10.7 SUMMARY 336 Exhibit 10.5 (continued) Classif1cation Table CD Predicted Observed Ev~N~ NO EVE~7 Total +---------------------+ 59 EVENT 45 79 NO EVENT 12 67 138 +---------------------+ Total 57 81 Sens1tivity= 76.3% Specificity= 84.S~ Correct= 81.2\\ False Positive Rate= 21.1% False Negative Rate= 17.3% .rdata [Ia]. The same infonnation is given by the value of 44.034 with 5 df and is statistically significant (p = .0001), suggesting that there is a relationship between the independent variables and the log of odds of a mutual fund being attractive [Ib]. All the variables are significant at p S .10 [3]. The parameter estimates suggest that. as expected. the effects of SIZE, TOTRET, and YIEW on the log of odds of a mutual fund being most attractive are positive, and the effects of SCH,4RGE and EXPENRAT are negative. The most influential variable for log of odds is EXPENRAT of the mutual fund. The log of odds of a mutual fund being most attractive, after controlling for the effects of all other measures, decreases by 1.4361 for a unit increase in EXPENRAT. On the other hand, after controlling for the effects of all other variables, the odds of a mutual fund being most attractive change by a factor of only .238 (i.e., e- \\,4361) for a unit increase in £XPENRAT. The model appears to have good predictive validity. The classification rate of 81.2% is substantially greater than the naive prediction rate of 57.2% (i.e., 79/138) [5]. Re- call that the logistic regression procedure uses the jackknifed estimate of classification rate. The nonjackknifed estimate of classification rate is 82.6%. Using Eqs. 8.18 and 8.20, the Z· statistic is equal to 7.076. suggesting that the cIas~ification rate of 81.2% is statistically significant. All the statistics for the association between probabilities and observed responses are also high, once again suggesting that the model has good pre- dictive validicy [4]. 10.7 SUMMARY This chapter provided a discussion of logistic regression, which is an alternative procedure to discriminant analysis. As opposed to discriminant analysis, logistic regression analysis does not make any assumptions regarding the distribution of the independent variables. Therefore, it is preferred to discriminant ~nalysis when the independent variables are a combination of cate- gorical and continuous because in such cases the multivariate normality assumption is clearly violated. Logistic regression analysis is also preferred when all the independent variables are continuous. but are not nonnally distributed. On the other hand. when the multivariate normality assumption is not violated then discriminant analysis is preferred because it is computationally more efficient than logistic regression analysis. In the next chapter we discuss multivariate analysis of variance (MANOVA). MANOVA is a generalization of analysis of variance and is very closely related to discriminant analysis. In fact. much of the output reported by discriminant analysis is also reported by MANOY.-\\.
336 CHAPl'ER 10 LOGISTIC REGRESSION QUESTIONS 10.1 Both logistic regression and multiple regression belong to the category of generalized linear models. In what ways are the fundamental assumptions associated with the two methods different? 10.2 Consider the following models for k independent variables (1) p - a + f3lxl + f32xZ + .. , + f3/iX/i In (1 ~ p) == a + {31XI + {32 X2 + ... + {3l:Xk , (2) where p is the probability of one of two possible outcomes. While the model in (2) is the logistic regression model. the model in (1) is often referred to as the linear proba- bility model. The difference between the two models lies in the transformation of the dependent variable from a simple probability measure to the log of the odds. What are the problems associated with the linear probability model that make this transfonnation attractive (thereby rendering the logistiC regression model a vast im- provement over the linear probability model)? 10.3 Use the data in the contingency Table Q10.1 to answer the questions that follow. Table QI0.1 Heart Disease Blood Cholesterol Present Absent (mg/lOO mJ) 5 30 9 26 < 200 14 20 18 16 200-225 23 8 226- 250 251-275 > 275 (a) What is the probability that (i) Heart disease is present? (ii) Blood cholesterol is less than 200 mg/ 100 ml? (iii) Blood cholesterol is greater than 225 mg,' 100 ml? (iv) Heart disease is absent given that blood cholesterol is between 200 and 225 mg.! 100 ml? (v) Heart disease is present given that blood cholesterol is greater than 250 mg i 100 ml? (vi) Blood cholesterol is greater than 275 mg.' 100 ml given that heart disease is absent? (b) What are the odds that (i) Heart disease is present (as against its being absent)? (ii) Blood cholesterol is Jess than 225 mg': tOO ml (as against its being greater than or equal to 225 mg,' 100 ml)? (iii) Heart disease is present given that blood cholesterol is greater than 275 mg.' 100 ml? (iv) Blood cholesterol is greater than 250 mg.' 100 ml given that heart disease is present? ]0.4 Refer to the admission dara in file ADMIS.DAT. Consider the applicants who were ad- mitted and those who were not admitted. Create a new variable that reflects the admission
QUESTIONS 337 status of the applicants. and let this variable take values of 0 and 1 respectively for admitted and not-admitted applicants. Recode the GPAs of the applicants to reffect a GPA category as follows: GPA GPA Category <2.50 1 2.51 to 3.00 2 3.01 to 3.50 3 4 > 3.50 Perfonn logistic regression on the data using the GPA category as the only independent variable (you will have to use a dummy variable coding to reflect the four GPA cate- gories). Discuss the effect of the GPA category on the admission status and comment on the accuracy of classification. Include the GMAT score as a second independent variable in the model and interpret the solution. Is there any improvement in the accuracy of classification? 10.5 Table QI0.2 presents data on 114 males between the ages of 40 and 65. These subjects were classified on their blood cholesterol level and subsequently on whether they devel- oped heart disease. . Table QI0.2 Heart Blood Cholesterol (mgl100 cc) Disease <200 200-219 220-259 > 259 Present 6 10 30 45 Absent 5 6 57 (a) Using hand calculations. compute the various probabilities and the log of odds. (b) Using the above log of odds compute the logistic regression equation and interpret it (do not use any logistic regression software). (c) Use a canned software (e.g.• SAS or SPSS) to estimate the logistic regression model and compare the results to that obtained above (you will have to use dummy variable coding for the cholesterol levels). (d) What conclusions can you draw from the classification table? Should the classifi- cation table be interpreted? Why or why not? 10.6 Refer to the data in file DEPRES.DAT (refer to file DEPRES.DOC for a description of the data). Using \"CASES\" as the dependent variable analyze the data by logistic regres- sion. Interpret the solution and compare with the results obtained by discriminant anal- ysis. 10.7 Table Q10.3 presents the data from a survey of 2409 individuals on their preferences for a particular branch of a savings institution. Each of the four variables has two possible responses: 1 indicates a positive answer and 2 indicates a negative answer. Using B as the dependenl variable, analyze the data by logistic regression. Interpret the solution and comment on the appropriateness of USing logistic regression as against discriminant analysis.
S38 CHAPTER 10 LOGISTIC REGRESSION Table QIO.S· Previous Recommend Connnient Familiarity Patronage Location (D) (B) (C) (A) 1.0 2.0 1 423 187 2 459 412 2 1 49 47 2 2 68 127 2 1 13 84 2 17 40,.,,7., 0 1 2 3 91 'Variable A is whether the person had previously patronized the branch; Variable B is whether the person strongly recommends the branch; Variable C is the person's opinion whether the branch is conveniently located; and Variable D is whether the person is familiar v·..ith the branch, Levell ofeach va9-abJe reflects a positive answer and level 2 a negative one. Source: Dillon, W.R.. & Goldstein. M. (1984). Multivariate A.nalysi5- Methods and Applications. John Wiley & Sons Inc., New York. 10.8 Table QlOA presents data on the drinking habits of 5468 high school students. Table QIO.4 Student Student Does Drinks Not Drink Both parents drink One parent drinks 398 1401 Neither parent drinks 421 1814 201 1233 Usc logistic regression to analyze and interpret the above data. 10.9 Table Q10.5 is based on records of accidents in 1988 compiled by the Department of Highway Safety and Motor Vehicles in the State of Florida. Table QIO.5 Safet~' Equipment Injury in Use Fatal Nonfatal None 1601 162527 Seat belt 510 412368 Source: Dep:ll1ment of Highway SafelY and Mo- tor Vehic1e~, State of Aorida. (a) Perfonn a contingency table analysis on the above data. (b) Use logistic regression to analyze (he data. (c) Compare the results obtained in (a) and (b).
AIO.I MAXIMUM LIKEUHOOD ESTIMATION 339 10.10 A sample of elderly people is given psychiatric examinations to detennine if symptoms ofsenility are present One explanatory variable is the score on a subtest of the Wechsler Adult Intelligence Scale (WAIS). Table Q1O.6 shows the data. Table QI0.iY XYX YX Y X Y X Y 9 1 7 1 7 0 17 0 13 0 13 1 5 1 16 0 14 0 13 0 6 1 14 1 9 0 19 0 9 0 8 1 13 0 9 0 9 0 15 0 10 1 16 0 11 0 11 0 10 0 4 1 10 0 13 0 14 0 11 0 14 1 12 0 15 0 10 0 12 0 8 1 11 0 13 0 16 0 4 0 11 1 14 0 10 0 10 0 14 0 7 1 15 0 11 0 16 0 20 0 9 1 18 0 6 0 14 0 'X - WAIS Score; Y - Senility, where 1 \"\" symptomS present and o = symptoms absent. Source: Agresti, A. (1990). Categorical Data Analysis. John Wiley & Sons, New York. (a) Analyze the data using logistic regression. Plot the predicted probabilities of senil- ity against the WAIS values. (b) Analyze the data using least-squares regression (i.e.• fit the linear probability model). Plot the predicted probabilities of senility against the WAlS values. (c) Compare the plots in (a) and (b) and discuss the appropriateness of using logistic regression versus simple regression. Appendix Maximum likelihood estimation is the most popular technique for estimating the parameters of the logistic regression model. In the following section we provide a brief discussion of the maximum likelihood estimation technique. AlO.1 MAXIl\\fiJM LIKELmOOD ESTIMATION The dependent variable in the logistic regression model is binary. which takes on two values. Let Y be the random binary variable whose value is zero or one. The probability P(Y = 1) is given by e$X (AW.l) P(Y = I) = p = -1--+-~-=x=· where Pis the vector of coefficients and X is the vector of independent variables. This equation can be rewritten as I nI--Pp- = pX. (A 10.2)
840 CHAPTER 10 LOGISTIC REGRESSION Equation AlD.2 represents the log of odds as a linear function of the independent variables. Unfortunately, the values for the dependent variable (i.e.. the log of odds) are not available, so the parameters of Eq. A lD.2 cannot be estimated directly. However. the likelihood function provides a solution to this problem. Each observation can be considered as a Bernoulli trial. That is. it is a binomial with the total number of trials equal to 1. Consequently. for the ith observation (AlO.3) Assuming that all the n observations are independent the likelihood function is given by f1n L = p;I'(1 - Pi)I-)', 0'-) D )1-'.(I :r. = BeX~X·)1, ( +Ie~x .. tAlD.4) 1 and the log of the likelihood function is given by 1 )n (eBX' L(l-InL = I = ?:Y; 1 + eBX)+ n Y;)(l + e~x . (AlD.5) 1'= J 1I , . . The estimate for the parameter vector Jl is obtained by maximizing Eq. AI0.5. The usual pro- cedure is to take the first-order derivath·es of the equation with respect to each independent variable. setting each equation to zero. and then solving the resulting e9uations. However, the resulting equations do not have an analytical solution. Consequently. Jl is obtained by maxi- mizing Eq. AIO.5 using efficient iterative techniques such as the Newton-Raphson method (see Haberman 1978 for a discussion of the Newton-Raphson method). AIO.2 ILLUSTRATIVE EXAMPLE Consider the example given in Section 10.2. The data are given in Table 10.1. Using Eq. A 10.5. the likelihood function is given by (AlO.6) Let us obtain the estimates of f30 and {31 by maximizing this equation using trial and error. Table Al 0.1 gives the value of the likelihood function for various values of /30 and {31. As can be seen. the likelihood function takes a maximum val ue of - ~ 932 when f30 = -1.7047 and /3( ~ 4.0073. and these estimates are the same as reponed in Exhibit 10.1 f3]. Also, notice that Table AIO.l Values of the Maximum Likelihood Function for Different Values of Po and Pl Po /31 -1.00 -1.70~7 -1.5 -1.0 2 -13.275 -11.996 -11.333 -10.518 3 -10.096 -9.539 -9.334 -9.469 4.0073 -9.044 -8.932 -8.987 -9.610 -9.1857 -9.277 -9.446 5 -10.272
AIO.2 ILLUSTRATIVE EXAMPLE 341 the value of 17.864 for -:- 2LogL reported in Exhibit 10.1 [~cl is twice the maximum value of logL in TabJ.e AIO.I (Le.• 17.864 = -2 x -8.932). Obviously, the trial-and-error method is an inefficient way of identifying values for the pa- rameters that would result in the maximum value for the likelihood function. As mentioned previously. the Newton-Raphson method is an efficient technique and it is the one employed by the logistic regression procedure in SAS.
