerful cause of gender prejudice, but your data analysis variable. Because univariate analysis does not contradicts that expectation. You should report the involve the relationships between two or more lack of correlation, since such information is useful variables, its purpose is descriptive rather than to other researchers who will conduct research on explanatory. this topic. While it would be more satisfying to dis- cover what causes prejudice, it’s very important to • Several techniques allow researchers to sum- know what doesn’t cause it. marize their original data to make them more The protection of subject privacy is as impor- manageable while maintaining as much of the tant in quantitative as in qualitative analysis. In the original detail as possible. Frequency distributions, former case, however, it’s often easier to collect and averages, grouped data, and measures of disper- record data in ways that make subject identifica- sion are all ways of summarizing data concerning tion more difficult. However, the first time public a single variable. officials demand that you reveal the names of ­student-subjects who reported using illegal drugs Subgroup Comparisons in a survey, this issue will take on more salience. (Don’t reveal the names, by the way. If necessary, • Subgroup comparisons can be used to describe burn the questionnaires—“accidentally.”) similarities and differences among subgroups with Main Points respect to some variable. Introduction Bivariate Analysis • Quantitative analysis involves the techniques • Bivariate analysis focuses on relationships be- by which researchers convert data to numerical tween variables rather than on comparisons of forms and subject them to statistical analyses. groups. Bivariate analysis explores the statistical Quantification of Data association between the independent variable and the dependent variable. Its purpose is usually ex- • Some data, such as age and income, are intrinsi- planatory rather than merely descriptive. cally numerical. • The results of bivariate analyses often are pre- • Often, quantification involves coding into catego- sented in the form of contingency tables, which are constructed to reveal the effects of the inde- ries that are then given numerical representations. pendent variable on the dependent variable. Introduction to Multivariate Analysis • Multivariate analysis is a method of analyzing the simultaneous relationships among several variables. It may also be used to understand the relationship between two variables more fully. • The logic and techniques involved in quantita- tive research can also be valuable to qualitative researchers. Sociological Diagnostics • Sociological diagnostics is a quantitative analy- sis technique for determining the nature of social problems such as ethnic or gender discrimination.

Online Study Resources ■ 439 Ethics and Quantitative Data Analysis 4. Using the hypothetical data in the following table, how would you construct and interpret tables • Unbiased analysis and reporting is as much an showing the following? ethical concern in quantitative analysis as in the a. The bivariate relationship between age and at- case of qualitative analysis. titude toward abortion • Subjects’ privacy must be protected in quantitative b. The bivariate relationship between political ori- entation and attitude toward abortion data analysis and reporting. c. The multivariate relationship linking age, politi- cal orientation, and attitude toward abortion Key Terms Age Political Attitude toward Frequency Orientation Abortion The following terms are defined in context in the chapter and at the bottom of the page where the term Young Liberal Favor 90 is introduced, as well as in the comprehensive glossary 10 at the back of the book. Young Liberal Oppose 60 40 average mean Young Conservative Favor 60 bivariate analysis median 40 codebook mode Young Conservative Oppose 20 contingency table multivariate analysis 80 continuous variable quantitative analysis Old Liberal Favor discrete variable standard deviation dispersion univariate analysis Old Liberal Oppose frequency distribution Old Conservative Favor Old Conservative Oppose Proposing Social Research: S P SS E x e r c i s e s Quantitative Data Analysis See the exercise for Chapter 16 (p. 495). See the booklet that accompanies your text for ­exercises using SPSS (Statistical Package for the Review Questions and Exercises S­ ocial Sciences). There are exercises offered for each chapter, and you’ll also find a detailed primer on 1. How might the various majors at your college using SPSS. be classified into categories? Create a coding system that would allow you to categorize them Online Study Resources according to some meaningful variable. Then ­create a different coding system, using a different Access the resources your instructor has assigned. For variable. this book, you can access: 2. How many ways could you be described in nu- CourseMate for The merical terms? What are some of your intrinsi- Practice of Social Research cally numerical attributes? Could you express some of your qualitative attributes in quantitative Login to CengageBrain.com to access chapter- terms? specific learning tools including Learning Objectives, Practice Quizzes, Videos, Internet Exercises, Flash Cards, 3. How would you construct and interpret a con- ­Glossaries, Web Links, and more from your Sociology tingency table from the following information: CourseMate. 150 Democrats favor raising the minimum wage, and 50 oppose it; 100 Republicans favor raising the minimum wage, and 300 oppose it?

440 ■ Chapter 14: Analyzing Quantitative Data If your professor has assigned Aplia homework: 1. Sign into your account. 2. After you complete each page of questions, click “Grade It Now” to see detailed explanations of every answer. 3. Click “Try Another Version” for an opportunity to improve your score. Visit www.cengagebrain.com to access your account and purchase materials.

CHAPTER 15 Origins and Paradigm of the Elaboration Model chapter o v er v i e w Introduction Interpretation Specification We’ll use the elaboration model to The Origins of the Refinements examine the fundamental logic of Elaboration Model   to the Paradigm multivariate and causal analysis. Exploring applications of this logic The Elaboration Paradigm Elaboration and Ex Post in the form of simple percentage Replication Facto Hypothesizing tables provides a foundation for Explanation making sense of more-complex analytic methods. Aplia for The Practice of Social Research After reading, go to “Online Study Resources” at the end of this chapter for

442 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Introduction analysis in social research. Especially through the use of contingency tables, this method portrays the This chapter addresses the logic of multivariate logical process of scientific analysis. Moreover, if you analysis in quantitative social research. It builds on can comprehend fully the use of the elaboration earlier discussions of causation among variables. model using contingency tables, you should greatly In Chapter 4, we looked at the criteria for causa- improve your ability to use and understand more- tion, and I introduced the idea of spuriousness. As sophisticated statistical t­echniques, such as partial we saw, sometimes there appears to be a causal regressions and log-linear models, for example. relationship between two variables (e.g., number of storks and birthrates), but a more careful analysis In a sense, this discussion of elaboration analysis shows that apparent relationship to be caused by is an extension of our earlier examination of spu- the influence of a third variable (e.g., rural/urban). riousness in Chapter 4. As you’ll recall, one of the Rural communities have higher birthrates and also criteria of causal relations in social research is that more storks than urban areas do. As we will see in the observed relationship between two variables this chapter, there are a number of other possible not be an artifact caused by some other variable. multivariate relationships. In the case of the positive relationship between the number of fire trucks responding to a fire and the To explore this topic, we are going to utilize a amount of damage done, for example, we saw that social science analysis perspective that is referred the size of the fire explained away the apparent re- to variously as the elaboration model, the inter- lationship between trucks and damage. The bigger pretation method, the Lazarsfeld method, or the the fire, the more trucks responding to it; and the ­Columbia school. Its many names reflect the fact bigger the fire, the more damage done. The logic that it aims at elaborating on an empirical rela- used in that hypothetical example was the same as tionship among variables in order to interpret that the logic of the elaboration model. As the early ex- relationship, in the manner developed by Paul amples that gave birth to the elaboration model will Lazarsfeld while he was a professor at Columbia illustrate, social research often reveals a counter­ University. As such, the elaboration model is one intuitive understanding of social life. method for doing multivariate analysis. Using both hypothetical and real examples, we’ll Researchers use the logic of elaboration model see that the testing of an observed relationship may to understand the relationship between two vari- result in a variety of discoveries and logical interpre- ables through the simultaneous introduction of tations. Spuriousness is only one of the possibilities. additional variables, though they may not always refer to the model by name. Though developed pri- The accompanying Tips and Tools feature “Why marily through the medium of percentage tables, Do Elaboration?” by one of the elaboration model’s it can be used with other statistical techniques, as creators, Patricia Kendall, provides another power- Chapter 16 will show. ful justification for using this model. I firmly believe that the elaboration model offers The Origins the clearest available picture of the logic of causal of the Elaboration Model elaboration model  A logical model for under- The historical origins of the elaboration model pro- standing the relationship between two variables vide a good illustration of how scientific research by controlling for the effects of a third. Principally works in practice. As I mentioned in Chapter 1, ­developed by Paul Lazarsfeld. The various outcomes during World War II Samuel Stouffer organized of an elaboration analysis are replication, explana- and headed a special social research branch within tion, interpretation, and specification. the U.S. Army. Throughout the war, this group

The Origins of the Elaboration Model ■ 443 Tips and Tools Why Do Elaboration? League colleges (regardless of their financial circumstances or academic qualifications and regardless of the desire of the colleges to accept them) Patricia L. Kendall and the control group to other colleges and universities, wait 20 years Department of Sociology, Queens College, CUNY or so until the two groups have reached professional maturity, and then measure the relative success of the two groups. Certainly a bizarre process. There are several aspects of a true controlled experiment. The most crucial are (1) creating experimental and control groups that are identical Sociologists also investigate the hypothesis that coming from a bro- within limits of chance (this is done by assigning individuals to the two ken home leads to juvenile delinquency. How would we go about study- groups through processes of randomization: using tables of random ing this experimentally? If you followed the example above, you would numbers, flipping coins, etc.); (2) making sure that it is the experimenter see that studying this hypothesis through a true experiment would be who introduces the stimulus, not external events; and (3) waiting to see totally impossible. Just think of what the experimenter would have to do! whether the stimulus has had its presumed effect. The requirements of true experiments are so unrealistic in sociolog- We may have the hypothesis, for example, that attending Ivy ical research that we are forced to use other, and less ideal, methods in League colleges leads to greater success professionally than attending all but the most trivial situations. We can study experimentally whether other kinds of colleges and universities does. How would we study this students learn more from one type of lecture than another, or whether through a true experiment? Suppose you said,“Take a group of people a film changes viewers’attitudes. But these are not always the sorts of in their 40s, find out which ones went to Ivy League colleges, and see questions in which we are truly interested. whether they are more successful than those who went to other kinds of colleges.”If that is your answer, you are wrong. We therefore resort to approximations—generally surveys—that have their own shortcomings. However, the elaboration model allows us A true experiment would require the investigator to select to examine survey data, take account of their possible shortcomings, and s­ everal classes of high school seniors, divide each class at random into draw rather sophisticated conclusions about important issues. experimental and control groups, send the experimental groups to Ivy conducted a large number and variety of surveys higher morale than should those being trained among U.S. servicemen. Although the objectives in the South. of these studies varied somewhat, they generally focused on the factors affecting soldiers’ combat 3. Soldiers with more education should be more effectiveness. likely to resent being drafted into the army as enlisted men than should those with less Several of the studies examined morale in the education. military. Because morale seemed to be related positively to combat effectiveness, improving Each of these propositions made sense logically, morale would make the war effort more effective. and common wisdom held each to be true. Stouffer Stouffer and his research staff sought to uncover decided to test each empirically. To his surprise, some of the variables that affected morale. In none of the propositions was confirmed. part, the group sought to confirm empirically some commonly ­accepted propositions, including We discussed the first proposition in Chapter 1. the following: As you may recall, Stouffer found that soldiers s­erving in the Military Police (where p­ romotions 1. Promotions surely affect soldiers’ morale, so were the slowest in the army) had fewer com- soldiers serving in units with low promotion plaints about the promotion system than did those rates should have relatively low morale. serving in the Army Air Corps (where promo- tions were the fastest in the army). The other 2. Given racial segregation and discrimination propositions fared just as badly. African American in the South, African American soldiers being soldiers serving in northern training camps and trained in northern training camps should have those s­erving in southern training camps seemed

