540 15. GEE Examples Introduction In this chapter, we present examples of GEE models applied to three datasets containing correlated responses. Abbreviated The examples demonstrate how to obtain odds ratios, con- Outline struct confidence intervals, and perform statistical tests on the regression coefficients. The examples also illustrate the effect of selecting different correlation structures for a GEE model applied to the same data, and compare the results from the GEE approach with a standard logistic regression approach in which the correlation between responses is ignored. The outline below gives the user a preview of the material to be covered by the presentation. A detailed outline for review purposes follows the presentation. I. Overview (page 542) II. Example I: Infant Care Study (pages 542–550) III. Example II: Aspirin–Heart Bypass Study (pages 551–555) IV. Example III: Heartburn Relief Study (pages 555–557) V. Summary (page 557)
Objectives Objectives 541 Upon completing this chapter, the learner should be able to: 1. State or recognize examples of correlated responses. 2. State or recognize when the use of correlated analysis techniques may be appropriate. 3. State or recognize examples of different correlation structures that may be used in a GEE model. 4. Given a printout of the results of a GEE model: i. State the formula and compute the estimated odds ratio ii. State the formula and compute a confidence interval for the odds ratio iii. Test hypotheses about the model parameters using the Wald test, generalized Wald test, or Score test, stating the null hypothesis and the distribution of the test statistic, and corresponding degrees of freedom under the null hypothesis 5. Recognize how running a GEE model differs from running a standard logistic regression on data with correlated dichotomous responses. 6. Recognize the similarities in obtaining and interpreting odds ratio estimates using a GEE model compared with a standard logistic regression model.
542 15. GEE Examples In this chapter, we provide examples of how the GEE approach is used to carry out logis- Presentation tic regression for correlated dichotomous responses. I. Overview FOCUS Examples: Modeling outcomes with dichotomous correlated responses Three examples are presented: We examine a variety of GEE models using three databases obtained from the following 1. Infant Care Study studies: (1) Infant Care Study, (2) Aspirin– 2. Aspirin–Heart Bypass Study Heart Bypass Study, and (3) Heartburn Relief 3. Heartburn Relief Study Study. II. Example 1: Infant Care Study Introduced in Chap. 14 In Chap. 14, we compared model output from two models run on data obtained from an infant care health intervention study in Brazil (Cannon et al., 2001). We continue to examine model output using these data, comparing the results of specifying different correlation structures. Response (D): Recall that the outcome of interest is a dicho- tomous variable derived from a weight- Weight-for-height standardized (z) for-height standardized score (i.e., z-score) obtained from the weight-for-height distribu- score if z < ---1 (‘‘Wasting’’) tion of a reference population. The dichoto- (1 mous outcome, an indication of “wasting,” is coded 1 if the z-score is less than negative 1, D¼ and 0 otherwise. 0 otherwise
Presentation: II. Example 1: Infant Care Study 543 Independent variables: The independent variables are BIRTHWGT (the weight in grams at birth), GENDER (1 ¼ BIRTHWGT (in grams) male, 2 ¼ female), and DIARRHEA, a dichoto- mous variable indicating whether the infant GENDER 8 had symptoms of diarrhea that month <> 1 (1 ¼ yes, 0 ¼ no). We shall consider DIAR- if symptoms RHEA as the main exposure of interest in this DIARRHEA ¼ >: present in analysis. Measurements for each subject were past month obtained monthly for a 9-month period. The 0 otherwise variables BIRTHWGT and GENDER are time- independent variables, as their values for a DIARRHEA given individual do not change month to month. The variable DIARRHEA, however, is Exposure of interest a time-dependent variable. Time-dependent variable Infant Care Study Model The model for the study can be stated as shown on the left. logit PðD ¼ 1jXÞ ¼ b0 þ b1BIRTHWGT þ b2GENDER þ b3DIARRHEA Five GEE models presented, with Five GEE models are presented and compared, different Ci: the last of which is equivalent to a standard logistic regression. The five models in terms 1. AR1 autoregressive of their correlation structure (Ci) are as fol- 2. Exchangeable lows: (1) AR1 autoregressive, (2) exchangeable, 3. Fixed (3) fixed, (4) independent, and (5) independent 4. Independent with model-based standard errors and scale 5. Independent (SLR) factor fixed at a value of 1 [i.e., a standard logistic regression (SLR)]. After the output for all five models is shown, a table is presented that summarizes the results for the effect of the variable DIARRHEA on the outcome. Addition- ally, output from models using a stationary 4-dependent and a stationary 8-dependent cor- relation structure is presented in the Practice Exercises at the end of the chapter. A GEE model using an unstructured correlation struc- ture did not converge for the Infant Care data- set using SAS version 9.2.
544 15. GEE Examples Output presented: Two sections of the output are presented for b^h; sb^h (empirical), and each model. The first contains the parameter estimate for each coefficient (i.e., beta), its esti- Wald test P-values mated standard error (i.e., the square root of the estimated variance), and a P-value for the “Working” correlation matrix Wald test. Empirical standard errors rather (Ci) containing r^ than model-based are used for all but the last model. Recall that empirical variance estima- Sample: tors are consistent estimators even if the corre- K ¼ 168 infants, ni 9, but lation structure is incorrectly specified (see 9 infants “exposed cases”: Chap. 14). (i.e., D ¼ 1 and DIARRHEA ¼ 1 for any The second section of output presented for month) each model is the working correlation matrix (Ci). The working correlation matrix contains the estimates of the correlations, which depend on the specified correlation structure. The values of the correlation estimates are often not of primary interest. However, the examina- tion of the fitted correlation matrices serves to illustrate key differences between the underly- ing assumptions about the correlation struc- ture for these models. There are 168 clusters (infants) represented in the data. Only nine infants have a value of 1 for both the outcome and diarrhea variables at any time during their 9 months of measurements. The analysis, therefore, is strongly influenced by the small number of infants who are classi- fied as “exposed cases” during the study period.