CHAPTER 11 Multivariate Analysis of Variance Consider the following scenarios: • A marketing manager is interested in determining if geographic region (e.g.. north. south, east, and west) has an effect on consumers' taste preferences. purchase in- tentions, and attitudes toward the product. • A medical researcher is interested in determining whether personality (e.g.. Type A or Type B) has an effect on blood pressure, cholesterol. tension. and stress levels. • A political ana1yst is interested in determining if party affiliation (Democratic. Re- publican, Independent) and gender have any effect on voters' views on a number of issues such as abortion, taxes, economy. gun control, and deficit. For each of these examples we have categorical independent variable(s) with two or more levels, and a set of metric dependent variables. We are interested in deter- mining if the categorical independent variable(s) affect the metric dependent variables. MANOVA (multivariate analysis of variance) can be used to address each of the pre- ceding problems. In MANOVA the independent variables are categorical and the de- pendent variables are continuous. MANOVA is a multivariate extension of ANOVA (ana1ysis of variance) with the only difference being that in MANOVA there are mul- tiple dependent variables. The reader will notice that the objective of MANOVA is very similar to some of the objectives of discriminant analysis. Recall that in discriminant analysis one of the ob- jectives was to determine if the groups are significantly different with respect to a given set of \"'ariables. Although the two techniques are similar in this respect, there are some important differences. In this chapter we discuss the similarities and dissimilarities be- tween MANOVA and discriminant analysis. The statistical tests in MANOVA. and consequently in discriminant analysis. are based on a number of assumptions. These assumptions are discussed in Chapter 12. ILl GEOMETRY OF MANOVA A geometric illustration of MANOVA begins by first considering the case of one in- dependent variable at two levels and one dependent variable. The illustration is then extended to the case of two dependent variables, followed by a discussion of p depen- dent variables. Finally. we discuss the case of more than one independent variable and p dependent variables. 342
11.1 GEOMETRY OF MANOVA 343 11.1.1 One Independent Variable at Two Levels and One Dependent Variable As shown in Figure 11.1, the centroid or mean (i.e., f\\ and 1'2) of each group can be represented as a point in the one-dimensional space. If the independent variable has an effect on the dependent variable, then the means of the two groups are different (i.e., they are far apart) and the effect of the independent variable is measured by the dif- ference between the two means (i.e., the distance between the two points). The extent to which the means of the two groups are different (Le., far apart) can be measured by the euclidean distance between the centroids. However, as discussed in Chapter 3, Mahalanobis distance (MD) is the preferred measure of distance between two points.! The greater the MD between the two centroids, the greater the difference between the two groups with respect to Yand vice versa. Statistical tests are available to determine if the MD between the two centroids is large, i.e.. significant at a given alpha level. Thus, geometrically, MANOVA is concerned with determining whether the MD be- tween group centroids is significantly greater than zero. In the present case, because there are only two groups and one dependent variable, the problem reduces to compar- ing the means of two groups using a (-test. That is, a two-group independent sample t-test is a special case of MANOVA. 11.1.2 One Independent Variable at Two Levels and Two or More Dependent Variables First consider the case where we have only two dependent variables. Since the indepen- dent variable is at two levels, there are two groups. Let Y and Z be the two dependent variables and (YI,.ttl and (Y2•.t1), respectively. be the centroids of the two groups. As shown in Figure 11.2, the centroid of each group can be represented as a point or a Centroid for CentrOid for / Group I / Group 2 ----------e----------e--------------- y: Figure ll.l One dependent variable and one independent variable at two levels. Z / Centroid for Group I (Y1 ZI) / Centroid for Group 2 (Y2~) ~-----------------------Y Figure ll.2 Two dependent variables and one independent variable at two levels. I Recalllhat for uncorrelated variables the Mahalanobis distance reduces to the statistical distance. and if the variances of the variables are equal to one then the statistical distance is equal to the euclidean distance.
344 CHAPTER 11 1'.fiJLTIVARIATE ANALYSIS OF VARIANCE vector in the two-dimensional space defined by the dependent variables. Once again. the MD between the two points measures the distance between the centroids of the two groups. The larger the distance, the greater the difference between the two groups and vice versa Once again, geometrically. MANOVA reduces to computing the distance between the centroids of the two groups and detennining if the distance is statistically significant. In the case of p variables, the centroids of the two groups can be repre- sented as two points in the p-dimensional space and the problem reduces to determining .whether the distance between the two points is different from zero. 11.1.3 More Than One Independent Variable and p Dependent Variables Consider the example described at the beginning of the chapter in which a political an- alyst is interested in detennining the effect of two independent variables, voters' Part)' Affiliation and Gender. on voters' attitude towards a number of issues. In order to illus- trate the problem geometrically. let us assume that two dependent variables. Y and Z. are used to measure voters' attitude towards two issues, say tax increase and gun con- trol. Table 11.1 gives the means of the two dependent variables for the different cells. In the table, the first subscript refers to the level for Gender and the second subscript refers to the levels for Party Affiliation. The dot in the subscript indicates that the mean is computed across all levels of the respective subscript. For example, .211 is the mean for male Democrats and .2.1 is the mean for all Democrats (i.e .. male and female). There are three types of effects: (I) main effect of Gender; (2) main effect of Party Affilia- tion; and (3) the interaction effect of Gender and Party Affiliation. Panels I and II of Figure 11.3 give the geometrical representation of the main effects, and Panels III and IV. respectiveJy. give the geometrical representations in the absence and presence of interaction effects. In Panel!. the main effect of Gender is meal>ured by the distance between the two centroids. Similarly, in Panel II the main effect of Party Affiliation is measured by the distances between pairs of the three centroids. There will be three distances, each representing the distance between pairs of groups. In Panel III, the solid circles give the centroids for Democrats. the open circles give the centroids for Republicans. and the stars give the centroids for Independents. The distance between the solid circles is a measure of Gender effect for Democrats. Similarly, the distances between open circles and stars. respectively, are measures of the Gender effect for Republicans and Inde- pendents. If the effect of Gender is independent of Pal'~\\, Affiliation then. as depicted in Panel III. the vectors joining the respective centroids should be parallel. On the other Table 11.1 Cell Means Party Affiliation Gender Democrats Republicans Independents Mean Male til ZI:! rZIJ ZI. Female YI:! Mean f\" Yn I. Z:!I 2:::: Y11 2'1.' Z;!. t.1 YZ2 f23 f! f.1 t.:! Z.J t .. Y.:; 1\".. r.~
z • (rIo 2'0) Centroid for males y PanclJ z • (r.. i':.) ~troid forDemo.:ralS • (Y.2 Z:2) CCl1lfoid for Republicans • c1.3 i.3) C~lrOid for Independents y P:melll Z Ccnrroids for females P:mcllll y Cenrroids for males Z CcnU'Oids for females -...... I-_\"(Yn Zu) y Figure 11.3 PancllV More than one independent variable and two dependent variables. Panel ofI, Mam effect gender. Panel II, Main effect of party affiliation. Panel III, No Gender X party affiliation interaction effect. Panel Iv, Gender x party affiliation interaction effect.
348 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE hand, if the effect of Gender is not independent of Party Affiliation then, as depicted in Panel N, the vectors joining the respective points will not be parallel. The magnitude of the interaction effect between the two variables is indicated by the extent to which the vectors are nonparallel. The preceding discussion can easily be extended to more than two independent vari- ables and p dependent variables. The centroids will be points in the p-dimensional space. The distances between centroids will give the main effects, and the nonparal- lelism of the vectors joining the appropriate centroids gives the interaction effects. ll2 ANALYTIC COMPUTATIONS FOR TWO-GROUP MANOVA The data set given in Table 8.1. Chapter 8, is employed to illustrate MANOVA. This data set is chosen because it is small and can also be used to show the similarities between MANOVA and two-group discriminant analysis. 11.2.1 Significance Tests The first step is to detemline if the two groups are Significantly different with respect to the variables. That is, are the centroids of the two groups significantly different? This question is answered by conducting multivariate and univariate significance tests, which are discussed in the following pages. Multivariate Significance Tests The null and alternative hypotheses for multivariate statistical Significance testing in MANOVAare: Ho: (Jf.LL2l1!) = (JJLL'll2'!.) Ha: (fJ.L.2L1ll)# (1-L12) (11.1) 1-L22 where JLij is t'1e mean of the ith variable for the jth group. Note that the null hypothe- sis formally states that the difference between the centroids of the two groups is zero. Table 11.2 shows the various MANOVA computations for the data given in Table 8.1. The formulae used to compute the various statistics in Table Il.2 are given in Chapter 3. MD2 between the centroids or means of the two groups is 15.155 and is direct!y propor- tional to the (lifference between the two groups. MD2 can be transformed into various test statistics to determine if it is large enough to claim that the difference between the groups is statistically significant. In the case of two groups. MD2 and Hotelling's T2 are related as (11.2) T'1 can be transformed into an exact F-ratio as follows: =F +(11] III - P - 1) T':.. (11.3) (nj + n2 - p)2 which has an F distribution with p and (n, + n~ - p - 1) degrees of freedom. From Table 11.2, the values for T2 and F-rauo are. respectively. equal to 90.930 and 43.398. The F-ratio is statistically significant at p < .05 and the null hypothesis is rejected, i.e., the means of the two groups are significant]y different.
11.2 ANALYTIC COMPUTATIONS FOR TWO-GROUP MANOVA 34'1 Table 11.2 MANOVA Computations XI ~ (.191 .184) X2 - (.003 .(01) (XI - X2) = (.188 .183) SSCP ~ (.265 .250) SSCP... = (.053 .045) SSCPb ~ (.212 .205) .045 .062 ( .250 .261 .205 .199 S = (.00243 .00203 ) ... .00203 .00280 1. Multivariare Analysis (a) Statistical Significance Tests • MD1 =: (.188 .183)/S~I(.188 .183)' = 15.155. • T2 = (12 x 12)15.155 = 90 930 12 + 12 . • F = (12 + 12 - 2 - 1)90.930 = 43 398 (12+ 12-2)2 . • Eigenvalue of SSCPbSSCP: I = 4.124 • Pillai's Trace \"'\" .805 • Hotelling's Trace = 4.124 • Wilks' A - .195 • Roy's Largest Root = .805 • F-rario = 1 - .195 X 12 + 12 - 2- I = 43.346 .195 2 (b) Effect Size • Partial eta square = .805 2. Univariate Analysis Statistic EBITASS ROTC (a) Statistical Analysis (.191 - .003f- (.184 - .001f- MD'1 .00243 .00280 t = 14.545 11.960 (b) Effect Size l\"'xl? 12 x 12 x 14.545 Partial eta square - 24 - x 11.960 24 = 87.270 = 71.760 9.342 8,471 87.270 71.760 87.270 + 22 71.760 + 22 = .799 = .765 From the above discussion it is clear that in the case of two groups the objective of MANOVA is to obtain the difference (i.e., distance) between groups using a suitable measure (e.g., MD'!.) and assessing whether the difference is statistically significant. Besides MD2, other measures of the difference between the groups are available. In Chapter 8 it was seen that for a univariate case SSb/SSw or SSb X SS;l is one of the
848 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE measures for the difference between two groups and it is related to the t-value. T2, MD2, and the F-ratio. For multiple dependent variables the multivariate analog for the differences between groups is a function of the eigenvalue(s) of the SSCPb X SSCp;::l matrix.2 Some of the measures fonned using the eigenvalues are LK (11.4) Pillai's Trace = 1 Ai A' ''''1' + , LK Hotelling's Trace = Ai i\", 1 n--K 'I Wilks'A = ','''' 1 1 + Ai Roy's Largest Root = 1 Am;x . + max where Ai is the ith eigenvalue and K is the number of eigenvalues. Notice that all the measures differ with respect to how a single index is computed from the eigenvalues. Olson (1974) found that the test statistic based on Pillai' s trace was the most robust and had adequate power to detect true differences under different conditions, and therefore we recommend its use to test multivariate significance. It can be shown that for two groups all of the above measures are equivalent and can be transformed into T\"l or an exactF-ratio.3 For example, from Table 9.4, Chapter 9. the relationship between Wilks' A and F-ratio is (11.5) Table 11.2 gives the '.'alues for all the above statistics and the F-ratio. The F-ratio for all the statistics is significant at p < .05. That is, the two groups are significantly different with respect to the dependent variables. Univariate Significance Tests Having determined that the means of the two groups are significantly different, the next obvious question is: Which variables are responsible for the differences between the two groups? One suggested procedure is to compare the means of each variable for the two groups, That is. conduct a series of (-tests for comparing the means of two groups. Table 11.2 also gives MD2, T2, t-value. and the F-ratio for each variable.4 It can be seen that both EBITASS and ROTC significantly contribute to the differences between the two groups. 11.2.2 Effect Size The statistical significance tests determine whether the differences in the means of the groups are statistically significant. Once again, for large sample sizes even small dif- IThe number of eigen\\'a1ue$ is equal to min(G - I, p) where G andp. respectively. are the number ofgroups and number of dependent variables, ~ As shown laler in the chapter. the various measures are not tP,e same for more than two groups and can only be approximalely transformed into the F-ratio. \"The ,-\\'alue is equal to ,':T! and for two groups the F-ratio is equal to T~.