444 ■ Chapter 15: Origins and Paradigm of the Elaboration Model to differ little if at all in their general morale. And Finally, the concepts of reference group and l­ess-educated soldiers were more likely to resent relative deprivation seemed to explain the anomaly being drafted into the army than those with more of highly educated draftees accepting their induc- education were. tion more willingly than those with less education did. Stouffer reasoned as follows: Rather than trying to hide the findings or just running tests of statistical significance and publish-   1. A person’s friends, on the whole, have about ing the results, Stouffer asked, “Why?” He found the same educational status as that person does. the answer to this question within the concepts of reference group and relative deprivation. Put sim-   2. Draft-age men with less education are more ply, Stouffer suggested that soldiers did not evalu- likely to engage in semi-skilled production-line ate their positions in life according to absolute, occupations and farming than more educated objective standards, but rather on the basis of their men. position relative to others around them. The people they compared themselves with were in their ref-   3. During wartime, many production-line indus- erence group, and they felt relative deprivation if tries and farming are vital to the national inter- they didn’t compare favorably in that regard. est; workers in those industries and farmers are exempted from the draft. Following this logic, Stouffer found an answer to each of the anomalies in his empirical data.   4. A man with little education is more likely to Regarding promotion, he suggested that soldiers have friends in draft-exempt occupations than judged the fairness of the promotion system based a man with more education. on their own experiences relative to others around them. In the Military Police, where promotions   5. When each compares himself with his friends, were few and slow, few soldiers knew of a less- a less educated draftee is more likely to feel qualified buddy who had been promoted faster discriminated against than a draftee with more than they had. In the Army Air Corps, however, education. the rapid promotion rate meant that many sol- diers knew of less-qualified buddies who had been (Stouffer et al. 1949–1950: 122–27) promoted faster than seemed appropriate. Thus, ironically, the MPs said the promotion system was Stouffer’s explanations unlocked the mystery of generally fair, and the air corpsmen said it was not. the three anomalous findings. Because they were not part of a preplanned study design, however, he A similar analysis seemed to explain the case of lacked empirical data for testing them. Neverthe- the African American soldiers. Rather than compar- less, Stouffer’s logical exposition provided the basis ing conditions in the North with those in the South, for the later development of the elaboration model: African American soldiers compared their own sta- understanding the relationship between two vari- tus with the status of the African American civilians ables through the controlled introduction of other around them. In the South, where discrimination variables. was at its worst, they found that being a soldier insulated them somewhat from adverse cultural Paul Lazarsfeld and his associates at Columbia norms in the surrounding community. Whereas University formally developed the elaboration model southern African American civilians were grossly in 1946. In a methodological review of Stouffer’s discriminated against and denied self-esteem, good army studies, Lazarsfeld and Patricia Kendall used jobs, and so forth, African American soldiers had a the logic of the elaboration model to present hy- slightly better status. In the North, however, many pothetical tables that would have proved Stouffer’s of the African American civilians they encountered contention regarding education and acceptance held well-paying defense jobs. And with discrimi- of induction had the empirical data been available nation being less severe, being a soldier did not (Kendall and Lazarsfeld 1950). help one’s status in the community. The central logic of the elaboration model b­ egins with an observed relationship between two variables and the possibility that one variable may be causing the other. In the Stouffer example, the

The Origins of the Elaboration Model ■ 445 Table 15-1 Table 15-2 Summary of Stouffer’s Data on Education Hypothetical Relationship between Education and Acceptance of Induction and Deferment of Friends High Ed. Low Ed. Friends Deferred? High Ed. Low Ed. Should not have been deferred 88% 70% Yes 19% 79% Should have been deferred 12 30 No 81 21 100 100 100 100 (1,761) (1,876) (1,761) (1,876) Source: Tables 15-1, 15-2, 15-3, and 15-4 are reprinted with permission of Table 15-3 The Free Press, a Division of Simon & Schuster, Inc., from Continuities in Social R­ esearch: Studies in the Scope and Method of  “The American Soldier”by Hypothetical Relationship between Deferment of Friends Robert K. Merton and Paul Lazarsfeld. Copyright © 1950 by The Free Press. and Acceptance of One’s Own Induction C­ opyright renewed © 1978 by Robert K. Merton. All rights reserved. initial two variables were educational level and accep- Friends Deferred? tance of being drafted as fair. Because the soldiers’ ed- ucational levels were set before they were drafted Yes No (and thus having an opinion about being drafted) it would seem that educational level was the cause, or Should not have been deferred 63% 94% independent variable, and acceptance of induction was Should have been deferred 37 6 the effect, or dependent variable. As we just saw, 100 100 however, the observed relationship countered what (1,819) (1,818) the researchers had expected. represent inductees’ judgments of their own situ- The elaboration model examines the impact of ation, with the latter group feeling it was fair for other variables on the relationship first observed. them to have been drafted. Sometimes this analysis reveals the mechanisms through which the causal relationship occurs. Then, Kendall and Lazarsfeld created some Other times an elaboration analysis disproves the hypothetical tables to represent what the analysis existence of a causal relationship altogether. might have looked like had soldiers been asked whether most of their friends had been drafted or In the present example, the additional ­variable deferred. In Table 15-2, 19 percent of those with was whether or not a soldier’s friends were de- high education hypothetically said their friends ferred or drafted. In Stouffer’s speculative expla- were deferred, compared with 79 percent of the nation, this variable showed how it was actually soldiers with less education. logical that soldiers with more education would be the more accepting of being drafted: because it was Notice that the numbers of soldiers with high likely that their friends would have been drafted. and low education are the same as in Stouffer’s Those with the least education were likely to have real data. In later tables, you’ll see that the num- been in occupations that often brought deferments bers who accepted or resented being drafted re- from the draft, leading those drafted to feel they main true to the original data. Only the numbers had been treated unfairly. saying that friends were or were not deferred were made up. Kendall and Lazarsfeld began with Stouffer’s data showing the positive association between edu- Stouffer’s explanation next assumed that sol- cation and acceptance of induction (see Table 15-1). diers with friends who had been deferred would be In this and the following tables, “should have been more likely to resent their own induction than those deferred” and “should not have been deferred” who had no deferred friends would. Table 15-3

446 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Table 15-4 Hypothetical Data Relating Education to Acceptance of Induction through the Factor of Having Friends Who Were Deferred      Friends Deferred    No Friends Deferred High Ed.   Low Ed. High Ed.   Low Ed. Should not have been deferred 63% 63% 94% 95% Should have been deferred 37 37 6 5 100 100 100 100 100% = (335) (1,484) (1,426) (392) presents the hypothetical data that would have sup- On the other hand, among those with high ported that assumption. education the acceptance of induction is strongly related to whether or not friends were deferred: The hypothetical data in Tables 15-2 and 15-3 63 percent versus 94 percent. And the same is true would confirm linkages that Stouffer had specified among those with less education. The hypothetical in his explanation. First, soldiers with low education data in Table 15-4, then, would support Stouffer’s were more likely to have friends who were deferred contention that education affected acceptance of than soldiers with more education were. Second, induction only through the medium of having having friends who were deferred made a soldier friends deferred. Highly educated draftees were more likely to think he should have been deferred. less likely to have friends deferred and, by virtue Stouffer had suggested that these two relationships of that fact, were more likely to accept their own would clarify the original relationship between induction as fair. Those with less education were education and acceptance of induction. Kendall and more likely to have friends deferred and, by virtue Lazarsfeld created a hypothetical table that would of that fact, were less likely to accept their own confirm Stouffer’s explanation (see Table 15-4). induction. Recall that the original finding was that draftees Recognize that neither Stouffer’s explana- with high education were more likely to accept their tion nor the hypothetical data denied the reality induction into the army as fair than those with less of the original relationship. As educational level education were. In Table 15-4, however, we note increased, acceptance of one’s own induction also that level of education has no effect on the accep- increased. The nature of this empirical relationship, tance of induction among those who report having however, was interpreted through the introduction friends deferred: 63 percent among both educational of a third variable. The variable, deferment of friends, groups indicate that they accept their induction did not deny the original relationship; it merely (that is, they say they should not have been deferred). clarified the mechanism through which the origi- Similarly, educational level has no significant ­effect on nal relationship occurred. acceptance of induction among those who r­eported having no friends deferred: 94 and 95 percent say This, then, is the heart of the elaboration they should not have been deferred. model and of multivariate analysis. Having ob- served an empirical relationship between two test variable  A variable that is held constant in an variables (such as level of education and acceptance attempt to clarify further the relationship between of induction), we seek to understand the nature of two other variables. Having discovered a relationship that relationship through the effects produced by between education and prejudice, for example, we introducing other variables (such as having friends might hold sex constant by examining the relation- who were deferred). Mechanically, we accomplish ship between education and prejudice among men this by first dividing our sample into subsets on only and then among women only. In this example,

The Elaboration Paradigm ■ 447 control variable. In our example, having friends Figure 15-1 Ce deferred or not is the test variable, and the Bab sample is divided into those who have deferred Intervening Test Variable friends and those who do not. The relationship So between the original two variables (acceptance Figure 15-2 of induction and level of education) is then recom- 1-13 puted separately for each of the subsamples. The Antecedent Test Variable tables produced in this manner are called the Ce partial tables, and the relationships found in the the diagram. In fact, we have one independent Bab partial tables are called the partial relation- variable (the test variable) and two dependent ships, or partials. The partial relationships are variables. The incorrect terminology has been used So then compared with the initial relationship dis- only to provide continuity with the preceding ex- covered in the total sample, often referred to as ample. Because of their individual relationships 1-13 the zero-order relationship to indicate that to the test variable, the “independent” and “de- no test variables have been controlled for. pendent” variables are empirically related to each other, but there is no causal link between them. Although the elaboration was first demon- Their empirical relationship is merely a product strated through the use of hypothetical data, it laid of their coincidental relationships to the test vari- out a logical method for analyzing relationships able. (Subsequent examples will further clarify this among variables that have been actually measured. relationship.) As we’ll see, our first, hypothetical example de- scribes only one possible outcome in the elabora- Table 15-5 provides a guide to understanding an tion model. There are others. elaboration analysis. The two columns in the table indicate whether the test variable is antecedent or The Elaboration Paradigm intervening in the sense described previously. The left side of the table shows the nature of the partial This section presents guidelines for understanding an elaboration analysis. To begin, we must know partial relationship  In the elaboration model, whether the test variable is antecedent (prior in this is the relationship between two variables when time) to the other two variables or whether it is examined in a subset of cases defined by a third intervening between them, because these positions variable. Beginning with a zero-order relationship suggest different logical relationships in the multi­ between political party and attitudes toward abortion, variate model. If the test variable is intervening, for example, we might want to see whether the re- as in the case of education, deferment of friends, and lationship held true among both men and women acceptance of induction, then the analysis is based on (i.e., controlling for sex). The relationship found the model shown in Figure 15-1. The logic of this among men and the relationship found among multivariate relationship is that the independent women would be the partial relationships, some- variable (educational level) affects the intervening times simply called the partials. test variable (having friends deferred or not), which zero-order relationship  In the elaboration in turn affects the dependent variable (accepting model, this is the original relationship between two induction). If the test variable is antecedent to both the independent and dependent variables, a differ- ent model must be used (see Figure 15-2). Here the test variable affects both the “independent” and “dependent” variables. Realize, of course, that the terms independent variable and dependent