Presentation: II. Example 1: Infant Care Study 545 Model 1: AR1 correlation structure The parameter estimates for Model 1 (autore- gressive – AR1 correlation structure) are pre- Empirical Wald sented on the left. Odds ratio estimates are Variable Coefficient Std Err p-value obtained and interpreted in a similar manner as in a standard logistic regression. INTERCEPT À1.3978 1.1960 0.2425 BIRTHWGT À0.0005 0.0003 0.1080 GENDER 0.5546 0.9965 DIARRHEA 0.0024 0.8558 0.7958 0.2214 Effect of DIARRHEA: For example, the estimated odds ratio for the OdR ¼ expð0:2214Þ ¼ 1:25 effect of diarrhea symptoms on the outcome (a low weight-for-height z-score) is exp 95% CI ¼ exp½0:2214 Æ 1:96ð0:8558Þ (0.2214) ¼ 1.25. The 95% confidence interval ¼ ð0:23; 6:68Þ can be calculated as exp[0.2214 Æ 1.96 (0.8558)], yielding a confidence interval of (0.23, 6.68). Working correlation matrix: 9 Â 9 The working correlation matrix for each of these models contains nine rows and nine col- umns, representing an estimate for the month- to-month correlation between each infant’s responses. Even though some infants did not contribute nine responses, the fact that each infant contributed up to nine responses accounts for the dimensions of the working correlation matrix. AR1 working correlation matrix The working correlation matrix for Model 1 is shown on the left. We present only columns 1, (9 Â 9 matrix: only three columns shown) 2, and 9. However, all nine columns follow the same pattern. COL1 COL2 . . . COL9 The second-row, first-column entry of 0.5254 ROW1 1.0000 0.5254 . . . 0.0058 for the AR1 model is the estimate of the corre- ROW2 0.5254 1.0000 . . . 0.0110 lation between the first and second month ROW3 0.2760 0.5254 . . . 0.0210 measurements. Similarly, the third-row, first- ROW4 0.1450 0.2760 . . . 0.0400 column entry of 0.2760 is the estimate of the ROW5 0.0762 0.1450 . . . 0.0762 correlation between the first and third month ROW6 0.0400 0.0762 . . . 0.1450 measurements, which is assumed to be the ROW7 0.0210 0.0400 . . . 0.2760 same as the correlation between any two mea- ROW8 0.0110 0.0210 . . . 0.5254 surements that are 2 months apart (e.g., row 7, ROW9 0.0058 0.0110 . . . 1.0000 column 9). It is a property of the AR1 correla- tion structure that the correlation gets weaker Estimated correlations: as the measurements are further apart in time. r^ ¼ 0:5254 for responses 1 month apart (e.g., first and second) r^ ¼ 0:2760 for responses 2 months apart (e.g., first and third, seventh and ninth)
546 15. GEE Examples r^j; jþ1 ¼ 0:5254 Note that the correlation between measure- r^j; jþ2 ¼ ð0:5254Þ2 ¼ 0:2760 ments 2 months apart (0.2760) is the square r^j; jþ3 ¼ ð0:5254Þ3 ¼ 0:1450 of measurements 1 month apart (0.5254), whereas the correlation between measure- ments 3 months apart (0.1450) is the cube of measurements 1 month apart. This is the key property of the AR1 correlation structure. Model 2: Exchangeable Wald Next we present the parameter estimates and correlation structure p-value working correlation matrix for a GEE model using the exchangeable correlation structure Empirical 0.2463 (Model 2). The coefficient estimate for DIAR- Variable Coefficient Std Err 0.1237 RHEA is 0.6485. This compares with the 0.9623 parameter estimate of 0.2214 for the same INTERCEPT À1.3987 1.2063 0.3906 coefficient using the AR1 correlation structure BIRTHWGT À0.0005 0.0003 in Model 1. GENDER À0.0262 0.5547 DIARRHEA 0.7553 0.6485 b^3 for DIARRHEA ¼ 0.6485 (vs. 0.2214 with Model 1) Exchangeable working correlation There is only one correlation to estimate with matrix an exchangeable correlation structure. For this model, this estimate is 0.4381. The interpreta- COL1 COL2 . . . COL9 tion is that the correlation between any two outcome measures from the same infant is ROW1 1.0000 0.4381 . . . 0.4381 estimated at 0.4381 regardless of which ROW2 0.4381 1.0000 . . . 0.4381 months the measurements are taken. ROW3 0.4381 0.4381 . . . 0.4381 ROW4 0.4381 0.4381 . . . 0.4381 ROW5 0.4381 0.4381 . . . 0.4381 ROW6 0.4381 0.4381 . . . 0.4381 ROW7 0.4381 0.4381 . . . 0.4381 ROW8 0.4381 0.4381 . . . 0.4381 ROW9 0.4381 0.4381 . . . 0.4381 Only one r^: r^ ¼ 0:4381 Model 3: Fixed correlation Next we examine output from a model with a structure fixed, or user-defined, correlation structure (Model 3). The coefficient estimate and stan- Empirical Wald dard error for DIARRHEA are 0.2562 and Variable Coefficient Std Err p-value 0.8210, respectively. These are similar to the estimates in the AR1 model, which were INTERCEPT À1.3618 1.2009 0.2568 0.2214 and 0.8558, respectively. BIRTHWGT À0.0005 0.0003 0.1110 GENDER À0.0304 0.5457 0.9556 DIARRHEA 0.8210 0.7550 0.2562
Presentation: II. Example 1: Infant Care Study 547 Fixed structure: r prespecified, not A fixed correlation structure has no correlation estimated parameters to estimate. Rather, the values of the correlations are prespecified. For Model 3, In Model 3, r fixed at 0.55 for con- the prespecified correlations are set at 0.55 secutive months; 0.30 for noncon- between responses from consecutive months secutive months. and 0.30 between responses from nonconsecu- tive months. For instance, the correlation Fixed working correlation matrix between months 2 and 3 or months 2 and 1 is assumed to be 0.55, whereas the correlation COL1 COL2 . . . COL9 between month 2 and the other months (not 1 or 3) is assumed to be 0.30. ROW1 1.0000 0.5500 . . . 0.3000 ROW2 0.5500 1.0000 . . . 0.3000 ROW3 0.3000 0.5500 . . . 0.3000 ROW4 0.3000 0.3000 . . . 0.3000 ROW5 0.3000 0.3000 . . . 0.3000 ROW6 0.3000 0.3000 . . . 0.3000 ROW7 0.3000 0.3000 . . . 0.3000 ROW8 0.3000 0.3000 . . . 0.5500 ROW9 0.3000 0.3000 . . . 1.0000 Correlation structure (fixed) for This particular selection of fixed correlation Model 3: combines AR1 and values contains some features of an autore- exchangeable features gressive correlation structure, in that consecu- tive monthly measures are more strongly Choice of r at discretion of user, correlated. It also contains some features of but may not always converge an exchangeable correlation structure, in that, for nonconsecutive months, the order of mea- Allows flexibility specifying surements does not affect the correlation. Our complicated Ci choice of values for this model was influenced by the fitted values observed in the working correlation matrices of Model 1 and Model 2. The choice of correlation values for a fixed working correlation structure is at the discre- tion of the user. However, the parameter esti- mates are not guaranteed to converge for every choice of correlation values. In other words, the software package may not be able to pro- vide parameter estimates for a GEE model for some user-defined correlation structures. The use of a fixed correlation structure con- trasts with other correlation structures in that the working correlation matrix (Ci) does not result from fitting a model to the data, since the correlation values are all prespecified. How- ever, it does allow flexibility in the specification of more complicated correlation patterns.