11.2 AI.'I'ALYTIC COMPUTATIONS FOR TWO-GROUP MANOVA 349 ferences are statistically significant. Consequently, one would like to measure the dif- ferences between the groups and then decide if they are large enough to be practically meaningful. That is, one would also like to assess the practical significance of the differ- ences between the groups. Effect sizes can be used for such purposes. The effect size of any given independent variable or factor is the extent to which it affects the dependent variable{s). Univariate effect sizes are for the respective dependent variables, w~ereas multivariate effect sizes are for all the dependent variables combined. Discussion of univariate and multivariate effect sizes follows. Univariate Effect Size A number of related measures of effect size can be used. One common measure is MD2 / which is related to T2 and the F-ratio. A second and more popular measure of effect size is the partial eta square. which is equal to SSb/SSt. The advantage of using partial eta square (PES) is that it ranges between zero and one, and it gives the proportion of the total variance that is accounted for by the differences between the two groups. In Chapter 8, it was seen that for the univariate case SSb A = SSw' Or, r; (- 5 = 1 - A = 1 _ SS..... = SSb (11.6) SSt SSt' which is equal to PES. Since A can be transformed into an F-ratio, PES is also equal to Fxdfb (11. 7) F X dfb +dfw where dfb and dfw are, respectively, between-groups and within-group degrees of free- dom. Using Eq. 11.7 and infonnation from Table 11.2, the PESs for EBITASS and ROTC, respectively, are equal to .799 and .765. High values for the PESs suggest that a substantial proportion of the variance in the dependent variables is accounted for by the differences between the groups. Multivariate Effect Size The multivariate effect size is given by the difference between the centroids of the two groups. As discussed earlier, MD2 measures the distance between the two groups and hence it can be used as a measure for the muIcivariate effect size. The larger the distance. the greater the effect size. The most popular measure of effect size, however, is once again PES and it gives the amount of variance in all the dependent variables that is accounted for by group differences. PES can be computed using Eq. 11.6 orEq. 11.7. Using Eq. 11.6, the value of PES is equal to .805. This high value for PES suggests that a large proportion of the variance in EBITASS and ROTC is accounted for by the differences between the two groups. That is, the differences between the groups with respect to the dependent variables are meaningful. 11.2.3 Power The power of a test is its ability to correcdy reject the null hypothesis when it is false. That is, it is the probability of making a correct decision. The power of a test is directly
350 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE proportional to sample size and effect size, and inversely related to p-,·alue. Power of the test can be obtained from power tables using effect size, p-value. and the sample size. The use of power tables is not illustrated as it requires a number of power tables that are not available in standard textbooks. Furthermore, the power of tests can be requested as ~,part of the MANOVA output in SPSS, or one can use SOLO Power Analysis (1992), which is a software package for computing the power of a number of statistical tests. The interested reader:is referred to Cohen (1977) for further details on computation and '\"\"use of power tables. 11.2.4 Similarities between MANOVA and Discriminant Analysis One of the objectives of discriminant analysis is to identify a linear combination (called the discriminant function) of the variables that would give the maximum separation between the two groups. Next. a statistical test is performed to determine if the groups are significantly different with respect to the linear combination (discriminant scores). A significant difference between the groups with respect to the linear combination is equivalent to testing that the two groups are different with respect to the variables form- ing the linear combination. In MANOVA, we test whether the centroids ofthe two groups are significantly differ- ent. Although a linear combination, which provides the maximum separation between the two groups, is not computed in MANOVA. the multivariate significance tests dis- cussed earlier implicitly test whether the mean scores of the two groups obtained from such a linear combination are significantly different. Note that the null and the alterna- tive hypotheses given in Eq. 11.1 are the same as those gi ven in Section 8.3.2 of Chap- ter 8. Also the F-ratios for the univariate analysis reported in Table 11.2 are. within rounding errors, the same as those reported in Exhibit 8.1, Chapter 8 [4]. From the preceding discussion it is clear that in the case of one independent variable there is no difference between MANOVA and discriminant analysis. In the case of more than one independent variable, however. MANOVA provides additional insights into the effects of independent variables on dependent variables that are not provided by dis- criminant analysis. Further discussion ofthe additional insights provided by MANOVA are provided in Sections 11.4 and 11.5. In the following sections we discuss the resulting output from the MANOVA proce- dure in SPSS. We will first discuss the output for the data given in Table 8.1, which gives the financial ratios for most-admired and least-admired firms. MANOVA that assesses the differences between two groups is sometimes referred to as two-group MANOVA. Next we discuss the output for multiple groups. MANOVA for assessing the differences between three or more groups is referred to as multiple-group MANOVA. Finally, we discuss the use of MANOVA to assess the effect of two or more independent variables. !L3 TWO-GROUP MANOVA Table 11.3 gives the SPSS commands for the data set given in Table 8.1. The PRIl\\YT subcommand specifies the options for the desired output. The CELLINFO option requests group means and standard deviations: the HOMOGENEITY option requests printing of Box's M statistic for testing the equality of covariance matrices; the ERROR option requests that the SSCP matrices be printed: the SIGNIF option requests print- ing of multivariate and univariate significance tests. the hypothesized sum of squares, effect sizes. and the eigenvalues of the SSCPb x SSCP:' 1 matrix. The PO\\VER sub- command requests that power for the F-test and the (-test be reported for the specified
11.3 TWO-GROUP MA..'IolOVA 351 Table 11.3 SPSS Commands MANOVA E8ITASS, ROTC 8Y EXCEL~(1,2) /PRI~T=CELLINFO(MEANS) HOMOGENEITY (80XN) ERROR (SSCP) SIGNIF(MULTIV UNIV HYPOTH EFSIZE EIGEN) /POWER F(.OS) T(.OS) /DESIGN EXCELL FINISH p-values. The DESIGN subcommand specifies the effects (i.e., main and interaction effects) that the researcher is interested in testing. Since there is only one factor, only one main effect is specified. Exhibit 11.1 gives the output. Discussion of the output is brief because many of the reported statistics have been discussed in the previous section. The reader should compare the reported statistics in the output to the computed statistics in the previous section. 11.3.1 Cell Means and Homogeneity of Variances The means and standard deviation for the dependent variables are reported for each cell or group [1]. One of the assumptions of MANOVA is that the covariance matrices for the two groups are the same. Note that this assumption is the same as that made in discriminant analysis. The test statistic used is Box's M, which can be approximately transformed to an F-ratio. The F-ratio is significant at p < .05, suggesting that the co- variance matrices for the two groups are different [2]. Note that this test is also reported by the discriminant analysis procedure and the value reported here is the same as that reported in Exhibit 8.1, Chapter 8 [13]. As discussed in Chapter 12, the effect of viola- tion of this assumption is not appreciable because the two groups are equal in size. 11.3.2 Multivariate Significance Tests and Power In the present case there is on]y one main effect that pertains to the effect of EXCELL (finn's excellence) on finn's perfonnance as measured by the two variables. EBITASS and ROTC. A significant main effect implies that the two groups of firms (i.e., most- and least-admired) are significantly different with respect to these variables. All the test statistics indicate a statistically significant effect for EXCELL at p < .05 [4a]. Note that values of Wilks' A. eigenvalue. and canonical correlation [4a, ·k] are the same as those reported in Exhibit 8.1, Chapter 8 [8]. From Eq. 11.7, the PES is equal to PES = 43.301 x 2 805 =. , (43.301 x 2) + 21 and is the same as the effect size reported in the output [4b], and in Table' 11.2. The power for the multivariate tests is large, suggesting that the probability of rejecting the null hypothesis when it is false is very high [4b]. The output also reports the value of the noncentrality parameter [4b]. The noncentrality parameter is closely related to the effect size and the corresponding test statistic. For any given test statistic, the value of the noncentrality parameter is zero if the null hypothesis is true. For example, the noncentrality parameter for T2 is zero if the null hypothesis is true (i.e., the effect size is zero). As the effect size increases. the probability of rejecting the null also increases and so does the value of the noncentrality parameter. In a two-group case the value of
352 CHAPI'ER 11 MULTIVARIATE ANALYSIS OF VARIANCE Exhibit 11.1 MANOVA for most-admired and least-admired firms 0 ) CELL NUMBER 12 Variable EXCELL 12 Cell Means and Standard Deviations Variable EBITASS Variable ROTC FACTOR CODE Mean Std. Oev. N FACTOR CODE Mean Std. Dev. -N .191 .053 12 EXCELL 1 .003 .045 12 EXCELL .1., .183 .030 \"... L .097 .107 24 EXCELL EY.CELL 2 \"- .001 .069 12 Fer entire sample For ent~re salnple .092 .10';' :~ ~ultivariate test for Homogeneity of Dispersion matrices Boxs M = 21.50395 F WITH (3,87120) DF = 6.46365, P = .000 (Approx.) = =Ch~-Square w~th 3 DF 19.38614, P .000 (Approx.) ~ITHIN CELL: Sum-of-Squares and Cross-Products EBITASS ROTC EBITASS .053 ROTC .045 .062 ~.djEUFFsEtCTed.. EXCELL sis Sum-of-Squares and Cross-Products Hypothe ESITASS EBITASS ROTC ROTC .212 .199 .206 ~u1tivariate Tests of S~gnificance (S = 1, M = 0, N = 9 1/2) Test Name Value Exact F Hypoth. DF Error DF Sig. of F 21. 00 .000 Pilla~s .80484 43.30142 2.00 21. 00 .000 21. 00 .000 Eotellings 4.12394 43.30142 2.00 WJ.lks .19516 43.30142 2.00 Rays .80484 Note .. F statistics are exact. ~u1tJ.variate Effect Size and Observed Power at .0500 Level TEST NMot.E Effect Size Noncent . Power . 805 (All) 86.603 1. 00 @igenvalues and Canonical Corre1atlons Root No. Eigenvalue Pct. Cum. Pct. Canon Cor. 1 4.124 100.000 100.000 .897 @::-.ivariate F-tests with (1,22) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F EBITASS .212:)6 .00243 ROTC .19929 .05338 .21206 .De280 87.40757 .OOC .0Oc .06169 .19929 n. 06986 Variable ETA Square Noncent.. Power EBITASS .79692 87.40757 1. 00000 ROTC .76362 71.06966 1.00000
11.3 TWO-GROUP MAl\"iOVA 353 the noncentrality parameter for all the multivariate test statistics is approximately equal to the T2 statistic.S 11.3.3 Univariate Significance Tests and Power The hypothesized and error sums of squares for univariate tests reported in the output are? respectively, the between-groups and within-group sums of squares [5J. The sums of squares are taken from the diagonals of the respective SSCP matrices [3a. 3b]. The univariate F-ratios for both variables are significant at p < .05, implying that both vari- ables contribute to the differences between the groups [5]. Values for PES are the same as reported in Table 11.2 and are quite high, once again suggesting that a high proportion of the variance in the dependent variables is accounted for by the differences between the groups [5]. Note that the univariate F-ratios are exactly the same as those reported in Exhibit 8.1, Chapter 8 [4]. The power for the univariate tests pertaining to each in- dependent variable is very high. In the univariate case the value of the noncentrality parameter is equal to the respective F-ratio. 11.3.4 Multivariate and Univariate Significance Tests A multivariate test was first performed to determine if the centroids or the mean vectors of the two groups are significantly different, then a univariate test was done to deter- mine which variables contribute to the difference between the two groups. One might wonder why a multivariate test was necessary when it was followed by a univariate test. There are two important reasons for first conducting a multivariate test. First, if all the univariate tests are independent then the overall Type I error will be much higher than the chosen alpha. For example, if five independent univariate tests, each using an alpha level of .05, are performed then the probability that at least one of them is statistically significant due to chance alone will be equal to .226 [i.e., 1 - (1 - .05)5]. That is, the overall Type I error is not .05 but is .226. The actual Type I error will be larger if the tests are not independent. Second, as shown below, it is possible that the multivariate test is significant even though none of the univariate tests are significant. Consider the data set given in Table 11.4 and plotted in Figure 11.4. Exhibit 11.2 gives the partial SPSS output. The univariate tests indicate that none of the means are Table 11.4 Hypothetical Data to illustrate the Presence of Multivariate Significance in the Absence of Univariate Significance Mean Group 1 Group 2 Xl Xl Xl X2 13 45 25 55 47 56 6 11 87 6 12 89 3.80 6.00 7.60 6.40 5The formula for computing the noncentrality parameter for a two-group case is equal to [en - 3):' (n - 2)]T2.