448 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Table 15-5 To see what a replication looks like, turn back The Elaboration Paradigm to Tables 15-3 and 15-4. Imagine that our initial discovery was that having friends deferred strongly Test Variable influenced how soldiers felt about being drafted, as shown in Table 15-3. Had we first discovered this Partial Relationships Antecedent Intervening relationship, we might have wanted to see whether Compared with Original it was equally true for soldiers of different educa- Replication Replication tional backgrounds. To find out, we would have Same Relationship Explanation Interpretation made education our control or test variable. Less or none Specification Specification Split* Table 15-4 contains the results of such an ex- amination, though it is constructed somewhat dif- *One partial is the same or greater, and the other is less or none. ferently from what we would have done had we used education as the test variable. Nevertheless, relationships as compared with the original rela- we see in the table that having friends deferred or tionship between the independent and dependent not still influences attitudes toward being drafted variables. The body of the table gives the technical among those soldiers with high education and notations—replication, explanation, interpretation, those with low education. (Compare columns 1 and specification—assigned to each case. We’ll dis- and 3, then 2 and 4.) This result represents a rep- cuss each in turn. lication of the relationship between having friends deferred and attitude toward being drafted. Replication Researchers frequently use the elaboration Whenever the partial relationships are essentially model rather routinely in the hope of replicating the same as the original relationship, the term their findings among subsets of the sample. If we replication is assigned to the result, regardless of discovered a relationship between education and whether the test variable is antecedent or interven- prejudice, for example, we might introduce such ing. This means that the original relationship has test variables as age, region of the country, race, religion, been replicated under test conditions. If, in our pre- and so forth to test the stability of the original rela- vious example, education still affected acceptance tionship. If the relationship were replicated among of induction both among those who had friends young and old, among people from different parts deferred and those who did not, then we would of the country, and so forth, we would have grounds say the original relationship had been replicated. for concluding that the original relationship was a Note, however, that this finding would not confirm genuine and general one. Stouffer’s explanation of the original relationship. Having friends deferred or not would not be the Explanation mechanism through which education affected the acceptance of induction. Explanation is the term used to describe a spuri- ous relationship: an original relationship shown to replication  A technical term used in connection be false through the introduction of a test variable. with the elaboration model, referring to the elabora- This requires two conditions: (1) The test variable tion outcome in which the initially observed relation- must be antecedent to both the independent and ship between two variables persists when a control dependent variables. (2) The partial relationships variable is held constant, thereby supporting the idea must be zero or significantly less than those found that the original relationship is genuine. in the original. Several examples will illustrate this situation. explanation  An elaboration model outcome in which the original relationship between two vari- Let’s look at an example we touched on in ables is revealed to have been spurious, because the Chapter 4. There is an empirical relationship relationship disappears when an antecedent test ­between the number of storks in different areas variable is introduced.

The Elaboration Paradigm ■ 449 and the birthrates for those areas. The more storks Figure 15-3 Ce in an area, the higher the birthrate. This empiri- Bab cal relationship might lead one to assume that the The Facts of Life about Storks and Babies number of storks affects the birthrate. An anteced- So ent test explains away this relationship, however. Finally, let’s take a real research example. Years Rural areas have both more storks and higher ago, I found an empirical relationship between the 1-13 birthrates than urban areas do. Within rural areas, region of the country in which medical school fac- there is no relationship between the number of ulty members attended medical school and their storks and the birthrate; nor is there a relationship attitudes toward Medicare (Babbie 1970). To sim- within urban areas. plify matters, only the East and the South will be examined. Of faculty members attending eastern Figure 15-3 illustrates how the rural/urban medical schools, 78 percent said they approved of variable causes the apparent relationship between Medicare, compared with 59 percent of those at- storks and birthrates. Part I of the figure shows tending southern medical schools. This finding made the original relationship. Notice that all but one sense in view of the fact that the South seemed of the entries in the box for towns and cities with generally more resistant to such programs than the many storks have high birthrates and that all East did, and medical school training should pre- but one of those in the box for towns and cities sumably affect a doctor’s medical attitudes. However, with few storks have low birthrates. In percent- this relationship is explained away when we intro- age form, we say that 93 percent of the towns duce an antecedent test variable: the region of the and cities with many storks also had high birth- country in which the faculty member was raised. rates, contrasted with 7 percent of those with Of faculty members raised in the East, 89 percent at- few storks. That’s quite a large difference and tended medical school in the East and 11 percent in represents a strong ­association between the two the South. Of those raised in the South, 53 percent variables. Part II of the figure separates the towns from the cities (the rural from urban areas) and exam- ines storks and babies in each type of place sepa- rately. Now we can see that all the rural places have high birthrates, and all the urban places have low birthrates. Also notice that only one rural place had few storks and only one urban place had lots of storks. Here’s a similar example, also mentioned in Chapter 4 and at the beginning of this chap- ter. There is a positive relationship between the number of fire trucks responding to a fire and the amount of damage done. If more trucks respond, more damage is done. One might assume from this fact that the fire trucks themselves cause the d­ amage. However, an antecedent test variable, the size of the fire, explains away the original relation- ship. Large fires do more damage than small ones do, and more fire trucks show up at large fires than at small ones. Looking only at large fires, we would see that the original relationship vanishes (or ­perhaps reverses itself); and the same would be true looking only at small fires.

450 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Research in Real Life Attending an Ivy League College Table 1* and Success in Later Professional Life College Attended (X) Patricia L. Kendall Department of Sociology, Queens College, CUNY Later Professional Ivy League Other College Success (Y) College or University Probably the main danger for survey analysts is that a relationship they hope is causal will turn out to be spurious. That is, the original relation- Successful (25%) 1,300 (65%) 2,000 ship between X and Y is explained by an antecedent test factor. More Unsuccessful (75%) 1,700 (35%) 6,000 specifically, the partial relationships between X and Y reduce to 0 when Total (100%) 2,000 (100%) 8,000 that antecedent test factor is held constant. *I have had to invent relevant figures because the only published version of This was a distinct possibility in a major finding from a study West’s study contained no totals. See Ernest Havemann and Patricia Salter West, carried out several decades ago. One of my fellow graduate students at They Went to College (New York: Harcourt, Brace, 1952). Columbia University, Patricia Salter West, based her dissertation on ques- tionnaires obtained by Time Magazine from 10,000 of its male subscrib- Table 2 ers. Among many of the hypotheses developed by West was that male graduates of Ivy League schools (Brown, Columbia, Cornell, Dartmouth, Attendance at Ivy League Colleges According to Family Harvard, University of Pennsylvania, Princeton, and Yale) were more suc- Socioeconomic Status (SES) cessful in their later professional careers, as defined by their annual earn- ings, than those who graduated from other colleges and universities. Family SES (T) The initial fourfold table (Table 1) supported West’s expectation. College Attended (X)   High SES Low SES Although I made up the figures, they conform closely to what West actually found in her study. Having attended an Ivy League school seems Ivy League colleges 1,500 (33%) 500 (9%) to lead to considerably greater professional success than does being a Other colleges and universities 3,000 (67%) 5,000 (91%) graduate of some other kind of college or university. Total 4,500 (100%) 5,500 (100%) But wait a minute. Isn’t this a relationship that typically could According toTable 2, a third of those coming from families defined as be spurious? Who can afford to send their sons to Ivy League schools? wealthy, compared with 1 in 11 coming from less well-to-do backg­ rounds, Wealthy families, of course.† And who can provide the business and attended Ivy League colleges.Thus there is a very high correlation between professional connections that could help sons become successful in their careers? Again, wealthy or well-to-do families. In other words, the socioeconomic status of the student’s family may explain away the apparent causal relationship. In fact, some of West’s findings suggest that this might indeed be the case. attended medical school in the East and 47 percent quite likely to approve of Medicare, regardless of in the South. Moreover, the areas in which faculty where they attended medical school. Those raised members were raised related to attitudes toward in the South are relatively less likely to approve of Medicare. Of those raised in the East, 84 percent Medicare, but, again, the region of their medical ­approved of Medicare, as compared with 49 percent school training has little or no effect. These data of those raised in the South. indicate, therefore, that the original relationship between region of medical training and attitude Table 15-6 presents the three-variable relation- toward Medicare was spurious; it was due only to ship among (1) region in which raised, (2) region the coincidental effect of region of origin on both of medical school training, and (3) attitude toward region of medical training and attitude toward Medicare. Faculty members raised in the East are

The Elaboration Paradigm ■ 451 Table 3 Partial Relationships between X and Y with T Held Constant High Family SES (T) Low Family SES (T) Later Success (Y) Ivy League Other Ivy League Other College (X) College (X) College (X) College (X) Successful Not successful 1,000 (67%) 1,000 (33%) 300 (60%) 1,000 (20%) Total 500 (33%) 2,000 (67%) 200 (40%) 4,000 (80%) 3,000 (100%) 500 (100%) 5,000 (100%) 1,500 (100%) the two variables, X and T. (There is a similarly high correlation original relationship. Consider, for example, the intelligence of the stu- between family socioeconomic status [T ] and later p­ ro­fessional dents (as measured by IQ tests or SAT scores). Ivy League colleges pride success [Y ].) themselves on the excellence of their student bodies. They may therefore be willing to award merit scholarships to students with exceptional The magnitude of these so-called marginal correlations suggest qualifications but not enough money to pay tuition and board. Once that West’s hypothesis regarding the causal nature of having attended admitted to these prestigious colleges, bright students may develop the an Ivy League college might be incorrect; it suggests instead that the skills—and connections—that will lead to later professional success. socioeconomic status of the students’families accounted for the original Since West had no data on the intelligence of the men she studied, she relationship she observed. was unable to study whether the partial relationships disappeared once this test factor was introduced. We are not done yet, however. The crucial question is what hap- pens to the partial relationships once the test factor is controlled. These In sum, the elaboration paradigm permits the investigator to rule are shown in Table 3. out certain possibilities and to gain support for others. It does not permit us to prove anything. These partial relationships show that, even when family socio- economic status is held constant, there is still a marked relationship †Since she had no direct data on family socioeconomic status,West defined as wealthy between having attended an Ivy League college and success in later or having high socioeconomic status those who supported their sons completely during professional life. As a result, West’s initial hypothesis received support all four years of college. She defined as less wealthy or having low socioeconomic status from the analysis she carried out. those whose sons worked their way through college, in part or totally. Despite this, West had in no way proved her hypothesis. There are almost always additional antecedent factors that might explain the Medicare. When region of origin is held constant, the topic is still of vital interest to students: To what as in Table 15-6, the original relationship disap- extent does your professional success depend on pears in the partials. attending the “right” school? In the Research in Real Life feature “Attending Interpretation an Ivy League College and Success in Later Profes- sional Life,” Patricia Kendall, one of the founders of Interpretation is similar to explanation, except the elaboration model, recalls a study in which the for the time placement of the test variable and the researcher suspected an explanation but found a implications that follow from that difference. replication. Though the data are no longer current,

452 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Table 15-6 Here’s another example of interpretation. ­Researchers have observed that children from Region of Origin, Region of Medical School Training, ­broken homes are more likely to become de- and Attitude toward Medicare linquent than those from intact homes are. This relationship may be interpreted, however, Percent Who through the introduction of supervision as a test Approve of Medicare variable. Among children who are supervised, delinquency rates are not affected by whether Region in Which Raised or not their p­ arents are divorced. The same is true among those who are not supervised. It is East South the relationship ­between broken homes and the lack of supervision that produced the original Region of Medical East 84 50 relationship.   School Training South 80 47 Specification Source: Earl R. Babbie, Science and Morality in Medicine (Berkeley: University of California Press, 1970), 181. Sometimes the elaboration model produces par- tial relationships that differ significantly from each Interpretation represents the research outcome other. For example, one partial relationship is the in which a test or control variable is discovered to same as or stronger than the original two-variable be the mediating factor through which an indepen- relationship, and the second partial relationship is dent variable has its effect on a dependent variable. less than the original and may be reduced to zero. In The earlier example of education, friends deferred, the elaboration paradigm, this situation is referred to and acceptance of induction is an excellent illustra- as specification: We have specified the conditions tion of interpretation. In terms of the elaboration under which the original relationship occurs. model, the effect of education on acceptance of induction is not explained away; it is still a genuine Now recall the study, cited earlier in this book, relationship. In a real sense, educational differences of the sources of religious involvement (Glock, cause differential acceptance of induction. The in- Ringer, and Babbie 1967: 92). It was discovered that tervening variable, deferment of friends, merely helps among Episcopal church members, involvement to interpret the mechanism through which the re- decreased as social class increased. This finding is lationship occurs. Thus, an interpretation does not reported in Table 15-7, which examines mean levels deny the validity of the original causal relationship of church involvement among women parishioners but simply clarifies the process through which that at different levels of social class. relationship functions. Glock interpreted this finding in the context of interpretation  A technical term used in connec- others in the analysis and concluded that church tion with the elaboration model. It represents the involvement provides an alternative form of gra­ research outcome in which a control variable is tification for people who are denied gratification in discovered to be the mediating factor through which the secular society. This conclusion explained why an independent variable has its effect on a depen- women were more religious than men, why old dent variable. people were more religious than young people, and so forth. Glock reasoned that people of lower social specification  A technical term used in connection class (measured by income and education) had with the elaboration model, representing the elabo- fewer chances to gain self-esteem from the secular ration outcome in which an initially observed rela- society than people of higher social class did. To illus- tionship between two variables is replicated among trate this idea, he noted that social class was strongly some subgroups created by the control variable but related to the likelihood that a woman had ever held not among others. In such a situation, you will have an office in a secular organization (see Table 15-8). specified the conditions under which the original relationship exists: for example, among men but not among women.