548 15. GEE Examples Independent correlation structure: Next, we examine output from models that two models incorporate an independent correlation struc- Model 4. Uses empirical sb^; ture (Model 4 and Model 5). The key difference between Model 4 and a standard logistic f not fixed regression (Model 5) is that Model 4 uses the Model 5. Uses model-based sb^; empirical standard errors, whereas Model 5 uses the model-based standard errors. The f fixed at 1 other difference is that the scale factor is not preset equal to 1 in Model 4 as it is in Model 5. Sb affected These differences only affect the standard BUT errors of the regression coefficients rather than the estimates of the coefficients them- b not affected selves. Independent working correlation The working correlation matrix for an indepen- matrix dent correlation structure is the identity matrix – with a 1 for the diagonal entries and a 0 for COL1 COL2 . . . COL9 the other entries. The zeros indicate that the outcome measurements taken on the same ROW1 1.0000 0.0000 . . . 0.0000 subject are assumed uncorrelated. ROW2 0.0000 1.0000 . . . 0.0000 ROW3 0.0000 0.0000 . . . 0.0000 ROW4 0.0000 0.0000 . . . 0.0000 ROW5 0.0000 0.0000 . . . 0.0000 ROW6 0.0000 0.0000 . . . 0.0000 ROW7 0.0000 0.0000 . . . 0.0000 ROW8 0.0000 0.0000 . . . 0.0000 ROW9 0.0000 0.0000 . . . 1.0000 Measurements on same subject assumed uncorrelated. Model 4: Independent correlation The outputs for Model 4 and Model 5 (next structure page) are shown on the left. The corresponding coefficients for each model are identical as Empirical Std Wald expected. However, the estimated standard p- errors of the coefficients and the Variable Coefficient Err corresponding Wald test P-values differ for value the two models. INTERCEPT À1.4362 1.2272 0.2419 BIRTHWGT À0.0005 0.0003 0.1350 GENDER À0.0453 0.5526 0.9346 DIARRHEA 0.5857 0.1849 0.7764
Presentation: II. Example 1: Infant Care Study 549 Model 5: Standard logistic regres- sion (naive model) Model-based Wald Variable Coefficient Std Err p-value INTERCEPT À1.4362 0.6022 0.0171 BIRTHWGT À0.0005 0.0002 0.0051 GENDER À0.0453 0.2757 0.8694 DIARRHEA 0.4538 0.0871 0.7764 b^3 for DIARRHEA same but sb^ and In particular, the coefficient estimate for Wald P-values differ. DIARRHEA is 0.7764 in both Model 4 and Model 5; however, the standard error for DIAR- Model 4 vs. Model 5 RHEA is larger in Model 4 at 0.5857 compared Parameter estimates same with 0.4538 for Model 5. Consequently, the P- sb^ Model 4 > sb^ Model 5 values for the Wald test also differ: 0.1849 for Other data: possible that Model 4 and 0.0871 for Model 5. sb^ (empirical) < sb^ (model based) The other parameters in both models exhibit the same pattern, in that the coefficient esti- mates are the same, but the standard errors are larger for Model 4. In this example, the empiri- cal standard errors are larger than their model- based counterparts, but this does not always occur. With other data, the reverse can occur. Summary. Comparison of model A summary of the results for each model for results for DIARRHEA the variable DIARRHEA is presented on the left. Note that the choice of correlation struc- Correlation Odds 95% CI ture affects both the odds ratio estimates and structure ratio the standard errors, which in turn affects the width of the confidence intervals. The largest 1 AR(1) 1.25 (0.23, 6.68) odds ratio estimates are 2.17 from Model 4 and 2 Exchangeable 1.91 (0.44, 8.37) Model 5, which use an independent correlation 3 Fixed (user defined) 1.29 (0.26, 6.46) structure. The 95% confidence intervals for all 4 Independent 2.17 (0.69, 6.85) of the models are quite wide, with the tightest 5 Independent (SLR) 2.17 (0.89, 5.29) confidence interval (0.89, 5.29) occurring in Model 5, which is a standard logistic regres- sion. The confidence intervals for the odds ratio for DIARRHEA include the null value of 1.0 for all five models.