354 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARLI\\NCE X2 12 • • 10 0 8 0 • • Group 1 o Group2 60 • 00 4 • 2 Figure 1l.4 Presence of multivariate significance in the absence ofunivariate signif- icance. Exhibit 11.2 Multivariate significance, but no uni\\'ariate significance ~ITHIN CELLS Sum-of-Squares and Cross-Products Y1 Y2 Yl 34.BOO Y2 45.600 70.400 ~u1tivariate Tests of Signi!icance (S = 1, M ~ 0, N = 2 2/2) Test Name Value Exact F Hypo~h. DF Errcr DF Sig. of F Pillais .B0993 14.91429 2.00 :.00 .003 Hotellings 4.26123 14.91429 2.00 7.00 .003 Wilks .19001 14.91429 2.00 :.00 .003 Roys .B0993 Note .. F statistics are exact. ~nivariate F-tests with (I,B) D. F. Variable Hypoth. 55 Error SS Hypoth. MS Error ~5 F Sig. of F Yl 12.10000 4.35000 Y2 3.60000 34.80000 12.10000 2.78161 .134 S.8~OOO 70.40000 3.60000 .40909 .540 significantly different for the two groups at p < .05 [3J. However. (he multivariate test is significant at p < .05 [2]! Examination of Figure 11.4 gives a clue as to why this is the case. Note that there is not much separation between the two groups with respect to each variable: however. in the two-dimensional space the two groups are separated. Further insights regarding the presence of multivariate significance and absence of univariate significance can be gained by examining the pooled within-group SSCPK' matrix. which is equal to [1J SSCP = (34.800 45.600) ... 45.600 70.400' The error term. MSH • for the multivariate test in MANOVA is given by the determinant of tbe SSCPK• matrix. For the above matrix MS... = iSSCP\" I = 370.560. Now. if the
11.4 MULTIPLE-GROUP MANOVA 355 variables were not correlated and the difference between the means of the two groups were the same, then SSCPK' would be equal to ( 34.800 0) 70.400 o and MS\", = 2449.92, which is almost 6.6 times larger than if the variables were corre- lated. That is, the computation of the MSw for multivariate tests takes into account the correlation among the variables. In other words, the multivariate tests take into account the correlation among the variables, whereas univariate tests ignore this information in the data. U4 MULTIPLE-GROUP MANOVA Suppose a medical researcher hypothesizes that a treatment consisting of the simul- taneous administration of two drugs is more effective than a treatment consisting of the administration of only one of the drugs. A study is designed in which 20 subjects are randomly divided into four groups of five subjects each. Subjects in the first group are given a placebo, subjects in the second group are given a combination of the two drugs, subjects in the third group are given only one of the two drugs, and subjects in the fourth group are given the other drug. The effectiveness of the drugs (i.e., treatment elTectiveness) is measured by two response variables. Y1 and Y2. Table 11.5 gives the data and the group means. Note that this study manipulates one factor, labeled DRUG, and it has four levels. A one-factor study with more than two levels is often referred to as multiple-group MANOVA. Table 11.6 gives the SPSS commands and Exhibit 11.3 gives the partial output. Table 11.5 Data for Drug Effectiveness Study Treatments Means 1 2 3 4 Y1 Y2 Y1 Y2 Y1 Y2 Y1 Y2 12 89 2 -\\. -\\. 5 21 98 32 33 32 79 33 34 23 89 35 56 22 8 10 -\\. 6 57 22 89 34 45 Table 11.6 SPSS Commands for Drug Study MANOVA Yl Y2 BY DRUG(l,4) /PRINT=CELLr~TO(MEANS) HOMOGENEITY (BOXM) ERROR(SSCP,COV) SIGNIF(MULTIV,UNIV} /DESIGN=DRUG
356 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE Exhibit 11.3 MANOVA for drug study ~ultivaria~e test for Homogeneity of Dispersion matrices Bcxs M = 13.09980 =1.12256, P F WITH (9,2933) OF = =10.14325, P .343 (Approx.) =Chi-Square with 9 OF .339 (Approx.) ~ulEtFiFvEC~Tri.a• ~DeRUGTests of Significance (S ~ 2, M = 0, N = 6 1/2) Test Name Value Approx. F Hypoth. OF Error DF Sig. of F Pillais 1. 02715 5.~3102 6.0(' 32.00 .000 2B.OO .000 Hotellings 11. 41361 26.63176 6.00 30.00 .000 Wilks .07253 13.56607 6.00 Rays .91865 Note .. F statistic for WILKS' Lambda is exact. ~nivariate F-tests ~ith (3,16) D. F. Variable Hypoth. 55 Error SS Hypoth. MS Error MS F Sig. of F Yl 103.75000 10.00000 34.58333 .62500 130.00000 24.00000 43.33333 55.33333 .000 Y2 1.50000 28.88889 .000 11.4.1 Multivariate and Univariate Effects As Box's M statistic is not significant at p < .05, one fails to reject the null hypoth- esis suggesting that the covariance matrices of the groups are not different [1]. The multivariate effect of DRUG is significant at p < .05 [2a]. That is, the mean vectors of the four groups are significantly different. The univariate tests indicate that the four groups significantly differ with respect to both the dependent variables [2b]. Based on these results, the researcher can conclude that treatment groups are different with re- spect to their effectiveness. But. which pairs of groups or combinations of groups are different? For example, if the researcher wants to determine whether the effectiveness of the two drugs is different or the same then he/she would test whether groups 3 and 4 are different with respect to their effectiveness. Or, if the researcher is interested in determining whether the simultaneous administration of the drugs is different from the administration of only one drug then the approach would be to test for differences be· tween the effectivt·11ess of group 2 :md the average effectiveness of groups 3 and 4. Testing for differences between specific groups or combinations of groups is referred to as comparison or contrast testing. Notice that testing which groups are different with respect to a given set of variables was not an explicit objective of discriminant analy- sis.ln this respect. therefore, the objectives of discriminant analysis and MANOVA are different. 11.4.2 Orthogonal Contrasts Statistical significance testing of comparisons can be assessed by first fonning contrasts and then testing for their significance. A contrast is a linear combination of the group means of a given factor. It should be noted that it is a good statistical practice to perform contrast analysis detennined or stated a priori, rather than test all possible contrasts in search of significant effects~ For instance, let us assume that the researcher is interested in answering the following questions penaining to the study:
11.4 MULTIPLE-GROUP MA..'l'OVA 357 1. Is the effectiven~ss of the placebo (i.e., the first treatment or control group) different from the average effectiveness of the drugs given to the other three groups? A statistically significant difference would suggest that the drugs are effective either individually or when administered simultaneously. 2. Is the effectiveness of the two drugs administered to the second treatment group significantly different from the average effectiveness of the drugs administered to treatment groups 3 and 4? A significant difference would suggest that the effective- ness ofsimulraneously administering both drugs is different from the effectiveness of administering only one drug at a time. 3. Is the effectiveness of the drug given to the third treatment group significantly different from the effectiveness of the drug given to the fourth treatment group? A significant difference would suggest that the two drugs differ in their effectiveness. Each of these questions or hypotheses can be answered by forming a contrast and test- ing for its significance. In univariate significance tests, each contrast is tested separately for each dependent variable, whereas in multivariate significance tests each contrast is tested simultaneously for all the dependent variables. In order to simplify the discus- sion, we first discuss univariate significance tests, then multivariate significance tests. However, note that univariate contrasts should only be interpreted if the corresponding multivariate contrast is significant. Univariate Significance Tests for the Contrasts Consider the following linear combination: Cij = Cil /-Llj + C;'2/-L'J.j + ... + Cik/-Lkj where Cij is the ith contrast for the jth variable, CiA: are the coefficients of the con- trast, and /-Lkj is the mean of the kth group for the jth variable. Contrasts are said to be orthogonal if >.G for all i (11. 8) Cik = 0 k=1 and for all i =P I (11.9) where i and I are any two contrasts. For equal sample size the preceding equation re- duces to >G for all i ¥- I. (11.10) CikClk = 0 ~ k=1 From Eqs. 11.8 to 1L 10, two contrasts are orthogonal if the sum of the coefficients of each contrast is equal to zero, and the sum of the product of the corresponding coefficients of the two contrasts is also equal to zero. If these conditions do not hold, then the contrasts are correlated. In general, orthogonal contrasts are desired; however. the researcher is not constrained to only orthogonal contrasts. Correlated contrasts are discussed later. The total number of contrasts for any given factor or effect is equal to its degrees of freedom. However, there can be infinite sets of contrasts with each set consisting of the maximum number of allowable contrasts. For the present study, there
358 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE will be a maximum of three contrasts in each set: however, there can be infinite sets of contrasts, with each set consisting of three contrasts. Following is an example of the set of contrasts (for the dependent variable Y1) that would address the research questions posed earlier: = 111 /-1-21 + /-1-31 + /-1-41 ell f.Lll - 3/-1-21 - 3/-1-31 - 3/-1-41 = J.Lll - 3 (11.11) 2\" 2' =1 1 +/-1-31 J.L41 (11.12) /-1-21 - 2 e 2l = /-1-21 - f.131 - f.L41 e31 = /-1-31 - /-1-41, (11.13) The coefficients of this set of contrasts are presenred as Set 1 in Table 11.7. It can be readily checked that the three contrasts in Set 1 are orthogonal as the sum of the coeffi- cients of each contrast is equal to zero and the sum of the product of the corresponding coefficients of each pair of contrasts is also equal to zero. The table also presents three other sets of orthogonal contrasts. The specific set ofcontrasts that the researcher would test will depend on the questions that need to be addressed by the study. The null and the alternative hypotheses for testing the significance of univariate con- trasts are Ho : Cij = 0 Ha : eij -Iz. O. For example. the null and alternative hypotheses for C11 (i.e.. the first contrast for Y1) given by Eq. 11.11 are: Ho: Cll = 0 Ha : Cll :;1= O. which can be rewriuen as +H /-1-31 + /-1-41 a 3 H' - /-1-21 + /-L331 -+- /-1-41 . -4. /-1-21 o . /-1-11 - . /-1-11 -r- From these hypotheses, it is clear that the contrast essentially tests whether or not two means are significantly different. where each mean could be a weighted average of two or more means. Table 11.7 Coefficients for the Contrasts Groups Contrast 1 23 4 Set 1 Set 2 1 -1'3 -1/3 -1.'3 Set3 0 -1/2 -1/2 Set4 0 1 1 -1 -1/3 0 -1.2 -1.'3 -1/3 1 -1; 2 J 0 -1 0 0 -1 1 0 0 -1 -1 0 1 -1·'2 0 1' ') 1 .2 -} '2 I I- 0 0 -1 1 1 -] 0 -1/2 -1.;2 0 1'2 1/2
11.4 MULTIPLE--GROUP MA.'.IOVA 359 It can be shown that the standard error of contrast Cij is equal to (11.1~) where MSEj is the mean square error for the jth variable and its estimate is given by its pooled within-group variance. The resulting t-value is Jt = C/o} , (11.15) MSEj ')Gt .. l C'it;'.' nt or which can be rewritten as (11.16) In the univariate case, T2 is equal to the F-ratio, which has an F distribution with 1 and n - G degrees of freedom. Equation 1106 can also be written as (,GCi2j.'! 1 C~i'tlI ) L..k= (11.17) nk, F= MSE. 0 ) Note that in Eq. 11.16 the tenn CijMSEj1Cij is equal to MD2 and consequently T2, and the F-ratio is proportional to A1D2 or the sl:atistical distance between the two means. That is, once again the problem reduces to obtaining the distance between two means and determining if this distance is statistically significant. In Eq. 11.17. the nu- merator is the hypothesis mean square and. since the hypothesis to be tested has one degree of freedom, it is also the hypothesis sum of squares. and the denominator is the error mean square. Multivariate Significance Test for the Contrasts Multivariate contrasts are used to simultaneously test for the effects of all the dependent variables. A multivariate contrast is given by Ci = CilJ.Ll + ci2fJ.2 + ... + CijJ.Lk where ~is a vector of means for the kth group and Ci is the ith contrast vector. The nu~d alternative hypotheses for the multivariate significance test for the ith contrast are \\ \\ \\ Ha : Cj :r/= O. The test statistic, T2, is the multivariate analog ofEq. 11.6 and is given by (11.18)
380 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE where t~l is the inverse of the pooled within-group covariance matrix. T2 can be trans- formed into an F-ratio using F = (dft' - P + 1)T: (11.19) dft' xp ~~ has an F distribution withp and dJe + P - 1 degrees of freedom, where dfe is the error;degrees of freedom. Estimation and Significance Testing ofContrasts Using SPSS The MANOVA program can be used to estimate and conduct significance tests for a number of different types of contrasts. Table 11.8 gives the SPSS commands for form- ing and testing the set of contrasts given by Eqs. 11.11 to 11.13. The CONTRAST subcommand specifies that contrasts for the DRUG factor are to be formed and tested. Following the equals sign. the desired set of contrasts is specified. The set of contrasts given by Eqs. 11.11 to 11.13 are called Helmert contrasts. MANOVA automatically generates the co·efficients for Helmert and a number of other commonly used con- trasts (refer to the SPSS manual for other commonly used contrasts). The keyword HELMERT requests the forming and testing of Helmen contrasts. In Helmert contrasts, the first contrast tests for the statistical significance of the mean of the first group \\\\-ith the average ofthe means of the remaining groups. the second contrast tests for the signif- icance of the mean of the second group with the average of the means of the remaining groups. and so on. SPSS gives the user the option of specifying contrast coefficients using the SPECIAL keyword. il1ustrated later in this chapter. The PARAMETER op- tion requests the printing of the contrast estimates and their significance tests. In the DESIGN ~ubcommand, DRUG(i) refers to the ith contrast. For example. DRUG(l) is the first conu'ast (i.e., C1). DRUG(:!) is the second contrast (i.e.. C:!). and so on. Exhibit 11.4 gives the partial output. UNJv..I\\..RIATE SIGNIFICANCE TESTS. Consider the contrast given by Eq. 11.11, which is the first contrast and is represented in the output as Drog( 1). Substituting the means reported in Table 11.5, we get C 11 = 2 - 8 + ~ + 4 = - 3. 000 and _,>_9+4+5 __ C 12 - - 3 - 4.000. which are the same as those reponed in Exhibit 11.4 [4a. Sa]. From the output the MSt' for Yl and Y:. respectively, are equal to 0.625 and 1.500 [3d]. Using Eq. 11.14, the Table 11.8 SPSS Commands for Helmert Contrasts Y2 EY CEt;::;(:,4) \" (:V!~:?_:..s,:, (DP.UG) =E::~!~EF~: /??~:!\\'!~.s::;!~!F (~!U~::·,,·, :}l::':)