Table 15-7 The Elaboration Paradigm ■ 453 Social Class and Mean Church Involvement secular society, he used as a variable the holding among Episcopal Women of secular office. In this test, social class should be unrelated to church involvement among those who Social Class Levels had held such office. Low High Table 15-9 presents an example of a specification. 01234 Among women who have held office in secular organizations, there is essentially no relationship Mean involvement 0.63 0.58 0.49 0.48 0.45 between social class and church involvement. In e­ ffect, the table specifies the conditions under which Note: Mean scores rather than percentages have been used here. the original relationship holds: among those women Source: Tables 15-7, 15-8, and 15-9 are from Charles Y. Glock, Benjamin B. lacking gratification in the secular society. Ringer, and Earl R. Babbie, To Comfort and to Challenge (Berkeley: University of California Press, 1967). Used with permission of the Regents of the University The term specification is used in the elabora- of California. tion paradigm regardless of whether the test vari- able is antecedent or intervening. In either case, Table 15-8 the meaning is the same. We have specified the Social Class and the Holding of Office particular conditions under which the original in Secular Organizations ­relationship holds. Percent who have Low Social Class Levels High Refinements to the Paradigm held office in a 0 123 4 secular organization 46 47 54 60 83 The preceding sections have presented the primary logic of the elaboration model as developed by Table 15-9 L­ azarsfeld and his colleagues. Here we look at some logically possible variations, some of which can be Church Involvement by Social Class and Holding found in a book by Morris Rosenberg (1968). Secular Office First, the basic paradigm assumes an initial rela- Mean Church Involvement tionship between two variables. It might be useful, for Social Class Levels however, in a more comprehensive model to differ- entiate between positive and negative relationships. Low High Moreover, Rosenberg suggests using the elaboration 01234 model even with an original relationship of zero. He cites as an example a study of union membership Have held office 0.46 0.53 0.46 0.46 0.46 and attitudes toward having Jews on the union staff (see Table 15-10). The initial analysis indicated that Have not held office 0.62 0.55 0.47 0.46 0.40 length of union membership did not relate to the attitude: Those who had belonged to the union less Glock then reasoned that if social class were than four years were just as willing to accept Jews related to church involvement only by virtue of on the staff as were those who had belonged for the fact that lower-class women would be denied more than four years. The age of union members, o­ pportunities for gratification in the secular society, however, was found to suppress the relationship the original relationship should not hold among between length of union membership and attitude women who were g­ etting gratification. As a rough toward Jews. Overall, younger members were indicator of the receipt of gratification from the more favorable to Jews than older members were. At the same time, of course, younger members were not likely to have been in the union as long as the old members. Within specific age groups, however, those in the union longest were the most

454 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Table 15-10 Example of a Suppressor Variable I: No Apparent Relationship between Attitudes toward Jews and Length of Time in the Union Length of Time in the Union Percent who don’t care if there are Jews on the union staff Less than four years Four years or more 49.2 50.5 (126) (256) II: In Each Age Group, Length of Time in Union Increases Willingness to Have Jews on Union Staff Length of Time in the Union Less than four years Four years or more Percent who don’t care if there are Jews on the union staff, by age 29 years and under 56.4 62.7 30–49 years (78) (51) 50 years and older 37.1 48.3 (35) (116) 38.4 56.1 (13) (89) Source: Adapted from Morris Rosenberg, The Logic of Survey Analysis (New York: Basic Books, 1968), 88–89. Used by permission. supportive of having Jews on the staff. Age, in this suggests another dimension that could be added case, was a suppressor variable, concealing the to the paradigm. relationship between length of membership and at- titude toward Jews. Third, the limitation of the basic paradigm to partials that are the same as or weaker than the Second, the basic paradigm focuses on original neglects two other possibilities. A partial partials being the same as or weaker than the relationship might be stronger than the original. o­ riginal relationship but does not provide guide- Or, on the other hand, a partial relationship might lines for specifying what constitutes a significant be the reverse of the original—for example, nega- difference between the original and the partials. tive where the original was positive. When you use the elaboration model, you’ll ­ frequently find yourself making an arbitrary ­ Rosenberg provides a hypothetical example decision about whether a given partial is sig­ of the latter possibility by first suggesting that nificantly weaker than the original. This, then, a researcher might find that working-class re- spondents in his study are more supportive of suppressor variable  In the elaboration model, a the civil rights movement than middle-class test variable that prevents a genuine relationship respondents are (see Table 15-11). He further from appearing at the zero-order level. suggests that race might be a distorter variable distorter variable  In the elaboration model, a test in this ­instance, reversing the true relationship variable that reverses the direction of a zero-order between class and attitudes. Presumably, ­African relationship. ­American respondents would be more supportive of the movement than whites would, but A­ frican A­ mericans would also be overrepresented

The Elaboration Paradigm ■ 455 Table 15-11 of the women had been hired relatively recently, Example of a Distorter Variable (Hypothetical) when salaries were higher overall than in the ear- lier years when many of the men had been hired I: Working-Class Subjects Appear More Liberal on Civil Rights than Middle- (reported in E. Cook 1995). Class Subjects All these new dimensions further complicate Civil Rights Score Middle Class Working Class the notion of specification. If one partial is the same as the original, and the other partial is even High 37% 45% stronger, how should you react to that situation? Low 63 55 You’ve specified one condition under which the 100 100 original relationship holds up, but you’ve also 100% = (120) (120) specified another condition under which it holds even more clearly. II: Controlling for Race Shows the Middle Class to Be More Liberal than the Working Class Finally, the basic paradigm focuses primarily Social Class on dichotomous test variables. In fact, the elabora- tion model is not so limited—either in theory or in Blacks Whites use—but the basic paradigm becomes more com- plicated when the test variable divides the sample Civil Middle Working Middle Working into three or more subsamples. And the paradigm Rights Class Class Class Class becomes more complicated yet when more than Score one test variable is used simultaneously. High 70% 50% 30% 20% Low 30 50 70 80 I’m not saying all this to fault the basic elabo- 100 100 100 100 ration paradigm. To the contrary, I want to em- 100% = (20) (100) (100) (20) phasize that the elaboration model is not a simple algorithm—a set of procedures through which to Source: Morris Rosenberg, The Logic of Survey Analysis (New York: Basic Books, analyze research. Rather, it’s primarily a logical 1968), 94–95. Used by permission. device for assisting the researcher in understanding his or her data. A firm understanding of the elabo- among working-class respondents and under- ration model will make a sophisticated analysis represented among the middle class. Middle- easier. However, this model suggests neither which class African American respondents might be variables should be introduced as controls nor more supportive than working-class African definitive conclusions about the nature of elabora- ­Americans, however; and the same relationship tion results. For all these things, you must look to might be found among whites. Holding race your own ingenuity. Such ingenuity, moreover, will constant, then, the r­ esearcher would conclude come only through extensive experience. By point- that support for the civil rights movement was ing to oversimplifications in the basic elaboration greater among the middle class than among the paradigm, I’ve sought to bring home the point that working class. the model provides only a logical framework. You’ll find sophisticated analyses far more complicated Here’s another example of a distorter variable than the examples I’ve used to illustrate the basic at work. When Michel de Seve set out to exam- paradigm. ine the starting salaries of men and women in the same organization, she was surprised to find the At the same time, if you fully understand the women were receiving higher starting salaries, on basic model, you’ll understand other techniques the average, than their male counterparts were. such as correlations, regressions, and factor analy- The distorter variable was time of first hire. Many ses a lot more easily. Chapter 16 places such tech- niques as partial correlations and partial regressions in the context of the elaboration model.

456 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Elaboration and Ex Post Facto relationships that do not confirm some prior Hypothesizing hypothesis. Before we leave the discussion of the elaboration Surely, few researchers would now wish that model, we should look at it in connection with a Samuel Stouffer had hushed up his anomalous form of fallacious reasoning called ex post facto findings regarding morale among soldiers in the h­ ypothesizing. Although the social science litera- army. Stouffer noted peculiar empirical observa- ture presents a host of references warning against tions and set about hypothesizing the reasons for it, inexperienced researchers can sometimes be those findings. And his reasoning has proved in- confused about its implications. valuable to researchers ever since. The key is that his “after the fact” hypotheses could themselves “Ex post facto” means “after the fact.” When be tested. you observe an empirical relationship between two variables and then simply suggest a reason for that There is another, more sophisticated point relationship, that is sometimes called ex post facto to be made here, however. Anyone can gener- hypothesizing. You’ve generated a hypothesis link- ate hypotheses to explain observed empirical ing two variables after their relationship is already relationships in a body of data, but the elabora- known. You’ll recall, from an early discussion in tion model provides the logical tools for testing this book, that all hypotheses must be subject to those hypotheses within the same body of data. disconfirmation in order to be meaningful. Un- A good example of this testing may be found in less you can specify empirical findings that would the earlier discussion of social class and church disprove your hypothesis, it’s not really a hypothesis involvement. Glock explained the original re- as researchers use that term. You might reason, lationship in terms of social deprivation theory. therefore, that once you’ve observed a relationship If he had stopped at that point, his comments between two variables, any hypothesis regarding would have been interesting but hardly persua- that relationship cannot be disproved. sive. He went beyond that point, however. He noted that if the hypothesis was correct, then the This is a fair assessment if you’re doing nothing relationship between social class and church in- more than dressing up your empirical observations volvement should disappear among those women with deceptive hypotheses after the fact. Having who were receiving gratification from the secular observed that women are more religious than men, society—those who had held office in a secular you should not simply assert that women will be organization. This hypothesis was then subjected more religious than men because of some general to an empirical test. Had the new hypothesis not dynamic of social behavior and then rest your case been confirmed by the data, he would have been on the initial observation. forced to reconsider. The unfortunate spin-off of the injunction These additional comments should further against ex post facto hypothesizing is its inhibition illustrate the point that data analysis is a con- of good, honest hypothesizing after the fact. Inex- tinuing process, demanding all the ingenuity perienced researchers are often led to believe that and perseverance you can muster. The image of they must make all their hypotheses before exam- a researcher carefully laying out hypotheses and ining their data—even if that process means mak- then testing them in a ritualistic fashion results ing a lot of poorly reasoned ones. Furthermore, only in ritualistic research. they’re led to ignore any empirically observed In case you’re concerned that the strength of ex ex post facto hypothesis  A hypothesis created post facto proofs seems to be less than that of the after confirming data have already been collected. It traditional kinds, let me repeat the earlier assertion is a meaningless construct because there is no way that “scientific proof” is a contradiction in terms. for it to be disconfirmed. Nothing is ever proved scientifically. Hypotheses, explanations, theories, or hunches can all escape a stream of attempts at disproof, but none can be