550 15. GEE Examples Impact of misspecification Typically, a misspecification of the correlation structure has a stronger impact on the stan- (usually) Sb dard errors than on the odds ratio estimates. In this example, however, there is quite a bit of OR variation in the odds ratio estimates across the five models (from 1.25 for Model 1 to 2.17 for For Models 1–5: Model 4 and Model 5). OdR range ¼ 1:25À3:39 OdR range suggests model This variation in odds ratio estimates suggests instability. a degree of model instability and a need for cautious interpretation of results. Such evi- Instability likely due to small num- dence of instability may not have been appar- ber (nine) of exposed cases. ent if only a single correlation structure had been examined. The reason the odds ratio var- ies as it does in this example is probably due to the relatively few infants who are exposed cases (n ¼ 9) for any of their nine monthly measurements. Which models to eliminate? It is easier to eliminate prospective models than to choose a definitive model. The working Models 4 and 5 (independent): correlation matrices of the first two models Evidence of correlated presented (AR1 autoregressive and exchange- observations able) suggest that there is a positive correlation between responses for the outcome variable. Therefore, an independent correlation struc- ture is probably not justified. This would elim- inate Model 4 and Model 5 from consideration. Model 2 (exchangeable): The exchangeable assumption for Model 2 may If autocorrelation suspected be less satisfactory in a longitudinal study if it is felt that there is autocorrelation in the responses. If so, that leaves Model 1 and Model 3 as the models of choice. Remaining models: similar results: Model 1 and Model 3 yield similar results, with an odds ratio and 95% confidence interval of Model 1 (AR1) 1.25 (0.23, 6.68) for Model 1 and 1.29 (0.26, OdRð95% CIÞ ¼ 1:25ð0:23; 6:68Þ 6.46) for Model 3. Recall that our choice of correlation values used in Model 3 was influ- Model 3 (fixed) enced by the working correlation matrices of OdR ð95% CIÞ ¼ 1:29ð0:26; 6:46Þ Model 1 and Model 2.
Presentation: III. Example 2: Aspirin–Heart Bypass Study 551 III. Example 2: Aspirin–Heart Bypass Study Data source: Gavaghan et al., 1991 The next example uses data from a study in Sydney, Australia, which examined the efficacy Subjects: 214 patients received up of aspirin for prevention of thrombotic graft to 6 coronary bypass grafts. occlusion after coronary bypass grafting (Gavaghan et al., 1991). Patients (K ¼ 214) Randomly assigned to treatment were given a variable number of artery bypasses (up to six) in a single operation, and group: if daily aspirin randomly assigned to take either aspirin (ASPI- 1 if daily placebo RIN ¼ 1) or a placebo (ASPIRIN ¼ 0) every day. One year later, angiograms were per- ASPIRIN ¼ formed to check each bypass for occlusion 0 (the outcome), which was classified as blocked (D ¼ 1) or unblocked (D ¼ 0). Additional cov- Response (D): Occlusion of a bypass ariates include AGE (in years), GENDER (1 ¼ male, 2 ¼ female), WEIGHT (in kilograms), graft 1 year later and HEIGHT (in centimeters). D¼ 1 if blocked 0 if unblocked Additional covariates: AGE (in years) GENDER (1 ¼ male, 2 ¼ female) WEIGHT (in kilograms) HEIGHT (in centimeters) Correlation structures to consider: In this study, there is no meaningful distinc- tion between artery bypass 1, artery bypass 2, Exchangeable or artery bypass 3 in the same subject. Since Independent the order of responses within a cluster is arbi- trary, we may consider using either the exchan- geable or independent correlation structure. Other correlation structures make use of an inherent order for the within-cluster responses (e.g., monthly measurements), so they are not appropriate here. Model 1: interaction model The first model considered (Model 1) allows for interaction between ASPIRIN and each of the Interaction terms between ASPIRIN other four covariates. The model can be stated and the other four covariates as shown on the left. included. logit PðD ¼ 1jXÞ ¼ b0 þ b1ASPIRIN þ b2AGE þ b3GENDER þ b4WEIGHT þ b5HEIGHT þ b6ASPIRIN Â AGE þ b7ASPIRIN Â GENDER þ b8ASPIRIN Â WEIGHT þ b9ASPIRIN Â HEIGHT
552 15. GEE Examples Exchangeable correlation structure Notice that the model contains a term for ASPI- RIN, terms for the four covariates, and four Wald product terms containing ASPIRIN. An exchangeable correlation structure is speci- Empirical Std p- fied. The parameter estimates are shown on the left. Variable Coefficient Err value The output can be used to estimate the odds INTERCEPT À1.1583 2.3950 0.6286 ratio for ASPIRIN ¼ 1 vs. ASPIRIN ¼ 0. If ASPIRIN 0.3934 3.2027 0.9022 interaction is assumed, then a different odds AGE 0.0118 0.3777 ratio estimate is allowed for each pattern of GENDER À0.0104 0.3216 0.0035 covariates where the covariates interacting WEIGHT À0.9377 0.0088 0.4939 with ASPIRIN change values. HEIGHT 0.0151 0.4421 ASPIRIN 0.0061 0.0185 0.7087 0.0116  AGE 0.0069 0.5848 0.0926 ASPIRIN 0.9836 0.0137 0.2848  GENDER ASPIRIN À0.0147 0.0218 0.6225  WEIGHT À0.0107 ASPIRIN  HEIGHT Odds ratio(ASPIRIN ¼ 1 vs. ASPIRIN ¼ 0) The odds ratio estimates can be obtained by separately inserting the values ASPIRIN ¼ 1 odds ¼ expðb0 þ b1ASPIRIN þ b2AGE and ASPIRIN ¼ 0 in the expression of the þ b3GENDER þ b4WEIGHT odds shown on the left and then dividing one þ b5HEIGHT þ b6ASPIRIN  AGE odds by the other. þ b7ASPIRIN  GENDER þ b8ASPIRIN  WEIGHT þ b9ASPIRIN  HEIGHTÞ This yields the expression for the odds ratio, Separate OR for each pattern of also shown on the left. covariates: OR ¼ expðb1 þ b6AGE þ b7GENDER þ b8WEIGHT þ b9HEIGHTÞ AGE ¼ 60, GENDER ¼ 1, WEIGHT The odds ratio (comparing ASPIRIN status) for ¼ 75 kg, HEIGHT ¼ 170 cm a 60-year-old male who weighs 75 kg and is OdRASPIRIN ¼ 1 vs: ASPIRIN ¼ 0Þ 170 cm tall can be estimated using the output as 0.32. ¼ exp½0:3934 þ ð0:0069Þð60Þ þ ð0:9836Þð1Þ þ ðÀ0:0147Þð75Þ þ ðÀ0:0107Þð170Þ ¼ 0:32 Chunk test A chunk test can be performed to determine if H0: b6 ¼ b7 ¼ b8 ¼ b9 ¼ 0 the four product terms can be dropped from the model. The null hypothesis is that the betas for the interaction terms are all equal to zero. Likelihood ratio test Recall for a standard logistic regression that for GEE models the likelihood ratio test can be used to simulta- neously test the statistical significance of sev- eral parameters. For GEE models, however, a likelihood is never formulated, which means that the likelihood ratio test cannot be used.