11.4 MlJLTIPLE-GROUP M&'lJOVA 361 Exhibit 11.4 Helmert contrasts for drug study ~FFECT ., DRUG(3) Multivariate Tests af Significance (5 = 1, M = 0, N = 6 1/2) Test Name Value Exact F Hypath. ...F Errar DF 5ig. of F 15.00 .:75 PHlais .20747 1. 96335 2.:)0 15.00 .j,75 15.00 .175 Hotellings .26178 1.96335 2.00 Wilks .79253 1. 96335 2.00 Roys .20747 Note .. F statistics are exact. ~nivariate F-tests with (1,16) D. F. - -var~ble Hypoth. S5 Errar 55 Hypath. MS Error M5 F Sig . .\"If F Yl 2.50000 10.00000 2.50000 .62500 4.00000 .063 1. SOOCO Y2 2.50000 24.00000 2.50000 1. 66667 .215 ~FFECT .. DRUG(2) =Multivariate Tests af Significance (S = 1, ~ = 0, N 6 1/2) Test Name Value Exact F Hypath. DF Error: DF 5ig. of F 15.00 .000 Pillais .87605 53.01047 2.00 15.00 .000 15.00 .000 Ho~ellings 7.06806 53.01047 2.00 Wilks .12395 53.01047 2.00 Rays .87605 Note .. F statistics are exact. ~nivariate F-tests w1th (1,16) D. F. Variable Hypath. 55 Error SS Hypath. M5 Error MS 51g. of F 67.50000 10.00000 67.50000 .62500 108.00000 .000 Yl 67.50000 24.00000 67.50000 .000 1.50000 45.00000 y2 ~FFECT .. DRUG (1) Multivariate Tests af Significance (S = I, M = 0, ~ = 6 1/2) Test Name Value Exact F Bypoth. DF Error DF 51g. of F 15.00 .000 Pillais .80330 30.62827 2.0:J 15.00 .000 15.00 .000 Hatellings 4.08377 30.62827 2.00 Wilks .19670 30.62827 2.00 Rays .80330 Note .. F statistics are exact. ~nivariare F-tests virh (1.16, D. F. ~ ® Variable Bypoth. 55 Error S5 Hypoth. MS Error M5 F 5ig. of F Y1 33.75000 10.00000 33.75000 .62500 Y2 60.00000 24.00000 60.00000 54.00000 .000 1.50000 40.00000 .000 (continued)
882 CHAPTER i1 MULTIVARIATE ANALYSIS OF VARIANCE Exhibit llA (continued) ~stimates for Y1 --- IndividUal univariate .9500 confidence intervals Parameter @® @ Sig. t Lower -95% CL-Upper Coeff. Std. Err. t-Value DRUG (1) -3.0000000 .40825 -7.34847 .00000 -3.86545 -2.13455 2 4.50000000 .43301 10.39230 5.41795 -1.0000000 .50000 -2.00000 .00000 3.58205 .05995 DRUG (2) 3 .06277 -2.05995 CL-Upper DRUG (3) 4 0stima~es for Y2 --- Individual univariate .9500 confidence intervals Parameter @® @ Sig. t Lower -95% Coeff. Std. Err. t-Value DRUG (1) -4.0000000 .63246 -6.32456 .00001 -5.34075 -2.65925 2 4.50000000 -1.0000000 .67082 6.70820 .00001 3.07792 5.92208 DRUG (2) 3 .77460 -1. 29099 .21505 -2.64207 .64207 DRUG (3) 4 standard error fa ell is I (' O..6..\".5 5-1 + 9-1 x 5-1 + 9-1 x 5-1 + 9-1 x 5-1) ==·408, and the standard error for C12 is 1.500 (5-1 + 9-1 X 5-1 +9-1 X5-1 +9-1 X5-1) \"== 632 which is also the same as reported in the output [4b, 5b]. The (-values reported in the output for CIl and C12. respectively, are -7.352 (-3/.408) and -6.329 (-4/.632) [4c, 5c]. Contrasts ell and Cn are statistically significant at p < .05, from which the researcher can conclude that the drugs are effective when administered either individ- ually or when administered simultaneously. Univariate significance tests for the contrasts are also reported in another section of the output. The reported test statistics are computed using Eq. 11.17. Once again, consider contrast ell. The numerator of Eq. 11.17 is equal to -32 1 1 1 1 1 1 1)=33.750 (5- +9- x5- +9- x5- +9- x5- and is equal to tlte reported hypothesis mean square [3cJ. and the denominator is equal to 0.625. which is the error mean square [3dJ. The F-ratio for the contrast is therefore equal to 54.000 (i.e., 33.750/ '.625) [3e], The only difference between the significance tests reported in [4] and [3b] is that in the fonner part of the output. the contrast estimates are also reported
11.4 MULTIPLE-GROUP MANOVA 363 The univariate contrasts for DRUG(3) (cf. Eq. 11.13) are not significant at p = .05, which leads to the conclusion that the effectiveness of the two drugs is the same [lb, 4. 5J. The significant univariate contrasts for DRUG(2) (cf. Eq. 11.12) suggest that simultaneously administering both drugs is more effective than administering only one drug at a time [2b, 4, 5]. The final conclusion is that the two drugs are equally effective, but the effectiveness of a treatment consisting of administering both drugs simultaneously is much greater than administering each drug separately. MULTIVARIATE SIGNIFICANCE TESTS. The multivariate estimate for the contrast, DRUG(I), is given by C1= (22) - ~ (8 9) - ~ (34) - ~ (4 5) =(-3-4), which can be easily obtained from the corresponding univariate estimates for the con- trasts [4a. 5a]. The T2 for the above contrast is (see Eq. 11.18) (! !r2 = + .!. x + .!. x ! + ! X!)-l (-3 - 4):£-1(-3 - 4)' r 4)'= 3.750(-3 - 4)(:~~ i~3~ (-3 - = 65.318, 5959595 W and the corresponding F-ratio reported in the output is (see Eq. 11.19): C61~ ~; 1)65.318 = 30.618, with 2 and 15 degrees of freedom, and is significant at p < .05 [3aJ. All the multivariate significance tests indicate that at an alpha level of .05, contrasts Cl and C2 are signif- icant [3a, 2a} and contrast C3 is not significant [tal. The overall conclusion reached is the same as discussed in the previous section. Once again, note that univariate analysis of contrasts should only be done if the corresponding multivariate significance tests are significant. Correlated Contrasts Suppose the researcher is interested in comparing the mean effectiveness of each treat- ment group with that of the placebo or control group (Le., group 1). Table 11.9 gives the coefficients for the set of conrrasts that would achieve that Objective. Note that the contrasts are not orthogonal because the sum of the product of the corresponding co- efficients for any pair of contrasts is not equal to zero. Table 11.10 gives the SPSS commands for testing the significance of the contrasts. Table 11.9 Coefficients for Correlated Contrasts Contrast Coefficients '\" C.: DRUG(l) 1 23 0 C2: DRUG(2) 0 C3: DRUG(3) -1 1 0 1 -1 0 1 -1 0 0
864 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE Table n.lO SPSS Commands for Correlated Contrasts MANOVJ.. Yl Y2 BY DRUG (1 , 4 \" /CON7RAST=SPECIAL(1 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0 1) /PRIN7'=SIGNIF(MUL'!'!V,UNIV) /METHOD=SSTYPE(UNIQUE) DESIGN {SOLUTION) /DESIGN-DRUG(3) DRUG (2) DRUG(l) The CONlRAST command specifies the coefficients for the contrasts, which are followed by the SPECIAL keyword. The first line gives the coefficients for the con- stant, which must be all ones. (The constant is common for all contrasts and is not interpreted.) Lines 2 through 4 give the coefficients for the three contrasts given in Ta- ble 11.9. That is, the second line represents contrast DRUG(1). the third line represents contrast DRUG(2), and the fourth line represents contrast DRUG(3). The METHOD command specifies the procedure or method to be used for computing or extracting the sum of squares-unique and sequential being the two available methods. For orthogo- nal contrasts, both procedures give identical results. However, for correlated contrasts the results and the subsequent conclusions depend on the method used for extracting the sums of squares. In order to illustrate the interpretational complexity of correlated contrasts, we will compare the results obtained from unique and sequential methods. Table 11.11 summarizes the 'significance results from the resulting output, which is not reproduced. From the multivariate significance tests for the unique method. contrasts DRUG(1) (i.e.. Cd and DRUG(3) (i.e., C3) are significant at p = .05 and contrast DRUG(2) (i.e., C2 ) is not significant at p = .05. That is, the mean effectiveness of groups 2 and 4 is significantly different from group 1. Now suppose that the sequential method, instead of the unique method, is specified for extracting the sum of squares. The command for requesting the sequential method is; METHOD = SSTYPE(SEQUENTIAL). Exhibit 11.5 gives the partial output and Table 11.11 also summarizes the significance test results. SPSS prints a warning message indicating that use of the sequential method for nonorthogonal designs may Table 11.11 Summary of Significant Tests Contrast Effects Controlled Multivariate Univariate Significance or PartiaIled Significance Yz }'l : Unique Method .000 DRUG(J) DRUG(2), DRUG(3) .055 .000 .000 DRVG(2) DRUG(3). DRUGO) .000 .063 .020 DRUG(3) DRUG(I). DRUG(2) .001 .001 Sequential Method .000 .000 .000 DRVG(l) DRUG(2). DRUG(3) .002 .000 .040 DRUG(2) DRUG(3) .682 .426 1.000 DRUG(3)
11.4 MULTIPLE-GROUP MA..\"'10VA 386 Exhibit U.5 SPSS output for correlated contrasts using the sequential method ~>warninq # 12189 >You are using SEQUE~TIAL Sums cf Sq~ares with a potentially >nonorthcqonal pa=tition of an ef!ec=. Either the design is >unbalanced, you have specified a noncrthogonal contrast for >the partitioned factor (default DEV:ATION, SIMPLE, or >REPEATED), or you have specified a SPECIAL contrast for the >partitioned factor. If you must interpret SEQUENTIAL F-tests >for nonorthogonally part~tioned effects, see the solution >matrix to determine the actual hypotheses tested. Default. >UNIQUE Sums of Squares a=e directly interpretable for >nonorthogonal partitions. O Solution Matrix for Between-SubJects Design P~~TER 2 l-DRUG FACTOR Constant Drug (3) Drug (2) Drug (1) 1 1234 1 1.118 -.645 -.913 -1.581 2 1.118 -.645 -.913 1. 581 3 1.118 -.645 1. 826 .000 4 1.118 1. 936 .000 .OOC not be appropriate [I]. The results are very different from those obtained for the unique method. From the multivariate tests, it is now concluded that contrasts DRUG(l) and DRUG(2) are significant at p = .05 and contrast DRUG(3) is not significant at p = .05. That is, the treatment effectiveness of group 4 is not different from group 1, whereas the unique method concluded that it was different. Also, it is now concluded that DRUG(2) is significant at p = .05; that is, group 3 is significantly different from group 1, whereas the unique method concluded that it was not different. What is the reason for these drastic differences in the results? The obvious answer is that there is correlation among the contrasts. As we now discuss, the two methods differ with respect to how the sums of squares are extracted. In the unique method the sums of squares are computed after the effects of all other contrasts are removed or partialled out, irrespective of the order of the contrasts spec- ified in the DESIGN subcommand. For example. contrast DRUG(l) is tested after the effects of the other contrasts. DRUG(2) and DRUG(3). are removed. In the sequential method, however, the partialling of the effect of other contrasts depends on the order in which the contrasts are specified in the DESIGN subcommand. The sum of squares for each contrast is extracted after the effect of the contrasts specified to its left have been partialled out. For example. for the DESIGN = DRUG(3) DRUG(2) DRUG(l) statement the sum of squares for DRUG(1) is computed after the effects of DRUG(2) and DRUG(3) have been partialled out, and therefore the respective sums of squares reflect the effect of DRUG( 1) after the effects of all other contrasts have been taken into consideration. On the other hand, the effect of DRUG(3) is computed without par- tialling out the effects of other contrasts, and therefore the computed sum of squares includes not only the effect of DRUG(1). but also the effects of other contrasts that are correlated with tills contrast. The preceding analysis implies that the hypotheses tested may not correspond to the hypotheses specified by the contrast statement. as the computed sum of squares also
366 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE includes the effect ofother contrasts. SPSS gives the option of printing a solution matrix that contains information about the actual hypotheses tested. The solution matrix can be obtained by specifying DESIGN(SOLUTION) option in the PRINT subcommand. Exhibit 11.5 [2] gives the solution matrix. The columns of the solution matrix represent the contrasts and correspond to the contrasts specified in the DESIGN subcommand. From the solution matrix it is clear that the contrasts tested are not the same as those in- tended by the researcher. For example, DRUG(3) actually tests the difference between group 4 and the average of groups 1, 2, and 3. and not the difference between:groups 4 and 1 as intended by the researcher. Similarly. DRUG(2) tests the difference between group 3 and the average of groups 1 and 1 and not groups 3 and 1. The question then becomes: Which of the two methods for extracting the sum of squares should be used? The sequential method has the desirable property tha't the sum of squares of each effect and the error sum of squares add up to the total sum of squares. However. the sequential method has the undesirable property that the actual contrasts tested may not be the same as those specified by the researcher. The unique method partials out the effect of all other contrasts and therefore the contrasts tested are the same as those intended. But. it has the undesirable property that the sum of squares of each effect and the error sum of squares do not add up to the total sum of squares. However. the emphasis should be on testing the correct contrasts and not whether the sum of squares of all the different contrasts add up to the total sum of squares. Therefore, the unique method should be preferred over the sequential method. Fortunately. the unique method is the default method for extracting the sum of squares in SPSS. 1L5 MANOVA FOR TWO INDEPENDENT VARIABLES OR FACTORS Suppose the advertising department has prepared three ads for introducing a new prod- uct and it is interested in identifying the best ad. The first ad uses a humorous appeal, the second uses an emotional appeal. and the third uses a comparative approach. It is further believed that respondent's gender could have an effect on his/her preference for the type of ad. An experiment is conducted in which 12 males and 12 females are Table 11.12 Data for the Ad Study\" Type of Ad Gender Humorous Emotional Comparative Means Male Y1 881010 5577 2244 (6.000) (3.000) (9.000) 3467 1232 6.000 Y:! 67910 (5.000) (2.000) 5.000 (8.000) 44:!2 JO 1088 (9.000) 5.000 F.e'Tlale Y1 2244 (3.000) 10967 (3.000) (8.000) 4.333 Y~ J 223 263 I 5.000 (2.000) (3.00()) 6.000 4.667 5.000 Means Y1 6.000 4.500 Y:! 5.000 4.000 Q Numbers in parentheses are cell means.