Key Terms ■ 457 proved in any absolute sense. The acceptance of a antecedent to the other two variables or interven- hypothesis, then, is really a function of the extent ing between them. to which it has been tested and not disconfirmed. No hypothesis, therefore, should be considered • The outcome of an elaboration analysis may be sound on the basis of one test—whether the hy- pothesis was generated before or after the obser- replication (whereby a set of partial relationships vation of empirical data. With this in mind, you is essentially the same as the corresponding zero- should not deny yourself some of the most fruit- order relationship), explanation (whereby a set of ful avenues available to you in data analysis. You partial relationships is reduced essentially to zero should always try to reach an honest understand- when an antecedent variable is held constant), ing of your data, develop meaningful theories for interpretation (whereby a set of partial relation- more general understanding, and not worry about ships is reduced essentially to zero when an inter- the manner of reaching that understanding. vening variable is held constant), or specification (whereby one partial relationship is reduced, Main Points ideally to zero, and the other remains about the same as the original relationship or is stronger). Introduction • A suppressor variable conceals the relationship • The elaboration model is a method of multivariate between two other variables; a distorter variable analysis appropriate for social research. It is pri- causes an apparent reversal in the relationship be- marily a logical model that can illustrate the basic tween two other variables (from negative to posi- logic of other multivariate methods. tive or vice versa). The Origins of the Elaboration Model Elaboration and Ex Post Facto • Paul Lazarsfeld and Patricia Kendall used the logic Hypothesizing of the elaboration model to present hypothetical • Ex post facto hypothesizing, or the development tables regarding Samuel Stouffer’s work regard- ing education and acceptance of induction in the of hypotheses “predicting” relationships that have army. already been observed, is invalid in science, be- cause disconfirming such hypotheses is impossible. • A partial relationship (or “partial”) is the observed Although nothing prevents us from suggesting reasons that observed relationships may be the relationship between two variables within a sub- way they are, we should not frame those reasons group of cases based on some attribute of the test in the form of “hypotheses.” More important, one or control variable. observed relationship and possible reasons for it may suggest hypotheses about other relationships • A zero-order relationship is the observed relation- that have not been examined. The elaboration model is an excellent logical device for this kind of ship between two variables without a third vari- unfolding analysis of data. able being held constant or controlled. Key Terms The Elaboration Paradigm The following terms are defined in context in the • The basic steps in elaboration are as follows: (1) A chapter and at the bottom of the page where the term is introduced, as well as in the comprehensive relationship is observed to exist between two vari- glossary at the back of the book. ables, (2) a third variable (the test variable) is held constant in the sense that the cases under study are distorter variable replication subdivided according to the attributes of that third elaboration model specification variable, (3) the original two-variable relationship ex post facto hypothesis suppressor variable is recomputed within each of the subgroups, and explanation test variable (4) the comparison of the original relationship with interpretation zero-order relationship the relationships found within each subgroup (the partial relationship partial relationships) provides a fuller understand- ing of the original relationship itself. • The logical relationships of the variables dif- fer depending on whether the test variable is

458 ■ Chapter 15: Origins and Paradigm of the Elaboration Model Proposing Social Research: Social Sciences). There are exercises offered for T h e E l a b o r at i o n M o d e l each chapter, and you’ll also find a detailed primer on using SPSS. See the exercise for Chapter 16 (p. 495). Review Questions and Exercises Online Study Resources 1. Review the Stouffer-Kendall-Lazarsfeld example Access the resources your instructor has assigned. For of education, friends deferred, and attitudes this book, you can access: ­toward being drafted. Suppose they had begun with an association between friends deferred and CourseMate for The attitudes toward being drafted, and then they had Practice of Social Research controlled for education. What conclusion would they have reached? Login to CengageBrain.com to access chapter-specific learning tools including Learning Objectives, Practice 2. In your own words describe the elaboration logic Quizzes, Videos, Internet Exercises, Flash Cards, Glossaries, of (a) replication, (b) interpretation, (c) explana- Web Links, and more from your Sociology CourseMate. tion, and (d) specification. If your professor has assigned Aplia homework: 3. Review the box on Ivy League colleges and suc- 1. Sign into your account. cess in later professional life. In your own words, 2. After you complete each page of questions, click explain what Patricia Kendall means when she says, “Despite this [support from the analysis of “Grade It Now” to see detailed explanations of partial relationships], West had in no way proved every answer. her hypothesis.” What conclusions can one rea- 3. Click “Try Another Version” for an opportunity to sonably draw from West’s study? improve your score. Visit www.cengagebrain.com to access your account 4. Construct hypothetical examples of suppressor and purchase materials. and distorter variables. 5. Search the web for a research report on the dis- covery of a spurious relationship. Give the web address of the document and quote or paraphrase what was discovered. S P SS E x e r c i s e s See the booklet that accompanies your text for exercises using SPSS (Statistical Package for the

CHAPTER 16 Methods of Statistical Analysis chapter o v er v i e w Introduction Other Multivariate Techniques Statistics allow researchers Descriptive Statistics to summarize data, measure Data Reduction Path Analysis associations between variables, Measures of Association and draw inferences from samples Regression Analysis Time-Series Analysis to populations. Getting acquainted with a few simple statistics frequently Inferential Statistics Factor Analysis used in social research is less Univariate Inferences painful (and less threatening to Tests of Statistical Analysis of Variance your social life) than you might   Significance believe. The Logic of Statistical Discriminant Analysis   Significance Chi Square Log-Linear Models t-Test Some Words of Caution Odds-Ratio Analysis Geographic Information   Systems (GIS) Aplia for The Practice of Social Research After reading, go to “Online Study Resources” at the end of this chapter for

460 ■ Chapter 16: Methods of Statistical Analysis Introduction statistics. There is a good chance (i.e., probability) that you will need to take a statistics course as part It has been my experience over the years that of your program of study, and I want the discus- many students are intimidated by statistics. Some- sions of this chapter to give you a running start on times statistics makes them feel they’re that course if you do need (or want) to take it. • A few clowns short of a circus We’ll be looking at two types of statistics: de- • Dumber than a box of hair scriptive and inferential. Descriptive statistics is a • A few feathers short of a duck medium for describing data in manageable forms. • All foam, no beer Inferential statistics, on the other hand, assists re- • Missing a few buttons on their remote control searchers in drawing conclusions from their obser- • A few beans short of a burrito vations; typically, this involves drawing conclusions • As screwed up as a football bat about a population from the study of a sample • About as sharp as a bowling ball drawn from it. After that discussion, I’ll briefly • About four cents short of a nickel introduce you to some of the analytic techniques • Not running on full thrusters* you may come across in your reading of the social science literature. Many people are intimidated by quantitative research because they feel uncomfortable with Descriptive Statistics mathematics and statistics. And indeed, many re- search reports are filled with unspecified computa- As I’ve already suggested, descriptive statistics tions. The role of statistics in social research is often present quantitative descriptions in a manageable important, but it’s equally important to see this role form. Sometimes we want to describe single vari- in its proper perspective. ables, and sometimes we want to describe the as- sociations that connect one variable with another. Empirical research is first and foremost a logical Let’s look at some of the ways to do these things. rather than a mathematical operation. Mathemat- ics is merely a convenient and efficient language Data Reduction for accomplishing the logical operations inherent in quantitative data analysis. Statistics is the applied Scientific research often involves collecting large branch of mathematics especially appropriate for masses of data. Suppose we surveyed 2,000 people, a variety of research analyses. This textbook is not asking each of them 100 questions—not an unusu- intended to teach you statistics or torture you with ally large study. We would then have a staggering them. Rather, I want to sketch out a logical context 200,000 answers! No one could possibly read all within which you might learn and understand those answers and reach any meaningful conclu- sion about them. Thus, much scientific analysis *  Thanks to the many contributors to humor lists on involves the reduction of data from unmanageable the Internet. details to manageable summaries. descriptive statistics  Statistical computations To begin our discussion, let’s look briefly at the describing either the characteristics of a sample or raw-data matrix created by a quantitative research the relationship among variables in a sample. De- project. Table 16-1 presents a partial data matrix. scriptive statistics merely summarize a set of sample Notice that each row in the matrix represents a observations, whereas inferential statistics move person (or other unit of analysis), each column beyond the description of specific observations to represents a variable, and each cell represents the make inferences about the larger population from coded attribute or value a given person has on a which the sample observations were drawn.

Table 16-1 Descriptive Statistics ■ 461 Partial Raw-Data Matrix Political Political Religious Importance of Person A Sex Age Education Income Occupation Affiliation Orientation Affiliation Religion Person B 32412304 Person C 1 42441112 Person D 1 25522423 Person E 2 54432224 Person F 1 37861151 2 13353511 2 given variable. The first column in Table 16-1 rep- and extent of the relationship between education resents a person’s sex. Let’s say a “1” represents and prejudice. male and a “2” represents female. This means that persons A and B are male, person C is female, and Notice, for example, that 23 people (1) have no so forth. education and (2) scored high on prejudice; 77 people (1) had graduate degrees and (2) scored In the case of age, person A’s “3” might mean low on prejudice. 30–39 years old, person B’s “4” might mean 40–49. However age has been coded (see Chapter 14), the Like the raw-data matrix in Table 16-1, this code numbers shown in Table 16-1 describe each of matrix provides more information than can eas- the people represented there. ily be comprehended. A careful study of the table shows that as education increases from “None” to Notice that the data have already been reduced “Graduate Degree,” there is a general tendency for somewhat by the time a data matrix like this one prejudice to decrease, but no more than a general has been created. If age has been coded as sug- impression is possible. For a more precise summary gested previously, the specific answer “33 years of the data matrix, we need one of several types old” has already been assigned to the category of descriptive statistics. Selecting the appropriate “30–39.” The people responding to our survey may measure depends initially on the nature of the two have given us 60 or 70 different ages, but we’ve variables. now reduced them to 6 or 7 categories. We’ll turn now to some of the options avail- Chapter 14 discussed some of the ways of fur- able for summarizing the association between two ther summarizing univariate data: averages such as the mode, median, and mean and measures of dis- Table 16-2 persion such as the range, the standard deviation, Hypothetical Raw Data on Education and Prejudice and so forth. It’s also possible to summarize the as- sociations among variables. Educational Level Measures of Association Prejudice None Grade High Graduate School School College Degree The association between any two variables can also High 23 be represented by a data matrix, this time produced Medium 11 34 156 67 16 by the joint frequency distributions of the two vari- Low  6 23 ables. Table 16-2 presents such a matrix. It provides 21 123 102 77 all the information needed to determine the nature 12 95 164