Presentation: III. Example 2: Aspirin–Heart Bypass Study 553 Two tests: There are two other statistical tests that can be utilized for GEE models. These are the Score test generalized Score test and the generalized Generalized Wald test Wald test. The test statistic for the Score test relies on the “score-like” generalized estimat- Under H0, both test statistics ing equations that are solved to produce the approximate w2 with parameter estimates for the GEE model (see df ¼ # of parameters tested. Chap. 14). The test statistic for the generalized Wald test generalizes the Wald test statistic for a single parameter by utilizing the variance– covariance matrix of the parameter estimates. The test statistics for both the Score test and the generalized Wald test follow an approxi- mate chi-square distribution under the null with the degrees of freedom equal to the num- ber of parameters that are tested. Chunk test for interaction terms: The output for the Score test and the generalized Wald test for the four interaction Type DF Chi-square P-value terms is shown on the left. The test statistic for the Score test is 3.66 with the corresponding Score 4 3.66 0.4544 p-value at 0.45. The generalized Wald test Wald 4 3.53 0.4737 yields similar results, as the test statistic is 3.53 with the p-value at 0.47. Both tests indi- Both tests fail to reject H0. cate that the null hypothesis should not be rejected and suggest that a model without the interaction terms may be appropriate. Model 2: No interaction model The no interaction model (Model 2) is pre- (GEE) sented at left. The GEE parameter estimates using the exchangeable correlation structure logit PðD ¼ 1jXÞ are also shown. ¼ b0 þ b1ASPIRIN þ b2AGE þ b3GENDER þ b4WEIGHT þ b5HEIGHT Model 2 Output (Exchangeable) Empirical Wald Variable Coefficient Std Err p-value INTERCEPT À0.4713 1.6169 0.7707 ASPIRIN À1.3302 0.1444 0.0001 AGE À0.0086 0.0087 0.3231 GENDER À0.5503 0.2559 0.0315 WEIGHT À0.0007 0.0066 0.9200 HEIGHT 0.0105 0.4448 0.0080
554 15. GEE Examples Odds ratio The odds ratio for aspirin use is estimated at OdRASPIRIN ¼ 1 vs: ASPIRIN ¼ 0 ¼ expðÀ1:3302Þ exp(À1.3302) ¼ 0.264, which suggests that aspirin is a preventive factor toward throm- ¼ 0:264 botic graft occlusion after coronary bypass grafting. Wald test The Wald test can be used for testing the hypothesis H0: b1 ¼ 0. The value of the z test H0: b1 ¼ 0 statistic is À9.21. The P-value of 0.0001 indi- cates that the coefficient for ASPIRIN is statis- Z ¼ À1:3302 ¼ À9:21; P ¼ 0:0001 tically significant. 0:1444 Score test Alternatively, the Score test can be used to test the hypothesis H0: b1 ¼ 0. The value of the chi- H0: b1 ¼ 0 square test statistic is 65.34 yielding a similar Chi-square ¼ 65.84, P ¼ 0.0001 statistically significant P-value of 0.0001. Note: Note, however, that the w2 version of the Wald Z2 ¼ (À9.21)2 ¼ 84.82, text (i.e., Z2) differs from the Score statistic. so Wald 6¼ Score Exchangeable working correlation The correlation parameter estimate obtained matrix from the working correlation matrix is À0.0954, which suggests a negative association COL1 COL2 . . . COL6 between reocclusion of different arteries from the same bypass patient compared with reoc- ROW1 1.0000 À0.0954 . . . À0.0954 clusions from different patients. ROW2 À0.0954 1.0000 . . . À0.0954 ROW3 À0.0954 À0.0954 . . . À0.0954 ROW4 À0.0954 À0.0954 . . . À0.0954 ROW5 À0.0954 À0.0954 . . . À0.0954 ROW6 À0.0954 À0.0954 . . . 1.0000 r^ ¼ À0:0954 Model 3: SLR (naive model) The output for a standard logistic regression (SLR) is presented on the left for comparison Model-based Wald with the corresponding GEE models. The parameter estimates for the standard logistic Variable Coefficient Std Err p-value regression are similar to those obtained from the GEE model, although their standard errors INTERCEPT À0.3741 2.0300 0.8538 are slightly larger. ASPIRIN À1.3410 0.1676 0.0001 AGE À0.0090 0.0109 0.4108 GENDER À0.5194 0.3036 0.0871 WEIGHT À0.0013 0.0088 0.8819 HEIGHT 0.0133 0.5580 SCALE 0.0078 0.0000 1.0000
Presentation: IV. Example 3: Heartburn Relief Study 555 Comparison of model results for A comparison of the odds ratio estimates with ASPIRIN 95% confidence intervals for the no-interaction models of both the GEE model and SLR is Correlation Odds shown on the left. The odds ratio estimates structure ratio 95% CI and 95% confidence intervals are very similar. This is not surprising, since only a modest Exchangeable 0.26 (0.20, 0.35) amount of correlation is detected in the work- (GEE) 0.26 (0.19, 0.36) ing correlation matrix ðr^ ¼ À0:0954Þ. Independent (SLR) In this example, predictor values In this example, none of the predictor variables did not vary within a cluster. (ASPIRIN, AGE, GENDER, WEIGHT, or HEIGHT) had values that varied within a clus- ter. This contrasts with the data used for the next example in which the exposure variable of interest is a time-dependent variable. IV. Example 3: Heartburn Relief Study Data source: Fictitious crossover The final dataset discussed is a fictitious cross- study on heartburn relief. over study on heartburn relief in which 40 sub- jects are given two symptom-provoking meals Subjects: 40 patients; 2 symptom- spaced a week apart. Each subject is adminis- provoking meals each; 1 of 2 tered an active treatment for heartburn treatments in random order (RX ¼ 1) following one of the meals and a stan- dard treatment (RX ¼ 0) following the other meal in random order. The dichotomous out- Treatment ðRXÞ ¼ 1 if active RX come is relief from heartburn, determined from a questionnaire completed 2 hours after 0 if standard RX each meal. Response (D): Relief from symptoms after 2 hours if yes D¼ 1 if no 0 Each subject has two observations There are two observations recorded for each subject: one for the active treatment and the RX ¼ 1 other for the standard treatment. The variable RX ¼ 0 indicating treatment status (RX) is a time- dependent variable since it can change values RX is time dependent: values within a cluster (subject). In fact, due to the change for each subject (cluster) design of the study, RX changes values in every cluster.