11.5 MANOVA FOR TWO INDEPENDENT VARIABLES OR FACTORS 367 Table 11.13 SPSS Commands for the Ad Study MANOVA Yl TO Y2 BY GENDER(1,2) AD(1,3) /PRINT=CELLINFO(MEANS SSCP) HOMOGENEITY (BOXM) ERROR (SSCP) SIGNIF(MULTIV UNIV HYPOTH EIGEN EFSIZE DIMENR) /DISCRIM=RAW /POWER F(.OS) T(.OS) /DESIGN GENDER AD GENDER BY AD exposed to one of the ads. The 12 males are divided randomly into three groups of four subjects each. Each group is exposed to a different ad and the respondents are asked to evaluate the ad with respect to how informative (YJ) and believable (Y2) the ad is on an II-point scale with I indicating low infonnativeness and low believability, and 11 indicating high informativeness and high believability. The procedure is repeated for the female sample. Table 11.12 gives the data and Table 11.13 gives the SPSS commands. The DISCRIM subcommand requests discriminant analysis for each effect and the printing of the raw (i.e., unstandardized) coefficients. The DESIGN statement gives the effects that the researcher is interested in. Since there are two factors, there are two main effects (i.e., GENDER and AD) and one two-way interaction (i.e., GENDER BYAD). Multivariate significance for each of these effects is tested using the previously described multivariate tests. Exhibit 11.6 gives the partial output. 11.5.1 Significance Tests for the GENDER xAD Interaction The first effect tested is the GENDE R X AD interaction. MANOVA labels the between- groups SSCP matrix for a given effect as the hypothesis SSCP matrix. The eigenvalues 11.4of the SSCPh x SSCp;:l matrix are used for computing the test statistics given by Eq. [Ic]. For example, A= (1 + ~.594)(1 + ~061) = 0.124, which is the same as reported in the output [Ia]. The F-ratios for all the tests indicate that the GENDER x AD interaction is statistically significant at an alpha of .05 [Ia]. The effect sizes are quite large and have high power [1 b]. Recall that in Section 11.2.4 we discussed the similarity between discriminant anal- ysis and MANOVA. The dimension reduction analysis pertains to the results that would be obtained if a discriminant analysis is done for the interaction part of the MANOVA problem. The number of functions that one can have depends on the rank of the SSCPII X SSCP;:: 1 matrix, which is equal to the degrees of freedom for the respective effect. Note that only the first discriminant function (i.e.• the first eigenvalue, or A) is statistically significant [Id], and accounts for 99% of the interaction effect [Ic]. The univariate tests for each measure of the GENDER x AD interaction effect are statistically significant and have high effect sizes and powers [Ie. If]. Interpretation ofthe GENDER x AD Interaction Results of discriminant analysis can be used to gain further insight into the nature of the interaction effect. Coefficients of the retained discriminant function(s) can be used
388 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE Exhibit 11.6 MANOVA for ad study EFFECT .• GENDER BY AD ~u1tivariate Tests of Significance (S = 2, M = -1/2, N = 7 1/2) Test Name -- .Value A'O'Orox\". F Hypoth. DF Error DF 5ig. of F PHlais .92596 7.75922 4.00 36.00 .000 32.00 .000 Hotellings 6.65546 26.62185 4.00 34.00 .000 Wilks .12~09 ~5. 62986 4.00 Roys .86632 Note .. F statistic for WILKS' L-ambda is exact. ~u1tivariate £ffect Size and Observed Power at .0500 Level TEST NAME Effect Size Noncent . P Pillais . 463 31. 037 Hotellings .769 206.487 .99 Wilks .648 62.519 1. 00 1.00 ~igenValUes and Canonical Correlations Root No. Eigenvalue Pet. . Cum. Pct. Canon Cor. 6.594 99.081 1 .061 99.081 .932 2 .919 100.000 .240 ~imension Reduct.ion ~~alys~s Roots Wilks L. F Bypoth. !)F Error DF Sig. of F 1 TO 2 2 TO 2 .12409 15.62986 4.UO 34.00 .000 .94236 1.10103 1.00 18.00 .308 @nivariate F-\"t.ests with (2,18) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F 5ig. of F Y1 156.00000 2~.00000 18.00000 1.33333 .000 Y2 149.33333 ~8.00000 74.66667 2.66667 58.50000 .000 28.00000 @Variable ETA Square No~cer.~. Power Yl .86667 217.00000 1.00000 Y2 .75676 56.00000 1.00000 @aw discriminant function coefficients Function No. Variable 1 Yl Y2 .98< -.lH (wntinued)
11.5 MANOVA FOR TWO INDEPENDENT VARIABLES OR FACTORS 369 Exhibit 11.6 (continued) @FF!:CT .. AD Multivariate Tests of Significance (S = 2, M = -1/2, N = 7 1/2) Test Name Value Approx. F Hypoth. ....,r- Error OF Sig. of F 36.00 .102 Fillais .37696 2.09032 4.00 32.00 .069 34.00 .082 Hotellings .60504 2.42017 4.00 Wilks .6230'; 2.26867 4.00 Rays .37696 Note .. F statist~c for WILKS' Lambda is exac~. ~FFECT .. GENDER = =Multivariate Tests of Significance (5 1, M : 0, N 7 1/2) Test Name Value Exact F Hypoth. OF Error OF 5ig. of F 17.00 .106 Pillais .23226 2.57143 2.00 17.00 .106 17.00 .106 Hotellings .30252 2.57143 2.00 Wilks .76,74 2.57143 2.00 Rays .23226 Note .. F stat~stics are exact. Table 11.14 Cell Means for Multivariate Gender x Ad Interaction Type of Ad Gender Humorous Emotional Comparative Male 7.944 5.334 2.724- Female 2.724 2.610 7.944 The discriminant function is (see Exhibit 11.6 [1 gJ) = .984 Y. - .114 Y2 Average discriminant score for cell (i.e., males, humorous ad) = .984 x 9 - .114 x 8 = 7.944. to fonn discriminant scores to represent the multivariate GENDER x AD interaction effect [1 gj. In the present case only one discriminant function is retained. Table 11.14 gives average discriminant scores for the various cells of the GENDER X AD interac- tion effect and a sample computation, and Figure 11.5 gives the plot of the scores. From the figure it is clear that ad preference is a function of respondents' gender. Specifically, males prefer the humorous ad whereas females prefer the comparative ad. Figure 11.5 also shows the univariate plot for the interaction effects. The nature of the interaction effect is the same for both the dependent measures. That is, the males prefer the hu- morous ad while the females prefer the comparative ad.
370 CHAPTER 11 MULTIVARIATE ANALYSIS OF VARIANCE y Multivariate 10 Humorous Emotional Comparative 8 Type of ad 6 (al 4 2 Univariate. r I r1 10 S v~.:.: 6 ~ ~ Humorous Emotional Comparative Type of ad fb) 4 2 HumD1'O~ Emotj,onal Comparative Typeo(ad (C') Figure 11.5 Gender x Ad Interaction. (a) Multivariate (b) Univariate, Y 1 (c) Uni- variate, Y 2 Significance Tests for Main Effects Multivariate statistics for the two main effects (i.e., GENDER and AD) indicate that none of them are statistically significant (2, 3J. The univariate effects are not shown as they are nonnally interpreted only if the corresponding multivariate tests are significant. Also, no further interpretation of the main effects is pro\\'ided as they are not significant. lL6 SUMMARY In this chapter \\\\'e discussed multivariate analysis of variance (MANOYA), a technique that delennines the effect of categorical independent variables on a number of continuous dependent variables. As such. it is a direct generalization of analysis of variance (ANOYA). It was seen that there is a close relationship between MAN OVA and discriminant analysis, especially for
QUESTIONS 371 two-group or multiple-group ~fANOVA. In two-group MA...\"lOVA there is only one independent variable and it is at two levels. For multiple-group !\\fANOVA the independent variable has more [han two levels. In the next chapter we discuss the assumptions made in MANOVA and discriminant analysis. the effects of violating these assumptions on [he results. and the available procedures for testing the assumptions. QUESTIONS 11.1 For the following two data sets: Group 1 Group 2 Group 1 Group 2 Yl Y2 Yl Yz Yt Yl Yl Yl 12 5 11 15 6 10 35 7 15 2 4.7 49 8 17 51 7 9.7 59 10 20 4 3.2 10 6 51 3 4.1 9 8.2 10 6 8 4.1 (a) State the null and the alternative hypotheses in words. (b) Replicate the calculations given in Table 11.2 of the text. (c) What conclusions can you draw from your calculations? (d) Will there be a difference in the results of univariate and multivariate significance tests? Explain. (Hint: Compute the correlation between Y 1 and Y2.) (e) If a two-group discriminant analysis is done. what would be the eigenvalue? (f) Analyze the data using two-group discriminant analysis and compare the results. 11.2 Table 7.19 of the text gives the nutrient data for various food items. Assuming that there are three clusters (the cluster memberships are given in Exhibit 7.5) conduct a MANOVA using the food nunients as the dependent variables. and interpret the results. Compare your results to those obtained by multiple-group discriminant analysis. 11.3 The management of a national department store chain developed a discriminant model to classify various departments (e.g.. men·s wear. appliances. electronics, etc.) into high-, medium-. and low-performance departments. File STORE.DAT gives the discriminant scores for four consecutive quarters for the departments classified into each category (Note: Higher scores imply better performance). Analyze the data using a repeated- measures MANOVA with the following SPSS commands: MANOVA QTR1, QTR2, QTR3, QTR4 BY PERFORM(1,3) /~TSFACTOR==QTR (4 ) /WSDESIG!-I==QTR /DESIGN=PERFORM (a) What conclusions can you draw from the analysis? (b) For each performance category plot the average score using the quarters on the X- axis. Based on a visual analysis. is there a trend? Is the trend different across the groups (Nore: Use the interaction between QTR and PERFORM)? Whatconc1usions can you draw from the trend? (c) Decompose the trend of each perfonnance category into linear. quadratic. and cubic trends. The following SPSS commands can be used for this purpose:
372 'CHAPTER 11 MULTIVARIATE ANALYSIS OF VARLo\\NCE SELECT IF (PERFORM E~ 1) QTR4 ~~OVA QTR1, QTR2, QTR3, /WSFACTOR=QTR(4) /WSDESIGN=QTR /CONTRAST=?OLYNOMIA~ What conclusions can you draw from the analysis? 11.4 Analyze the data in file FIN.OAT using industry as the grouping variable. Interpret the results, and compare the results to those obtained by multiple-group discriminant analysis. 11.5 Analyze the data given in files OEPRES.DAT and PHONE.OAT using MANOVA (use CASES as the grouping variable for data given in OEPRES.OAT. and the number of phones owned as the grouping variable for PHONE.DAT) and compare your results to the results obtained by discriminant analysis. lUi Consider the following data: Yl Yl l'3 Y.j Group 73 1 2 1 8 .2 3 3 1 915 5 1 95 3 4 .2 10 4 5 5 2 1I 8 7 6 .2 Run a MANOVA and interpret the results. What is the problem with the data and how would you rectify it'? 11.7 For each of the following contrasts indicate whether they are orthogonal or nonorthogonal and discuss the effects that are being tested. III ILl IL3 IL4 Contrast I 0 0 -1 1 1 0 -} 0 0 -1 0 -1 0.5 0 0 Contrast 2 0.5 0.33 0 0.5 0.33 -1 -I -0.5 0 -1 0 0.33 1 -1 0 0 Contrast 3 1 3 I -1 1 0 1 0 -3 Contmst 4 -} -3 -1 3
QUESTIONS 373 11.8 (a) Analyze ~e data in the table given below using MANOVA and interpret the solution. Factor A Factor B 1 12 2 Y1 Y1 Y1 Yl 3 2 8 64 5 7 4- 3 6 12 10 11 7 9 12 8 8 15 10 11 14 13 12 12 (b) Using appropriate contrasts. detennine if group 2 of factor A is significantly different from group 1 of factor A, and if group 3 of factor A is significantly different from group 1 of factor A. Are these contrasts orthogonal? Vlhy or why not? (c) Using appropriate contrasts, detennine if groups 2 and 3 of factor A are significantly different from each other. Are these contrasts orthogonal? Why or why not?