462 ■ Chapter 16: Methods of Statistical Analysis variables. Each of these measures of association is concepts in statistics, which perhaps helps to ac- based on the same model—proportionate count for the number of people who say of sta- reduction of error (PRE). tistics, “It’s all Greek to me.”) Lambda is based on your ability to guess values on one of the variables: To see how this model works, let’s assume that the PRE achieved through knowledge of values on I asked you to guess respondents’ attributes on the other variable. a given variable: for example, whether they an- swered yes or no to a given questionnaire item. To Imagine this situation: I tell you that a room assist you, let’s first assume you know the overall contains 100 people and I would like you to guess distribution of responses in the total sample—say, the gender of each person, one at a time. If half are 60 percent said yes and 40 percent said no. You men and half women, you’ll probably be right half would make the fewest errors in this process if the time and wrong half the time. you always guessed the modal (most frequent) ­response: yes. But suppose I tell you each person’s occupa- tion before you guess that person’s sex. What sex Second, let’s assume you also know the empiri- would you guess if I said the person was a truck cal relationship between the first variable and some driver? You would probably be wise to guess other variable: say, gender. Now, each time I ask “male”; although there are now plenty of women you to guess whether a respondent said yes or no, truck drivers, most are still men. If I said the next I’ll tell you whether the respondent is a man or a person was a nurse, you’d probably be wisest to woman. If the two variables are related, you should guess “female,” following the same logic. Although make fewer errors the second time. It’s possible, you would still make errors in guessing “sexes,” therefore, to compute the PRE by knowing the you would clearly do better than you would if you relationship between the two variables: the greater didn’t know their occupations. The extent to which the relationship, the greater the reduction of error. you did better (the proportionate reduction of error) would be an indicator of the association that This basic PRE model is modified slightly to exists between sex and occupation. take account of different levels of measurement— nominal, ordinal, or interval. The following sec- Here’s another simple hypothetical example tions will consider each level of measurement and that illustrates the logic and method of lambda. present one measure of association appropriate for Table 16-3 presents hypothetical data relating sex each. Bear in mind that the three measures dis- to employment status. Overall, we note that 1,100 cussed are only an arbitrary selection from among people are employed, and 900 are not employed. many appropriate measures. If you were to predict whether people were em- ployed, and if you knew only the overall distribu- Nominal Variables tion on that variable, you would always predict “employed,” because that would result in fewer If the two variables consist of nominal data (for errors than always predicting “not employed.” example, gender, religious affiliation, race), lambda ­Nevertheless, this strategy would result in (λ) would be one appropriate measure. (Lambda 900 errors out of 2,000 predictions. is a letter in the Greek alphabet corresponding to l in our alphabet. Greek letters are used for many Let’s suppose that you had access to the data in Table 16-3 and that you were told each person’s proportionate reduction of error (PRE)  A logi- sex before making your prediction of employment cal model for assessing the strength of a relationship status. Your strategy would change in that case. For by asking how much knowing values on one vari- every man you would predict “employed,” and for able would reduce our errors in guessing values on every woman you would predict “not employed.” the other. For example, if we know how much edu- In this instance, you would make 300 errors—the cation people have, we can improve our ability to 100 men who were not employed and the 200 estimate how much they earn, thus indicating there employed women—or 600 fewer errors than you is a relationship between the two variables. would make without knowing the person’s sex.

Descriptive Statistics ■ 463 Table 16-3 Let’s say we have a group of elementary stu- Hypothetical Data Relating Sex to Employment Status dents. It’s reasonable to assume that there is a re- lationship between their ages and their heights. Employed Men Women Total We can test this by comparing every pair of stu- Unemployed dents: Brett and Sophia, Brett and Terrell, Sophia Total 900 200 1,100 and Terrell, and so forth. Then we ignore all the 100 800 900 pairs in which the students are the same age 1,000 1,000 and/or the same height. We then classify each 2,000 of the remaining pairs (those who differ in both age and height) into one of two categories: those Lambda, then, represents the reduction in er- in which the older child is also the taller (“same” rors as a proportion of the errors that would have pairs) and those in which the older child is the been made on the basis of the overall distribution. shorter (“opposite” pairs). So, if Brett is older In this hypothetical example, lambda would equal and taller than Sophia, the Brett–Sophia pair is 0.67; that is, 600 fewer errors divided by the 900 counted as a “same.” If Brett is older but shorter total errors based on employment status alone. In than Sophia, then that pair is an “opposite.” this fashion, lambda measures the statistical asso- ciation between sex and employment status. To determine whether age and height are related to each other, we compare the number If sex and employment status were statistically of same and opposite pairs. If the same pairs out- independent, we would find the same distribution number the opposite pairs, we can conclude that of employment status for men and women. In this there is a positive association between the two case, knowing each person’s sex would not affect variables—as one increases, the other increases. the number of errors made in predicting employ- If there are more opposites than sames, we can ment status, and the resulting lambda would be conclude that the relationship is negative. If zero. If, on the other hand, all men were employed there are about as many sames as opposites, we and none of the women were employed, by know- can conclude that age and height are not related ing sex you would avoid errors in predicting em- to each another, that they’re independent of ployment status. You would make 900 fewer errors each other. (out of 900), so lambda would be 1.0—representing a perfect statistical association. Here’s a social science example to illustrate the simple calculations involved in gamma. Let’s Lambda is only one of several measures of say you suspect that religiosity is positively re- association appropriate for the analysis of two lated to political conservatism, and if Person A is nominal variables. You could look at any statis- more religious than Person B, you guess that A tics textbook for a discussion of other appropriate is also more conservative than B. Gamma is the measures. proportion of paired comparisons that fits this pattern. Ordinal Variables Table 16-4 presents hypothetical data relat- If the variables being related are ordinal (for ex- ing social class to prejudice. The general nature ample, social class, religiosity, alienation), gamma of the relationship between these two variables is (g) is one appropriate measure of association. Like that as social class increases, prejudice decreases. lambda, gamma is based on our ability to guess val- There is a negative association between social ues on one variable by knowing values on another. class and prejudice. However, whereas lambda is based on guessing exact values, gamma is based on guessing the or- Gamma is computed from two quantities: (1) dinal arrangement of values. For any given pair of the number of pairs having the same ranking on cases, we guess that their ordinal ranking on one the two variables and (2) the number of pairs hav- variable will correspond (positively or negatively) ing the opposite ranking on the two variables. The to their ordinal ranking on the other. pairs having the same ranking are computed as

464 ■ Chapter 16: Methods of Statistical Analysis Table 16-4 Table 16-5 Hypothetical Data Relating Social Class to Prejudice Gamma Associations among the Semantic Differentiation Prejudice Lower Class Middle Class Upper Class Items of the Sanctification Scale Low 200 400 700 Useful Honest Superior Kind Friendly Warm Medium 500 900 400 High 800 300 100 Good 0.79 0.88 0.80 0.90 0.79 0.83 Useful 0.84 0.71 0.77 0.68 0.72 Honest 0.83 0.89 0.79 0.82 follows. The frequency of each cell in the table is Superior 0.78 0.60 0.73 multiplied by the sum of all cells appearing below and to the right of it—with all these products being Kind 0.88 0.90 summed. In Table 16-4, the number of pairs with the same ranking would be 200(900 + 300 + 400 Friendly 0.90 + 100) + 500(300 + 100) + 400(400 + 100) + 900(100), or 340,000 + 200,000 + 200,000 + Source: Helena Znaniecki Lopata,“Widowhood and Husband Sanctification,” 90,000 5 830,000. Journal of Marriage and the Family (May 1981): 439–50. The pairs having the opposite ranking on the direction of the relationship. (A negative lambda two variables are computed as follows: The fre- would indicate that you made more errors in pre- quency of each cell in the table is multiplied by dicting values on one variable while knowing val- the sum of all cells appearing below and to the ues on the second than you made in ignorance of left of it—with all these products being summed. the second, and that’s not logically possible.) In Table 16-4, the numbers of pairs with opposite rankings would be 700(500 + 800 + 900 + 300) Table 16-5 is an example of the use of gamma + 400(800 + 300) + 400(500 + 800) + 900(800), in social research. To study the extent to which or 1,750,000 + 440,000 + 520,000 + 720,000 5 widows sanctified their deceased husbands, Hel- 3,430,000. Gamma is computed from the numbers ena Lopata (1981) administered a questionnaire of same-ranked pairs and opposite-ranked pairs as to a probability sample of 301 widows. In part, the follows: questionnaire asked the respondents to character- ize their deceased husbands in terms of the follow- same 2 opposite ing semantic differentiation scale: gamma 5 same 1 opposite Characteristic In our example, gamma equals (830,000 2 3,430,000) divided by (830,000 1 3,430,000), or Positive Negative 20.61. The negative sign in this answer indicates Extreme Extreme the negative association suggested by the initial in- spection of the table. Social class and prejudice, in Good 1 2 3 4 5 6 7 Bad this hypothetical example, are negatively associated Useful 1 2 3 4 5 6 7 Useless with each other. The numerical figure for gamma Honest 1 2 3 4 5 6 7 Dishonest indicates that 61 percent more of the pairs examined Superior 1 2 3 4 5 6 7 Inferior had the opposite ranking than the same ranking. Kind 1 2 3 4 5 6 7 Cruel Friendly 1 2 3 4 5 6 7 Unfriendly Note that whereas values of lambda vary from Warm 1 2 3 4 5 6 7 Cold 0 to 1, values of gamma vary from 21 to 11, rep- resenting the direction as well as the magnitude of Respondents were asked to describe their de- the association. Because nominal variables have no ceased spouses by circling a number for each pair ordinal structure, it makes no sense to speak of the of opposing characteristics. Notice that the series of

Descriptive Statistics ■ 465 numbers connecting each pair of characteristics is would minimize your errors by always guessing the an ordinal measure. mean value of the variable. Although this practice produces few if any perfect guesses, the extent of Next, Lopata wanted to discover the extent to your errors will be minimized. Imagine the task which the several measures were related to one of guessing peoples’ incomes and how much bet- another. Appropriately, she chose gamma as the ter you would do if you knew how many years of measure of association. Table 16-5 shows how she education they had as well as the mean incomes presented the results of her investigation. for people with 0, 1, 2 (and so forth) years of education. The format presented in Table 16-5 is called a correlation matrix. For each pair of measures, Lopata In the computation of lambda, we noted the has calculated the gamma. Good and Useful, for ex- number of errors produced by always guessing the ample, are related to each other by a gamma equal modal value. In the case of r, errors are measured to 0.79. The matrix is a convenient way of present- in terms of the sum of the squared differences be- ing the intercorrelations among several variables, tween the actual value and the mean. This sum is and you’ll find it frequently in the research litera- called the total variation. ture. In this case, we see that all the variables are quite strongly related to one another, though some To understand this concept, we must expand pairs are more strongly related than others. the scope of our examination. Let’s look at the logic of regression analysis and discuss correlation within Gamma is only one of several measures of as- that context. sociation appropriate for ordinal variables. Again, any introductory statistics textbook will give you a Regression Analysis more comprehensive treatment of this subject. The general formula for describing the association Interval or Ratio Variables between two variables is Y 5 f(X). This formula is read “Y is a function of X,” meaning that values of If interval or ratio variables (for example, age, Y can be explained in terms of variations in the val- ­income, grade point average, and so forth) are being ues of X. Stated more strongly, we might say that X associated, one appropriate measure of association causes Y, so the value of X determines the value of is Pearson’s product-moment correlation (r). The Y. Regression analysis is a method of determin- derivation and computation of this measure of as- ing the specific function relating Y to X. There are sociation are complex enough to lie outside the several forms of regression analysis, depending on scope of this book, so I’ll make only a few general the complexity of the relationships being studied. comments here. Let’s begin with the simplest. Like both gamma and lambda, r is based on Linear Regression guessing the value of one variable by knowing another. For continuous interval or ratio variables, The regression model can be seen most clearly in however, it’s unlikely that you could predict the the case of a linear regression analysis, in which precise value of the variable. On the other hand, a perfect linear association between two variables predicting only the ordinal arrangement of values on the two variables would not take advantage of regression analysis  A method of data analysis the greater amount of information conveyed by an in which the relationships among variables are interval or ratio variable. In a sense, r reflects how ­represented in the form of an equation, called a closely you can guess the value of one variable r­egression equation. through your knowledge of the value of another. linear regression analysis  A form of statistical To understand the logic of r, consider the way analysis that seeks the equation for the straight line you might hypothetically guess values that particu- that best describes the relationship between two lar cases have on a given variable. With nominal ratio variables. variables, we’ve seen that you might always guess the modal value. But for interval or ratio data, you