556 15. GEE Examples Model 1 For this analysis, RX is the only independent variable considered. The model is stated as logit PðD ¼ 1jXÞ ¼ b0 þ b1RX shown on the left. With exactly two observa- ni ¼ 2 : ½AR1; exchangeable; tions per subject, the only correlation to con- sider is the correlation between the two or unstructured responses for the same subject. Thus, there is only one estimated correlation parameter, ) same 2 Â2 Ci which is the same for each cluster. As a result, \" # using an AR1, exchangeable, or unstructured 1r correlation structure yields the same 2 Â 2 Ci ¼ working correlation matrix (Ci). r1 Exchangeable correlation structure The output for a GEE model with an exchange- able correlation structure is presented on the Empirical Wald p- left. Variable Coefficient Std Err value INTERCEPT À0.2007 0.3178 0.5278 RX 0.3008 0.3868 0.4368 Scale 1.0127 Á Á OdR ¼ expð0:3008Þ ¼ 1:35 The odds ratio estimate for the effect of 95% CI ¼ ð0:63; 2:88Þ treatment for relieving heartburn is exp(0.3008) ¼ 1.35 with the 95% confidence Exchangeable Ci COL2 interval of (0.63, 2.88). The working correlation matrix shows that the correlation between COL1 0.2634 responses from the same subject is estimated 1.0000 at 0.2634. ROW1 1.0000 ROW2 0.2634 SLR (naive) model A standard logistic regression is presented for comparison. The odds ratio estimate at Model- Wald exp(0.3008) ¼ 1.35 is exactly the same as was obtained from the GEE model with the Variable Coefficient based Std Err p-value exchangeable correlation structure; however, the standard error is larger, yielding a larger INTERCEPT À0.2007 0.3178 0.5278 95% confidence interval of (0.56, 3.25). RX 0.3008 0.4486 0.5826 Although an odds ratio of 1.35 suggests that Scale 1.0000 the active treatment provides greater relief for Á Á heartburn, the null value of 1.00 is contained in the 95% confidence intervals for both models. OdR ¼ expð0:3008Þ ¼ 1:35 95% CI ¼ ð0:56; 3:25Þ
V. SUMMARY Presentation: V. Summary 557 ü Chapter 15: GEE Examples These examples illustrate the GEE approach for modeling data containing correlated dichotomous outcomes. However, use of the GEE approach is not restricted to dichotomous outcomes. As an extension of GLM, the GEE approach can be used to model other types of outcomes, such as count or continuous out- comes. This presentation is now complete. The focus of the presentation was on several examples used to illustrate the application and interpre- tation of the GEE modeling approach. The examples show that the selection of different correlation structures for a GEE model applied to the same data can produce differ- ent estimates for regression parameters and their standard errors. In addition, we show that the application of a standard logistic regression model to data with correlated responses may lead to incorrect inferences. We suggest that you review the material cov- ered here by reading the detailed outline that follows. Then, do the practice exercises and test. Chapter 16: Other Approaches to The GEE approach to correlated data has been Analysis of Correlated Data used extensively. Other approaches to the anal- ysis of correlated data are available. A brief overview of several of these approaches is pre- sented in the next chapter.