CHAPTER 12 Assumptions As is the case with most of the multivariate techniques, properties of the statistical tests in MANOVA and discriminant analysis are derived or obtained by making a number of assumptions. In most empirical studies some or all of the assumptions made will be violated to varying degrees, which could affect the properties of the statistical tests. In this chapter we discuss these assumptions, and the effects of their violation on the statistical tests. Also. we discuss suggested procedures to test whether the data meet the assumptions, appropriate data transfonnations that can be used to make the data confonn to the assumptions, and steps that can be taken to mitigate the effect of violation of the assumptions on statistical tests. The assumptions are: 1. The data come from a multivariate normal distribution. 2. The covariance matrices for all the groups are equal. 3. The observations are independent. In other words, each observation is independent of other observations. Violation of the above assumptions can affect the significance and power of the statis- tical tests. Before we discuss the effects of each of these assumptions and the available tests for checking if an assumption is violated, we provide a brief discussion of signif- icance and power of test statistics. 12.1 SIGNIFICANCE AND POWER OF TEST STATISTICS Two types of errors are commonly made while testing the null and alternative hypothe- ses: Type I and Type II errors. Type I error, usually labeled as the significance or alpha level of a given test statistic, is the probability of falsely rejecting the null hypothesis due to chance. The researcher typically selects the desired or nominal alpha level (Le., the value for Type I error) to test the hypotheses. A nominal alpha level of, say, .05 means that if the study were replicated several times, one would expect to falsely reject the nuB hypothesis in about 5% of the studies due to chance alone. However, if some of the assumptions are violated then the actual number of times the null hypothesis will be falsely rejected could be more or less than the nominal alpha level. For example, it is quite possible that violating the multivariate normality assumption could result in an actual alpha level of .20 even though the nominal or desired alpha level selected was .05. 374
12.3 TESTL.\"iG UNIVARIATE NORMALITY 375 Type II error, usually represented by f3, is the probability of failing to reject the null hypothesis when in fact it is false. The power of a test is given by 1 - {3 and it is the probability ofcorrectly rejecting the null hypothesis when it ;sfalse. If the power is low, then the probability of finding statistically significant results decreases. Obviously, [he researcher would like to have a small alpha level and a high power. Therefore. it is important to know how the alpha level and power of the test statistic are affected by violation of the assumptions. 12.2 NORMALITY ASSUMPTIONS Almost all parametric statistical techniques assume that the data come from a multi- variate normal distribution. Research has found that for univariate (e.g., ANOYA) and multivariate techniques (e.g., MANOVA and discriminant analysis), violation of the normality assumption does not have an appreciable effect on the Type I error (Glass, Peckham, and Sanders 1972; Everitt 1979; Hopkins and Clay 1963; Mardia 1971; and Olson 1974). As discussed in Chapter 8, violation of the nonnality assumption does have an effect on the classification rates. Also, as discussed in the following, violation of the normality assumption does affect the power of the test statistic. A univariate normal distribution has zero skewness and a kurtosis of three. Some- times the kurtosis is normalized by subtracting three so that its value is zero for the normal distribution. Henceforth we will use the term kurtosis to refer to its normalized value. That is, a univariate normal distribution has zero skewness and zero kurtosis. A negatively skewed distribution has a skewness of less than zero and a positively skewed distributior. has a skewness of greater than zero. Similarly, a multivariate distribution is said to be skewed if its multivariate measure of skewness is not equal to zero. Re- search has shown that the power of a test is not affected by violation of the normality assumption if the nonnormality is solely due to skewness. A distribution is said to be leptokurtic (i.e., peaked) if its kurtosis is positive and platykurtic (i.e.• fiat) if its kurtosis is negative. Kurtosis does seem to have an effect on the power of a test statistic; however, the effect is more severe for platykurtic dis- tributions than for leptokurtic distributions. Olson (1974) found that for MANOVA the power of the test decreased substantially for platykurtic distributions. Furthermore, the severity of the effect increases as the assumption is violated for more than one cell or group. Since normality affects the power of the test. it is advisable to determine if the normality assumption has been violated. In the following section we discuss tests for assessing univariate normality, and in Section 12.4 we discuss tests for assessing multivariate normality. 12.3 TESTING UNIVARIATE NORMALITY Tests of univariate normality are discussed for several reasons. First, tests of multivari- ate normality are more complex and difficult, and understanding them is facilitated by an understanding of univariate tests. Second, although it is possible that the multivariate distribution may nor be normal even though all the marginal distributions are nonnal. such cases are rare. As stated by Gnandesikan (1977), it is only in rare cases that mul- tivariate nonnormality will not be detected by univariate nonnality tests. Finally, if the data do not come from a multivariate nonnal distribution then one would like to further
376 CHAPI'ER 12 ASSUMPTIONS investigate which variable's distribution is not nonnal. Such an investigation is neces- sary if one wants to transfonn the data for achieving normality. To test for univariate nonnality, one can employ graphical or analytical tests, as described in the following pages. 12.3.1 Graphical Tests A number of graphical tests such as the stem-and-Ieaf plot. the box-and-whiskers plot. and the Q-Q plot have been proposed. Of these, the Q-Q plot is the most popular and is the plot discussed in this chapter. For discussion of other tests the interested reader is referred to Tukey (1977). Use of the Q-Q plot is illustrated with hypothetical data simulated from a nonnal distribution having a mean of 10 and a standard deviation of 2. The Q-Q plot is obtained as follows: 1. Order the obsen'ations in ascending order such that XI < X2 ... < Xn • where n is the number of observations, The ordered observations are given in Column 2 of Table 12.1. Given that the value of each observation is unique, as is usually the case for continuous variables, then exactly j observations will be less than or equal to Xj. Each ordered observation therefore represents a sample quantile. 2. The proportion of observations that are less than Xj is estimated by (j - .5), 'n. The quantity .5 is subtracted for continuity correction. Column 3 of Table 12. I gives the proportion of observations that would be less than Xj. For each j, these proportions are assumed to be percentiles or probability levels for the cumulative standard nonnal distribution and the corresponding Z-values give the expected or theoretical quantiles. for a nonna! distribution. The Z-values can be obtained from the cumulative normal distribution table or from the PROBIT function in SAS. Table 12.1 Hypothetical Data Simulated from Normal Distribution Obsen'ation Ordered Probability Lc\\'el Z-\\'alue Number (j) Value (Xj ) or Percentile (4) (1) (2) (3) ...,. 4.813 0.033 -1.834 6.937 0.100 0.167 -1.~82 3 7.027 0.:B3 -0.967 4 7.804 0.300 -0.728 0.367 -0.524 5 8.560 0.433 -0.341 0.500 -0.168 6 8.727 0.567 0.633 0.000 7 9.754 0.700 0.168 0.767 0.341 8 9.996 0.833 0.524 0.900 O.72S 9 10.053 0.967 0.967 1.281 10 10.139 1.834 11 10.202 12 10.240 13 11.918 14 11.943 15 12.027
12.3 TESITNG UNIVARIATE NORMALITY 817 3. Column 4 of Table 12.1 gives the Z values. The plot between the ordered obser- vations or values (i.e., X j) and theoretical quantiles (e.g., Z) is called the Q-Q plot and is shown in Figure 12.1. A linear plot indicates that the distribution is normal and a nonlinear plot indicates that the distribution is nonnormal. The plot in Figure 12.1 is approximating linear, suggesting that the data in Table 12.1 are normally disrributed. The plot is not completely linear because of the small sample size. For a large sample size the plot would have been linear, as the simulated data do come from a normal disrribution. To illustrate the plot for a nonnormal distribution. the data in Table 12.1 were transformed as Y = ~. The plot of the transformed data is given in Figure 12.2. It is clear that the Q-Q plot in Figure 12.2 deviates substantially from linearity. Obviously, the test based on the Q-Q plot is subjective, for the researcher has to visu- ally establish whether the plot is linear or not For this purpose one could use \"training plots\" given in Daniel and Wood (1980) to assess the linearity of the plot Alterna- tively, and more preferably, the linearity of the Q-Q plot can be assessed by computing the correlation coefficient between the sample (i.e., Xj) and theoretical quantiles, and comparing it with the critical value given in Table T.5 in the Statistical Tables follow- ing Chapter 14. Values in Table T.5 give the percent points of the cumulative sampling distribution of the correlation between sample values and theoretical quantiles obtained empirically by FilIiben (1975). The correlation coefficients for the plots in Figures 12.1 and 12.2, respectively, are .967 and .814. It can be seen that the correlation of .967 is well above the critical value of .937 for alpha level of .05 and n = 15, suggesting that the respective Q-Q plot is linear and. therefore, the data set in Table 12.1 does come from a normal disrriburion. However, the transfonned dara do not come from a normal distribution as the correlation of .814 for the plot in Figure 12.2 is not greater than the critical value of .937. 3.--------------------------------. • •II 1- ---.•••I . -~ 0 f--------------.-....-....-._----------------I / . .-1- /\" I•/ -21-- • ~~~I--~I~~--~~~/--J~~l--~l--l~~/--~~ 4 5 6 7 8 9 10 11 12 13 14 15 16 Ordered observations (X) Figure 12.1 Q.Q Plot for data in Table 12.1.