466 ■ Chapter 16: Methods of Statistical Analysis Figure 16-1 city’s population and its crime rate. As was the case in our previous example, the values of Y (crime Simple Scattergram of Values of X and Y rates) generally correspond to those of X (popula- tions), and as values of X increase, so do values of exists or is approximated. Figure 16-1 is a scat- Y. However, the association is not nearly as clear as tergram presenting in graphic form the values it is in Figure 16-1. of X and Y as produced by a hypothetical study. It shows that for the four cases in our study, the In Figure 16-2 we can’t superimpose a straight values of X and Y are identical in each instance. line that will pass through all the points in the scat- The case with a value of 1 on X also has a value of tergram. But we can draw an approximate line 1 on Y, and so forth. The relationship between the showing the best possible linear representation two variables in this instance is described by the of the several points. I’ve drawn that line on the equation Y 5 X; this is called the regression equation. graph. Because all four points lie on a straight line, we could superimpose that line over the points; this is You may (or may not) recall from algebra that the regression line. any straight line on a graph can be represented by an equation of the form Y 5 a 1 bX, where X and The linear regression model has important de- Y are values of the two variables. In this equation, scriptive uses. The regression line offers a graphic a equals the value of Y when X is 0, and b repre- picture of the association between X and Y, and the sents the slope of the line. If we know the values of regression equation is an efficient form for sum- a and b, we can calculate an estimate of Y for every marizing that association. The regression model value of X. has inferential value as well. To the extent that the regression equation correctly describes the general We can now say more formally that regression association between the two variables, it may be analysis is a technique for establishing the regres- used to predict other sets of values. If, for example, sion equation representing the geometric line that we know that a new case has a value of 3.5 on X, comes closest to the distribution of points on a we can predict the value of 3.5 on Y as well. graph. The regression equation provides a math- ematical description of the relationship between the In practice, of course, studies are seldom lim- variables, and it allows us to infer values of Y when ited to four cases, and the associations between we have values of X. Recalling Figure 16-2, we variables are seldom as clear as the one presented could estimate crime rates of cities if we knew their in Figure 16-1. populations. A somewhat more realistic example is pre- To improve your guessing, you construct a sented in Figure 16-2, representing a hypothetical regression line, stated in the form of a regression relationship between population and crime rate in equation that permits the estimation of values on small- to medium-size cities. Each dot in the scat- one variable from values on the other. The general tergram is a city, and its placement reflects that format for this equation is Y 5 a 1 b(X), where a and b are computed values, X is a given value on ootnheevr.aTrihaeblvea, laCuneeds noYfgaisaatnghdeebesLatrieme acaotremndpivnuatlgeudetoonmthinei- mize the diffeBreanbcebsieb:etTwheeenPraactcutaicl evaoluf es of Y and the correspSooncdiianlgReestsimeaartcesh(,Y13)/beased on the known value1o-f13X3.-T0h49e7s9u-m6 of sFqigu.a1re6d-1differences between actual and estimated values of Y is called the unexplained variation because it represents errors that still exist even when estimates are based on known values of X. The explained variation is the difference be- tween the total variation and the unexplained

Descriptive Statistics ■ 467 Figure 16-2 A Scattergram of the Values of Two Variables with Regression Line Added (Hypothetical) variation. Dividing the explained variation by Multiple Regression the total variation produces a measure of the pro- portionate reduction of error corresponding to the Very often, social researchers find that a given de- similar quantity in the computation of lambda. pendent variable is affected simultaneously by sev- In the present case, this quantity is the correla- eral independent variables. Multiple regression tion squared: r2. Thus, if r 5 0.7, then r2 5 0.49, analysis provides a means of analyzing such situa- meaning that about half the variation has been tions. This was the case when Beverly Yerg (1981) explained. In practice, we compute r rather set about studying teacher effectiveness in physical than r2, because the product-moment correla- education. She stated her expectations in the form tion can take either a positive or a negative sign, of a multiple regression equation: depending on the direction of the relationship between the two variables. (Computing r2 and F 5 b0 1 b1I 1 b2X1 1 b3X2 1 b4X3 1 b5X4 1 e, taking a square root would always produce a where positive quantity.) You can consult any standard statistics textbook for the method of computing r, F 5 Final pupil-performance score although there are many data analysis programs available to do this. I 5 Initial pupil-performance score Unfortunately—or perhaps fortunately—social X1 5 Composite of guiding and supporting life is so complex that the simple linear regression practice model often does not sufficiently represent the state of affairs. As we saw in Chapter 14, it’s pos- X2 5 Composite of teacher mastery of content sible, using percentage tables, to analyze more than two variables. As the number of variables increases, X3 5 Composite of providing specific, Ctasekn- g a g e Learning such tables become increasingly complicated and related feedback hard to read. The regression model offers a useful Babbie: The Practice of alternative in such cases. Social Research, 13/e multiple regression analysis  rAepforermsenotfinstg1a-tt1his3eti3c-a0l 4979-6 Fig. 16-2 analysis that seeks the equation impact of two or more independent variables on a single dependent variable.

468 ■ Chapter 16: Methods of Statistical Analysis X4 5 Composite of clear, concise task between education and prejudice separately for presentation each age group. b 5 Regression weight Partial regression analysis is based on this e 5 Residual same logical model. The equation summarizing the relationship between variables is computed on the (Adapted from Yerg 1981: 42) basis of the test variables remaining constant. As in the case of the elaboration model, the result may Notice that in place of the single X variable in then be compared with the uncontrolled relation- a linear regression, there are several X’s, and there ship between the two variables to clarify further are also several b’s instead of just one. Also, Yerg the overall relationship. has chosen to represent a as b0 in this equation but with the same meaning as discussed previously. Curvilinear Regression Finally, the equation ends with a residual factor (e), which represents the variance in Y that is not ac- Up to now, we’ve been discussing the association counted for by the X variables analyzed. among variables as represented by a straight line. The regression model is even more general than Beginning with this equation, Yerg calculated our discussion thus far has implied. the values of the several b’s to show the relative contributions of the several independent vari- You may already know that curvilinear func- ables in determining final student-performance tions, as well as linear ones, can be represented scores. She also calculated the multiple-correlation by equations. For example, the equation X2 1 Y2 coefficient as an indicator of the extent to which 5 25 describes a circle with a radius of 5. Raising all six variables predict the final scores. This follows variables to powers greater than 1 has the effect of the same logic as the simple bivariate correlation producing curves rather than straight lines. In the discussed earlier, and it’s traditionally reported as real world there is no reason to assume that the re- a capital R. In this case, R 5 0.877, meaning that lationship among every set of variables will be lin- 77 percent of the variance (0.8772 5 0.77) in final ear. In some cases, then, curvilinear regression scores is explained by the six variables acting in analysis can provide a better understanding of em- concert. pirical relationships than any linear model can. Partial Regression Recall, however, that a regression line serves two functions. It describes a set of empirical ob- In exploring the elaboration model in Chapter 15, servations, and it provides a general model for we paid special attention to the relationship be- making inferences about the relationship between tween two variables when a third test variable was two variables in the general population that the held constant. Thus, we might examine the effect observations represent. A very complex equation of education on prejudice with age held constant, might produce an erratic line that would indeed testing the independent effect of education. To do pass through every individual point. In this sense, so, we would compute the tabular relationship it would perfectly describe the empirical observa- tions. There would be no guarantee, however, that partial regression analysis  A form of regression such a line could adequately predict new observa- analysis in which the effects of one or more variables tions or that it in any meaningful way represented are held constant, similar to the logic of the elabora- the relationship between the two variables in tion model. general. Thus, it would have little or no inferential curvilinear regression analysis  A form of regres- value. sion analysis that allows relationships among vari- ables to be expressed with curved geometric lines Earlier in this book, we discussed the need for instead of straight ones. balancing detail and utility in data reduction. Ul- timately, researchers attempt to provide the most faithful, yet also the simplest, representation of

Inferential Statistics ■ 469 their data. This practice also applies to regression Inferential Statistics analysis. Data should be presented in the simplest fashion that best describes the actual data; as such, Many, if not most, social science research projects linear regressions are the ones most frequently involve the examination of data collected from a used. Curvilinear regression analysis adds a new sample drawn from a larger population. A sample option to the researcher in this regard, but it does of people may be interviewed in a survey; a sample not solve the problems altogether. Nothing does of divorce records may be coded and analyzed; a that. sample of newspapers may be examined through content analysis. Researchers seldom if ever study Cautions in Regression Analysis samples just to describe the samples per se; in most instances, their ultimate purpose is to make asser- The use of regression analysis for statistical infer- tions about the larger population from which the ences is based on the same assumptions made for sample has been selected. Frequently, then, you’ll correlational analysis: simple random sampling, the wish to interpret your univariate and multivariate absence of nonsampling errors, and continuous in- sample findings as the basis for inferences about terval data. Because social science research seldom some population. completely satisfies these assumptions, you should use caution in assessing the results in regression This section examines inferential statistics— analyses. the statistical measures used for making inferences from findings based on sample observations to a Also, regression lines—linear or curvilinear— larger population. We’ll begin with univariate data can be useful for interpolation (estimating cases and move to multivariate. lying between those observed), but they are less trustworthy when used for extrapolation (estimating Univariate Inferences cases that lie beyond the range of observations). This limitation on extrapolations is important Chapter 14 dealt with methods of presenting uni- in two ways. First, you’re likely to come across variate data. Each summary measure was intended regression equations that seem to make illogical as a method of describing the sample studied. Now predictions. An equation linking population and we’ll use such measures to make broader assertions crimes, for example, might seem to suggest that about a population. This section addresses two uni- small towns with, say, a population of 1,000 should variate measures: percentages and means. produce 123 crimes a year. This failure in predictive ability does not disqualify the equation but drama- If 50 percent of a sample of people say they tizes that its applicability is limited to a particular had colds during the past year, 50 percent is also range of population sizes. Second, researchers our best estimate of the proportion of colds in the sometimes overstep this limitation, drawing infer- total population from which the sample was drawn. ences that lie outside their range of observation, (This estimate assumes a simple random sample, of and you’d be right in criticizing them for that. course.) It’s rather unlikely, however, that precisely 50 percent of the population had colds during the The preceding sections have introduced some year. If a rigorous sampling design for random se- of the techniques for measuring associations among lection has been followed, however, we’ll be able variables at different levels of measurement. Mat- to estimate the expected range of error when the ters become slightly more complex when the two sample finding is applied to the population. variables represent different levels of measurement. Though we aren’t going to pursue this issue in this inferential statistics  The body of statistical textbook, UCLA provides an excellent resource computations relevant to making inferences from online at http://www.ats.ucla.edu/stat/mult_pkg/ findings based on sample observations to some larger whatstat/default.htm, adapting the work of Dr. population. James Leeper at the University of Alabama.