558 15. GEE Examples Detailed I. Overview (page 542) Outline II–IV. Examples (pages 542–557) A. Three examples were presented in detail: i. Infant Care Study ii. Aspirin–Heart Bypass Study iii. Heartburn Relief Study B. Key points from the examples: i. The choice of correlation structure may affect both the coefficient estimate and the standard error of the estimate, although standard errors are more commonly impacted ii. Which correlation structure(s) should be specified depends on the underlying assumptions regarding the relationship between responses (e.g., ordering or time interval) iii. Interpretation of regression coefficients (in terms of odds ratios) is the same as in standard logistic regression V. Summary (page 557)
Practice Exercises 559 Practice The following printout summarizes the computer output Exercises from a GEE model run on the Infant Care Study data and should be used for Exercises 1–4. Recall that the data contained monthly information for each infant up to 9 months. The logit form of the model can be stated as follows: logit PðXÞ ¼ b0 þ b1BIRTHWGT þ b2GENDER þ b3DIARRHEA: The dichotomous outcome is derived from a weight-for- height z-score. The independent variables are BIRTHWGT (the weight in grams at birth), GENDER (1 ¼ male, 2 ¼ female), and DIARRHEA (a dichotomous variable indicat- ing whether the infant had symptoms of diarrhea that month; coded 1 ¼ yes, 0 ¼ no). A stationary 4-dependent correlation structure is specified for this model. Empirical and model-based standard errors are given for each regression parameter estimate. The working correlation matrix is also included in the output. Empirical Std Model-based Std Err Variable Coefficient Err 0.8747 INTERCEPT À2.0521 1.2323 0.0002 BIRTHWGT À0.0005 0.0003 0.3744 GENDER 0.5472 0.2841 DIARRHEA 0.5514 0.8722 0.1636 Stationary 4-Dependent Working Correlation Matrix COL1 COL2 COL3 COL4 COL5 COL6 COL7 COL8 COL9 ROW1 1.0000 0.5449 0.4353 0.4722 0.5334 0.0000 0.0000 0.0000 0.0000 ROW2 0.5449 1.0000 0.5449 0.4353 0.4722 0.5334 0.0000 0.0000 0.0000 ROW3 0.4353 0.5449 1.0000 0.5449 0.4353 0.4722 0.5334 0.0000 0.0000 ROW4 0.4722 0.4353 0.5449 1.0000 0.5449 0.4353 0.4722 0.5334 0.0000 ROW5 0.5334 0.4722 0.4353 0.5449 1.0000 0.5449 0.4353 0.4722 0.5334 ROW6 0.0000 0.5334 0.4722 0.4353 0.5449 1.0000 0.5449 0.4353 0.4722 ROW7 0.0000 0.0000 0.5334 0.4722 0.4353 0.5449 1.0000 0.5449 0.4353 ROW8 0.0000 0.0000 0.0000 0.5334 0.4722 0.4353 0.5449 1.0000 0.5449 ROW9 0.0000 0.0000 0.0000 0.0000 0.5334 0.4722 0.4353 0.5449 1.0000
560 15. GEE Examples 1. Explain the underlying assumptions of a stationary 4-dependent correlation structure as it pertains to the Infant Care Study. 2. Estimate the odds ratio and 95% confidence interval for the variable DIARRHEA (1 vs. 0) on a low weight- for-height z-score (i.e., outcome ¼ 1). Compute the 95% confidence interval in two ways: first using the empirical standard errors and then using the model- based standard errors. 3. Referring to Exercise 2: Explain the circumstances in which the model-based variance estimators yield con- sistent estimates. 4. Referring again to Exercise 2: Which estimate of the 95% confidence interval do you prefer? The following output should be used for Exercises 5–10 and contains the results from running the same GEE model on the Infant Care data as in the previous questions, except that in this case, a stationary 8-dependent correla- tion structure is specified. The working correlation matrix for this model is included in the output. Variable Coefficient Empirical Std Err INTERCEPT À1.4430 BIRTHWGT À0.0005 1.2084 GENDER 0.0003 DIARRHEA 0.0014 0.5418 0.3601 0.8122 Stationary 8-Dependent Working Correlation Matrix COL1 COL2 COL3 COL4 COL5 COL6 COL7 COL8 COL9 ROW1 1.0000 0.5255 0.3951 0.4367 0.4851 0.3514 0.3507 0.4346 0.5408 ROW2 0.5255 1.0000 0.5255 0.3951 0.4367 0.4851 0.3514 0.3507 0.4346 ROW3 0.3951 0.5255 1.0000 0.5255 0.3951 0.4367 0.4851 0.3514 0.3507 ROW4 0.4367 0.3951 0.5255 1.0000 0.5255 0.3951 0.4367 0.4851 0.3514 ROW5 0.4851 0.4367 0.3951 0.5255 1.0000 0.5255 0.3951 0.4367 0.4851 ROW6 0.3514 0.4851 0.4367 0.3951 0.5255 1.0000 0.5255 0.3951 0.4367 ROW7 0.3507 0.3514 0.4851 0.4367 0.3951 0.5255 1.0000 0.5255 0.3951 ROW8 0.4346 0.3507 0.3514 0.4851 0.4367 0.3951 0.5255 1.0000 0.5255 ROW9 0.5408 0.4346 0.3507 0.3514 0.4851 0.4367 0.3951 0.5255 1.0000 5. Compare the underlying assumptions of the stationary 8-dependent correlation structure with the unstruc- tured correlation structure as it pertains to this model. 6. For the Infant Care data, how many more correlation parameters would be included in a model that uses an unstructured correlation structure rather than a sta- tionary 8-dependent correlation structure?
Practice Exercises 561 7. How can the unstructured correlation structure be used to assess assumptions underlying other more constrained correlation structures? 8. Estimate the odds ratio and 95% confidence interval for DIARRHEA (1 vs. 0) using the model with the stationary 8-dependent working correlation structure. 9. If the GEE approach yields consistent estimates of the “true odds ratio” even if the correlation structure is misspecified, why are the odds ratio estimates differ- ent using a stationary 4-dependent correlation struc- ture (Exercise 2) and a stationary 8-dependent correlation structure (Exercise 8). 10. Suppose that a parameter estimate obtained from running a GEE model on a correlated data set was not affected by the choice of correlation structure. Would the corresponding Wald test statistic also be unaffected by the choice of correlation structure?