378 CHAPTER 12 ASSUMPI'IONS 3r-------------------------------~ 2- I- !/-! -I r I ,•.-------------------- •I •NO~.----------------------------~ ./-,•• --I \"j- i - 2•- -~~------~~I------~J~--------I~~O------~200 Ordered obserntions (Y} Figure 12.2 Q-Q Plot for transformed data. 12.3.2 Analytical Procedures for Assessing Univariate Normality Some of the analytical procedures or tests for assessing normality are the chi-square goodness of fi~ the Kolmogorov-Smimov test. and the Shapiro-Wilk test. Simulation studies conducted by Wilk, Shapiro. and Chen (1968) concluded that the Shapiro-\\ViIk test was the most powerful test in assessing univariate normality. In case the data do not come from a normal distribution, further assessment can be done by examining the skewness and kurtosis of the distribution. The EXAMINE procedure in SPSS can be used to obtain the preceding statistics. The EXAMINE procedure also can be used to obtain the Q-Q plot discussed in the previous section. In the following section we use the data set given in Table 12.2, which gives the financial ratios for most- and least- admired finns, to discuss the resulting output from the EXAMINE procedure. This data set has been used in Chapter 8 to illustrate two-group discriminant analysis. 12.3.3 Assessing Univariate Normality Using SPSS The SPSS commands for the EXAMINE procedure to assess the normality assumption for EBITASS are given in Table! 12.3. The EXAMINE command requests infOImation for EBITASS to evaluate its distribution. The plot option specifies the type of plot de- sired. NPPLOT gives the Q-Q plot described earlier. The DESCRIPTIVE option in the STATISTICS subcommand requests printing of a number of descriptive statistics. Exhibit 12.1 gives the output SPSS refers to the Q-Q plOl as the normal plot. It can be seen that the plot is similar to that given in Figure 12.1. and its linearity suggests that EBITASS is normally dis- tributed [2a]. The detrended lIormal plot gives the plot of the residuals after removing the linearity effect [2b]. For a n,Jrmal distribution. the detrended plot should be random
12.3 TESTING UNIVARIATE NOR.\\IALITY 379 Table 12.2 Financial Data for Most-Admired and Least-Admired Firms Squared Mahalanobis Distance Obs ROTC EBITASS (ltl Ji!) 1 0.182 0.158 1.289 1.150 2 0.206 0.210 1.098 3.6-+8 3 0.188 0.207 0.901 3.058 4 0.236 0.280 3.101 2.856 5 0.193 0.197 4.802 0.390 6 0.173 0.227 2.331 0.879 7 0.196 0.148 1.415 0.641 8 0.212 0.254 0.305 2.440 9 0.147 0.079 6.335 0.850 10 0.128 0.149 1.787 1.173 11 0.150 0.200 ~.918 12 0.191 0.187 0.643 13 -0.031 -0.012 1.318 0.672 14 0.053 0.036 15 0.036 0.038 16 -0.074 -0.063 17 -0.119 -0.054 18 -0.005 0.000 19 0.039 0.005 20 0.122 0.091 21 -0.072 -0.036 22 0.064 0.045 23 -0.024 -0.026 24 0.026 0.016 Table 12.3 SPSS Commands EXAMINE VARIASLES-EBITASS /PLOT=NPPLOT /STATISTICS-DESCRIPTIVE and centered around zero. This appears to be the case, suggesting once again that the dis- tribution of EBITA.SS is normal. Both the Shapiro-Wilk and the Kolmogorov-Smimov test statistics are not significant [3]. It is obvious that all the procedures suggest that the distribution of EBITASS is normaL If the distribution of EB/TASS were not normal. one would examine skewness and kurtosis to obtain further insights into the nature of nonnormality.:Skewness and kurto- sis, along with their standard errors, are printed for EBlTASS [1]. For large sample sizes (e.g., 25 or more) the standard errors can be used to compute the Z-values, which are .176 (.0833/472) for skewness and 1.562 (-1.433/.918) for kunosis [1]. Since both of these are less than the critical value of 1.96 for an alpha level of .05. it is concluded that the distribution of EBIT.4.SS is normal. For smaller samples the critical values obtained via simulation by D' Agostino and Tietjen (1971: 1973) can be used. These critical val- ues are reproduced in Table T.6 (Statistical Tables).
380 CHAPTER 12 ASSUMPI'IONS Exhibit 12.1 Univariate normality tests for data in Table 12.2 ~ EBITASS se s: 24.0 !-~~ss~ng cases; .0 Percent missing: .0 Valid ca Mean .0973 Std Err .0219 Min -.0630 Skewness .0833 I-lediar. .0115 May. .2800 S E Ske1ol.· .4723 .ossa Va:::-~ance 5% Trim .0962 Std Dev .1074 Range .3430 Kurtosis -1.4332 IQR .1960 S E KUrt .9176 @@ +-----------------------------+ +-----------------------------+ 1. 80 + '\" I .48 + I :;: T... ! 1 I '\" II '\" I '\" '\" '\" I 1.20 + ::: .32 + '\" '\" I .. '\" '\" I :;: I r '\" I I :;: I - I .60 + .16 + I ~ I I I \"'''' '\" '\" II '\" '\" '\" I I '\" '\" '\" '\" I Ir I .00 + •'\" I .00 + *'\" I •• * :;: I • II *• I I '\" II I •• I -.60 + * I -.16 + • I .. II I II -1.20 + I -.32 + I II I II -1. 60 + ::: -.45 + +--+-------+------+-------+---+ +--+-------+------+-------+---+ -.16 .00 .16 .32 -.16 .00 .16 .32 Norrr,a1 PI 0\"; Detrended Normal Plot CD Statist~c d: Significance Shapiro-Wi lks .!:301 24 .1010 K-S (Lilliefors) .1453 24 > .2000 12.4 TESTING FOR MULTIVARIATE NORMALITY There are very few tests for examining multivariate normality. The graphical test is similar to the Q-Q plot discussed for the univariate case. The analytical tests siI1)ply assess the multivariate measures of skewness and kurtosis. Unfortunately. not many programs have the option for computing these statistics.) Furthermore. the distribution ofthese test statistics1S not known. leading to their limited use for assessing multivariate nonnality. Consequently, only the graphical procedure is described below. The data set given in Table 12.2 is used to illustrate the graphical test. The first step is to compute squared Mahalanobis distance (M D2) of each observa- tion from the sample centroid. This distance is also reported in Table 12.2. It has been shown that when the parent population is normal and the sample size is sufficiently large I EQS. a covariance structure analysis program in B;\\1:DP. computes thel'e statistics.
12.4 TESTING FOR MULTIVARIATE NOR...'lALITY .381 (e.g.• 25 or more) these distances behave like a chi-square random variable (Johnson and Wichern 1988). This property can be used co obtain a chi-square plot as follows (see Gnandesikan 1977): DrI. First. order the M D2 from lowest to the highest such that M < M D~ < . .. < M D; where n is the number of observations. The ordered distances are presented in Column 2 of Table 12.4. 2. For each M D2. compute the (j - .5)/n percentile where j is the observation num- ber. These percentiles are reponed in Column 3 of Table 12.4. K3. The.r values for the percentiles are obtained from the distribution with p de- rgrees of freedom where p is the number of variables. The values can be obt::ti.ned r rfrom the tables or using the CINV function in SAS. The values are given in Column 4 of Table 12.4. K r4. M D2 and are then plotted. The plot is shown in Figure 12.3 and is similar to the Q-Q plot. The plot should be linear and any deviation from linearity indicates nonnormality. Table 12.4 Ordered Squared Mahalanobis Distance and Chi-Square Value Observation Ordered Squared Percentiles Chi-Square Number(j) MahaJanobis Distance Value (3) (4) (1) (M Jil) 0.021 0.042 1 (2) 0.063 0.129 2 0.104 0.220 3 0.305 0.146 0.315 4 0.390 0.188 0.415 5 0.641 0.229 0.521 6 0.643 0.271 0.632 7 0.672 0.313 0.7-1-9 8 0.850 0.35-1- 0.874 9 0.879 0.396 1.008 10 0.901 0.-1-38 1.151 11 1.098 0.-1-79 1.305 12 1.150 0.521 1.471 13 1.173 0.563 1.653 14 1.289 0.604 1.854 15 1.318 0.646 2.076 16 1.415 0.688 2.326 17 1.787 0.729 2.613 18 2.331 0.771 2.947 19 2.440 0.813 3.348 20 2.856 0.854 3.851 21 2.918 0.896 4.524 22 3.058 0.938 5.545 23 3.101 0.979 7.742 24 3.648 4.802 6.335
882 CHAPTER 12 ASSUMPTIONS 8 • 7 b7 6 • 5 ~• co :> e •~4 • 3• • -• 2 -. ,I -:--- - 00 :! 3 4 5 Ordered mahaianobis distance Figure 12.3 Chi-square plot for total sample. The plot in Figure 12.3 appears to be linear. from which we conclude that the assump- tion of multivariate normality is a reasonable one .<\\.S discussed earlier, one could com- pute the correlation coefficient of the plot and compa.re it with the critical values given in Table T.5 of the Statistical Tables. Although these critical values were obtained for univariate disUibutions, we feel that they provide a reasonable benchmark. The corre- lation coefficient for the plot in Figure 12.3 is 0.990. which is greater than the critical value of 0.957 for alpha = .05 and n = 24. from which it can be concluded that the data do come from a multivariate normal distribution. Unfortunately, none of the sta- tistical packages has a procedure to obtain the .i plots. Howe\\'er, PROC IML in SAS can be used to obtain these plots. The Appendix to this chapter gives the PROC IML program for obtaining the K plot. 12.4.1 Transformations If the data do not come from a normal distribution. one can transform the data such that the disUibution of the transformed variable is normal. In the multivariate case, each variable whose marginal distribution is not normal is transformed to make irs distri- bution normal. The type of transformation depends on the type of nonnormaIity with respect to skewness and kurtosis. However, in general, the square-root tr~nsformation works best for data based on counts. the logit transformation for proportions, and the Fisher's Z transformation for correlation coefficients. Table J2.5 gives various trans- formations that have been suggested to achieve normality. In situations where none of these transformations are appropria.te, one can use analytical procedures to identify the type of power transformation necessary to achieve normality. These procedures are discussed in Johnson and Wichern (1988).
12.5 EFFECT OF VIOLATINGTHE EQUAUTY OF COVARIANCE MATRICES ASSUMPTION 383 Tabu 12.5 Trarud'ormations To Achieve Normality Type of Scale Transformation Counts Proportions (p) Square-rooe transformation Correlations (r) (1Iogir(p) == O.5log ~ p) r)Fisher's Z = O.5log ( 11 +_ r 12.5 EFFECT OF VIOLATING THE EQUALITY OF COVARIANCE MATRICES ASSUMPTION In a univariate case (e.g.• ANOVA) the covariance matrix is a scalar and the assumption is met if the variance of the dependent variable is the same for all the cells. However, in the case of MANOVA and discriminant analysis, the equality of covariance matrices assumption is met only if the covariance matrices of all the cells are equal. Two ma- trices are said to be equal if and only if all the corresponding elements of the matrices are equal. For example, in the case of three dependent variables there will be six ele- ai.ments in the covariance matrix: three variances, u~, and u~. and three covariances, Ul~. U13, and U:!3. All the corresponding six elements of the matrices would have to be equal to satisfy the equality of covariance matrices assumption. Therefore, there is a greater chance that the equality of covariance matrices assumption will be violated in MANOVA and discriminant analysis than in ANOVA. Violation of the equality of covariance matrices assumption affects Type I and Type II errors. However, simulation studies have found that the effect is much more for a Type I error than for a Type II error. Consequently, most of our discussion pertains to how the significance level is affected (the Type I error). Research has shown that for equal cell sizes the significance level is not appreciably affected by unequal covariance matrices (Holloway and Dunn 1967; Hakstian. Roed. and Linn 1979; and Olson 1974). Therefore, every effon should be made to have equal cell sizes. However. for unequal cell sizes the significance level can be severely affected even for moderate differences in the covariance matrices. In the case of two groups. the following specific findings were obtained from simulation studies (Holloway and Dunn 1967; Hakstian, Roed, and Linn 1979). The test is liberal if the smaller group has more variability, that is, the actual alpha level of the test is more than the nominal alpha level. On the other hand. if the variability of the larger group is more than that of the smaller group then the test is conservative; that is. the actual alpha level will be less than the nominal alpha level. The implications of these findings are as follows: 1. If the test statistic is conservative due to unequal covariance matrices, then there is no need to be concerned about significant results because the results would still be significant after transforming the variables to achieve equality of covari- ance matrices. and consequently the conclusions of the study will not change. On the other hand, there is a need for concern about insignificant results. In this case,
384 CHAPTER 12 ASSUMPTIONS transformation of variables to obtain equal covariance matrices could result in sig- nificant findings that will change the conclusions of the study. 2. If the test is liberal due to unequal covariance matrices, then there is no need to be concerned about insignificant results as they will still be insignificant after the necessary transfonnations. But, one does need to be concerned about significant results. In this case the researcher may Dot be sure whether the significance is due to actual differences or due to chance because of the effect of the inequality of covariance matrices. As discussed in Chapter 8, violation of the equality of covariance matrices does affect the classification rates of discriminant analysis. Table 12.6 Data for Purchase Intention Study Segment (Group) 1 Segment (Group) 2 Y1 Y2 Y1 Y2 1.180 5.209 12.017 5.209 2.480 6.563 14.774 6.563 5.107 5.054 20.349 5.054 4.273 6.698 18.579 6.698 5.240 4.638 20.630 4.638 3.913 5.694 17.815 5.694 2.025 5.858 13.810 5.858 3.469 7.012 16.873 7.012 4.232 6.517 18.492 6.517 4.660 4.159 19.401 4.159 2.193 7.487 14.166 7.487 3.288 4.935 16.489 4.935 4.656 6.765 19.392 6.765 5.442 5.770 21.060 5.770 4.024 5.305 18.052 5.305 4.686 5.024 4.465 5.711 3.157 5.385 4.382 6.945 3.216 7.557 2.172 6.094 5.607 4.271 2.596 6.232 2.332 5.419 4.492 8.324 4.363 4.575 3.032 3.403 5.100 7.931 5.040 7.052 4.853 6.879
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