470 ■ Chapter 16: Methods of Statistical Analysis Chapter 5, on sampling theory, covered the pro- sampling with replacement, which is almost never cedures for making such estimates, so I’ll only review done—but this is probably not a serious problem. them here. In the case of a percentage, the quantity Although systematic sampling is used more fre- quently than random sampling, it, too, probably p3q presents no serious problem if done correctly. Stratified sampling, because it improves representa- n tiveness, clearly presents no problem. Cluster sam- pling does present a problem, however, because where p is a proportion, q equals (1 2 p), and n is the estimates of sampling error may be too small. the sample size, is called the standard error. As noted Quite clearly, street-corner sampling does not in Chapter 5, this quantity is very important in the warrant the use of inferential statistics. Finally, estimation of sampling error. We may be 68 percent the calculation of standard error in sampling as- confident that the population figure falls within sumes a 100 percent completion rate—that is, that plus or minus one standard error of the sample ­everyone in the sample completed the survey. figure; we may be 95 percent confident that it falls The seriousness of this problem increases as the within plus or minus two standard errors; and we completion rate decreases. may be 99.9 percent confident that it falls within plus or minus three standard errors. Third, inferential statistics are addressed to sampling error only, not nonsampling error such Any statement of sampling error, then, must as coding errors or misunderstandings of ques- contain two essential components: the confidence tions by respondents. Thus, although we might level (for example, 95 percent) and the confidence state correctly that between 47.5 and 52.5 percent ­interval (for example, 2.5 percent). If 50 percent of the population (95 percent confidence) would of a sample of 1,600 people say they had colds report having colds during the previous year, we during the year, we might say we’re 95 percent couldn’t so confidently guess the percentage who confident that the population figure is between had actually had them. Because nonsampling er- 47.5 percent and 52.5 percent. rors are probably larger than sampling errors in a respectable sample design, we need to be especially In this example we’ve moved beyond simply cautious in generalizing from our sample findings describing the sample into the realm of making es- to the population. timates (inferences) about the larger population. In doing so, we must take care in several ways. Tests of Statistical Significance First, the sample must be drawn from the popu- There is no scientific answer to the question of lation about which inferences are being made. A whether a given association between two vari- sample taken from a telephone directory cannot le- ables is significant, strong, important, interest- gitimately be the basis for statistical inferences about ing, or worth reporting. Perhaps the ultimate test the population of a city, but only about the popula- of significance rests in your ability to persuade tion of telephone subscribers with listed numbers. your audience (present and future) of the as- sociation’s significance. At the same time, there Second, the inferential statistics assume sev- is a body of inferential statistics to assist you in eral things. To begin with, they assume simple this regard called parametric tests of significance. As random sampling, which is virtually never the the name suggests, parametric statistics are those case in sample surveys. The statistics also assume that make certain assumptions about the param- eters describing the population from which the nonsampling error  Those imperfections of data sample is selected. They allow us to determine the quality that are a result of factors other than sam- statistical significance of associations. “Statisti- pling error. Examples include misunderstandings of cal significance” does not imply “importance” or questions by respondents and erroneous recordings by interviewers and coders. statistical significance  A general term referring to the likelihood that relationships observed in a sam- ple could be attributed to sampling error alone.

Inferential Statistics ■ 471 “significance” in any general sense. It refers simply 2. Assumptions regarding the representative- to the likelihood that relationships observed in a ness of samples selected through conventional sample could be attributed to sampling error alone. probability-sampling procedures Researchers often distinguish between statistical significance and substantive significance in this regard, 3. The observed joint distribution of sample ele- with the latter referring to whether the relationship ments in terms of the two variables between variables is big enough to make a mean- ingful difference. Whereas statistical significance Figure 16-3 represents a hypothetical popula- can be calculated, substantive significance is always tion of 256 people; half are women, half are men. a judgment call. The diagram also indicates how each person feels about seeing women as equal to men. In the dia- Although tests of statistical significance gram, those favoring equality have open circles, are widely reported in social science literature, the those opposing it have their circles filled in. logic underlying them is rather subtle and often misunderstood. Tests of significance are based on The question we’ll be investigating is whether the same sampling logic discussed elsewhere in this there is any relationship between sex and feel- book. To understand that logic, let’s return for a ings about equality for men and women. More moment to the concept of sampling error in regard specifically, we’ll see if women are more likely to univariate data. to favor equality than men are, because women would presumably benefit more from it. Take a Recall that a sample statistic normally pro- moment to look at Figure 16-3 and see what the vides the best single estimate of the corresponding answer to this question is. population parameter, but the statistic and the pa- rameter seldom correspond precisely. Thus, we re- The illustration in the figure indicates no rela- port the probability that the parameter falls within tionship between sex and attitudes about equality. a certain range (confidence interval). The degree Exactly half of each group favors equality and half of uncertainty within that range is due to normal opposes it. Recall the earlier discussion of propor- sampling error. The corollary of such a statement tionate reduction of error. In this instance, know- is, of course, that it is improbable that the param- ing a person’s sex would not reduce the “errors” eter would fall outside the specified range only as we’d make in guessing his or her attitude toward a result of sampling error. Thus, if we estimate equality. The table in Figure 16-3 provides a tabu- that a parameter (99.9 percent confidence) lies lar view of what you can observe in the graphic between 45 percent and 55 percent, we say by diagram. implication that it is extremely improbable that the parameter is actually, say, 90 percent if our Figure 16-4 represents the selection of a one- only error of estimation is due to normal sampling. fourth sample from the hypothetical population. In This is the basic logic behind tests of statistical terms of the graphic illustration, a “square” selec- significance. tion from the center of the population provides a representative sample. Notice that our sample con- The Logic of Statistical tains 16 of each type of person: Half are men and Significance half are women; half of each sex favors equality, and the other half opposes it. I think I can illustrate the logic of statistical significance best in a series of diagrams represent- The sample selected in Figure 16-4 would ing the selection of samples from a population. allow us to draw accurate conclusions about the Here are the elements in the logic: relationship between sex and equality in the larger 1. Assumptions regarding the independence of tests of statistical significance  A class of statistical two variables in the population study computations that indicate the likelihood that the relationship observed between variables in a sample can be attributed to sampling error only.

472 ■ Chapter 16: Methods of Statistical Analysis Figure 16-3 A Hypothetical Population of Men and Women Who Either Favor or Oppose Sexual Equality population. Following the sampling logic we saw shows, three-fourths of the women in the sample in Chapter 5, we’d note there was no relationship support equality, but only one-fourth of the men between sex and equality in the sample; thus, we’d do so. If we had selected this sample from a popu- conclude there was similarly no relationship in lation in which the two variables were unrelated to the larger population—because we’ve presumably each other, we’d be sorely misled by our sample. selected a sample in accord with the conventional rules of sampling. As you’ll recall, it’s unlikely that a properly drawn probability sample would ever be as inaccu- Of course, real-life samples are seldom such rate as the one shown in Figure 16-5. In fact, if we perfect reflections of the populations from which actually selected a sample that gave us the results they are drawn. It would not be unusual for us this one does, we’d look for a different explanation. to have selected, say, one or two extra men who Figure 16-6 illustrates the more likely situation. opposed equality and a couple of extra women who favored it—even if there was no relationship Notice that the sample selected in Figure 16-6 between the two variables in the population. Such also shows a strong relationship between sex and minor variations are part and parcel of probability equality. The reason is quite different this time. sampling, as we saw in Chapter 5. We’ve selected a perfectly representative sample, Figure 16-5, however, represents a sample that but we see that there is actually a strong reClaetionng-a g e L e a r n i n g falls far short of the mark in reflecting the larger ship between the two variables in the popBualabtiboniea:tThe Practice of population. Notice that it includes far too many supportive women and opposing men. As the table large. In this latest figure, women are moreSloikceilayl Research, 13/e ttohesuppoppourltateiqouna, laitnydththaensmamenplaerree:flTehcatts’sitt.1h-e13ca3s-e04i9n79-6 Fig. 16-3

Inferential Statistics ■ 473 Figure 16-4 A Representative Sample In practice, of course, we never know what’s simply put, there is a high probability of a small de- so for the total population; that’s why we select gree of unrepresentativeness and a low probability samples. So if we selected a sample and found the of a large degree of unrepresentativeness. strong relationship presented in Figures 16-5 and 16-6, we’d need to decide whether that finding ac- The statistical significance of a relationship curately reflected the population or was simply a observed in a set of sample data, then, is always ex- product of sampling error. pressed in terms of probabilities. “Significant at the .05 level (p  .05)” simply means that the prob- The fundamental logic of tests of ­statistical ability that a relationship as strong as the observed significance, then, is this: Faced with any discrep­ one can be attributed to sampling error alone is ancy between the assumed independence of no more than 5 in 100. Put somewhat differently, variables in a population and the observed distribu­ if two variables are independent of each other in tion of sample elements, we may explain that the population, and if 100 probability samples are d­ is­crepancy in either of two ways: (1) we may selected from that population, no more than 5 attribute it to an unrepresentative sample, or (2) of those samples should provide a relationship as we may reject the assumption of independence. strong as the one that has been observed. The logic and statistics associated with probability sampling methods offer guidance about the vary- There is, then, a corollary to confidenCceeinn­g a g e L e a r n i n g ing probabilities of varying degrees of unrepre- tervals in tests of significance, which repBreasebnbtsie: The Practice of sentativeness (expressed as sampling error). Most the probability of the measured associatiSonoscial Research, 13/e being due only to sampling error. This is called the

474 ■ Chapter 16: Methods of Statistical Analysis Figure 16-5 An Unrepresentative Sample level of significance. Like confidence intervals, frequently used in research reports: .05, .01, and levels of significance are derived from a logical .001. These mean, respectively, that the chances of model in which several samples are drawn from a obtaining the measured association as a result of given population. In the present case, we assume sampling error are 5/100, 1/100, and 1/1,000. that there is no association between the variables in the population, and then we ask what proportion of Researchers who use tests of significance nor- the samples drawn from that population would pro- mally follow one of two patterns. Some specify in duce associations at least as great as those measured advance the level of significance they’ll regard as in the empirical data. Three levels of significance are sufficient. If any measured association is statistically significant at that level, they’ll regard it as repre- level of significance  In the context of tests of sta- senting a genuine association between the two vari- tistical significance, the degree of likelihood that an ables. In other words, they’re willing to discount the observed, empirical relationship could be attribut- possibility of its resulting from sampling error only. able to sampling error. A relationship is significant at the .05 level if the likelihood of its being only Other researchers prefer to report the specific a f­unction of sampling error is no greater than level of significance for each association, disregard- ing the conventions of .05, .01, and .001. RCaethnerg a g e L e a r n i n g than reporting that a given association is sBiganbifibciaen:tThe Practice of

Inferential Statistics ■ 475 Figure 16-6 A Representative Sample from a Population in Which the Variables Are Related the .023 level, indicating the chances of its having sampling error alone. An example will illustrate resulted from sampling error as 23 out of 1,000. this procedure. Chi Square Let’s assume we’re interested in the possible re- lationship between church attendance and sex for Chi square (2) is a frequently used test of the members of a particular church. To test this re- significance in social science. It’s based on the lationship, we select a sample of 100 church mem- null hypothesis: the assumption that there is no bers at random. We find that our sample is made relationship between two variables in the total up of 40 men and 60 women and that 70 percent population (as you may recall from Chapter 3). of our sample say they attended church during the Given the observed distribution of values on the preceding week, whereas the remaining 30 percent two separate variables, we compute the conjoint say they did not. distribution that would be expected if there were no relationship between the two variables. The re- If there is no relationship between sex and sult of this operation is a set of expected f­requencies church attendance, then 70 percent of the men in for all the cells in the contingency table. We then the sample should have attended church during compare this expected distribution with the dis- the preceding week, and 30 percent should have tribution of cases actually found in the sample data, and we determine the probability that the stayed away. Moreover, women should haCveeant-g a g e L e a r n i n g tended in the same proportion. Table 16-6B(apbarbtie: The Practice of I) shows that, based on this model, 28 menSaoncdial Research, 13/e 42 women would have attended church, with