562 15. GEE Examples Test Questions 1–6 refer to models run on the data from the Heartburn Relief Study (discussed in Sect. IV). In that study, 40 subjects were given two symptom-provoking meals spaced a week apart. Each subject was administered an active treatment following one of the meals and a stan- dard treatment following the other meal, in random order. The goal of the study was to compare the effects of an active treatment for heartburn with a standard treatment. The dichotomous outcome is relief from heartburn (coded 1 ¼ yes, 0 ¼ no). The exposure of interest is RX (coded 1 ¼ active treatment, 0 ¼ standard treatment). Addition- ally, it was hypothesized that the sequence in which each subject received the active and standard treatment could be related to the outcome. Moreover, it was speculated that the treatment sequence could be an effect modifier for the association between the treatment and heartburn relief. Consequently, two other variables are considered for the analysis: a dichotomous variable SEQUENCE and the product term RX*SEQ (RX times SEQUENCE). The vari- able SEQUENCE is coded 1 for subjects in which the active treatment was administered first and 0 for subjects in which the standard treatment was administered first. The following printout summarizes the computer output for three GEE models run on the heartburn relief data (Model 1, Model 2, and Model 3). An exchangeable corre- lation structure is specified for each of these models. The variance–covariance matrix for the parameter estimates and the Score test for the variable RX*SEQ are included in the output for Model 1. Model 1 Coefficient Empirical Variable Std Err INTERCEPT À0.6190 0.4688 0.4184 0.5885 RX 0.8197 0.6495 0.7993 SEQUENCE À0.2136 RX*SEQ Empirical Variance Covariance Matrix RX*SEQ For Parameter Estimates INTERCEPT RX SEQUENCE INTERCEPT 0.2198 À0.1820 À0.2198 0.1820 À0.1820 0.3463 0.1820 À0.3463 RX À0.2198 0.1820 0.4218 À0.3251 SEQUENCE 0.1820 À0.3463 À0.3251 0.6388 RX*SEQ Score test statistic for RX*SEQ ¼ 0.07
Model 2 Coefficient Test 563 À0.5625 Variable 0.3104 Empirical INTERCEPT 0.7118 Std Err RX 0.4058 SEQUENCE Coefficient 0.3992 À0.2007 0.5060 Model 3 0.3008 Empirical Variable Std Err INTERCEPT 0.3178 RX 0.3868 1. State the logit form of the model for Model 1, Model 2, and Model 3. 2. Use Model 1 to estimate the odds ratios and 95% confi- dence intervals for RX (active vs. standard treatment). Hint. Make use of the variance–covariance matrix for the parameter estimates. 3. In Model 1, what is the difference between the working covariance matrix and the covariance matrix for parameter estimates used to obtain the 95% confidence interval in the previous question? 4. Use Model 1 to perform the Wald test on the interaction term RX*SEQ at a 0.05 level of significance. 5. Use Model 1 to perform the Score test on the interac- tion term RX*SEQ at a 0.05 level of significance. 6. Estimate the odds ratio for RX using Model 2 and Model 3. Is there a suggestion that SEQUENCE is con- founding the association between RX and heartburn relief. Answer this question from a data-based perspec- tive (i.e., comparing the odds ratios) and a theoretical perspective (i.e., what it means to be a confounder).
564 15. GEE Examples Answers to 1. The stationary 4-dependent working correlation struc- Practice Exercises ture uses four correlation parameters (a1, a2, a3, and a4). The correlation between responses from the same infant 1 month apart is a1. The correlation between responses from the same infant 2, 3, or 4 months apart is a2, a3, and a4, respectively. The correlation between responses from the same infant more than 4 months apart is assumed to be 0. 2. Estimated OR ¼ exp(0.1636) ¼ 1.18. 95% CI (with empirical SE): exp[0.1636 Æ 1.96(0.8722)] ¼ (0.21, 6.51); 95% CI (with model-based SE): exp[0.1636 Æ 1.96(0.2841)] ¼ (0.67, 2.06). 3. The model-based variance estimator would be a con- sistent estimator if the true correlation structure was stationary 4-dependent. In general, model-based vari- ance estimators are more efficient {i.e., smaller var½vdarðb^Þ} if the correlation structure is correctly specified. 4. The 95% confidence interval with the empirical stan- dard errors is preferred since we cannot be confident that the true correlation structure is stationary 4-dependent. 5. The stationary 8-dependent correlation structure uses eight correlation parameters. With nine monthly responses per infant, each correlation parameter represents the correlation for a specific time interval between responses. The unstructured correlation structure, on the other hand, uses a different correla- tion parameter for each possible correlation for a given infant, yielding 36 correlation parameters. With the stationary 8-dependent correlation struc- ture, the correlation between an infant’s month 1 response and month 7 response is assumed to equal the correlation between an infant’s month 2 response and month 8 response since the time interval between responses is the same (i.e., 6 months). The unstruc- tured correlation structure does not make this assumption, using a different correlation parameter even if the time interval is the same. 6. There are ð9Þð8Þ ¼ 36 correlation parameters using the 2 unstructured correlation structure on the infant care data and 8 parameters using the stationary 8-depen- dent correlation structure. The difference is 28 corre- lation parameters. 7. By examining the correlation estimates in the unstruc- tured working correlation matrix, we can evaluate which alternate, but more constrained, correlation structures seem reasonable. For example, if the
Answers to Practice Exercises 565 correlations are all similar, this would suggest that an exchangeable structure is reasonable. 8. Estimated OR ¼ exp(0.3601) ¼ 1.43. 95% CI: exp[0.3601 Æ 1.96(0.8122)] ¼ (0.29, 7.04). 9. Consistency is an asymptotic property. As the number of clusters approaches infinity, the odds ratio estimate should approach the true odds ratio even if the corre- lation structure is misspecified. However, with a finite sample, the parameter estimate may still differ from the true parameter value. The fact that the parameter estimate for DIARRHEA is so sensitive to the choice of the working correlation structure demonstrates a degree of model instability. 10. No, because the Wald test statistic is a function of both the parameter estimate and its variance. Since the variance is typically affected by the choice of cor- relation structure, the Wald test statistic would also be affected.
16 Other Approaches for Analysis of Correlated Data n Contents Introduction 568 Abbreviated Outline 568 Objectives 569 597 Presentation 570 Detailed Outline 589 Practice Exercises 591 Test 595 Answers to Practice Exercises D.G. Kleinbaum and M. Klein, Logistic Regression, Statistics for Biology and Health, 567 DOI 10.1007/978-1-4419-1742-3_16, # Springer ScienceþBusiness Media, LLC 2010
568 16. Other Approaches for Analysis of Correlated Data Introduction In this chapter, the discussion of methods to analyze out- come variables that have dichotomous correlated res- Abbreviated ponses is expanded to include approaches other than Outline GEE. Three other analytic approaches are discussed. These include the alternating logistic regressions algo- rithm, conditional logistic regression, and the generalized linear mixed model approach. The outline below gives the user a preview of the material to be covered by the presentation. A detailed outline for review purposes follows the presentation. I. Overview (page 570) II. Alternating logistic regressions algorithm (pages 571–575) III. Conditional logistic regression (pages 575–579) IV. The generalized linear mixed model approach (pages 579–587) V. Summary (page 588)
Objectives Objectives 569 Upon completing this chapter, the learner should be able to: 1. Contrast the ALR method to GEE with respect to how within-cluster associations are modeled. 2. Recognize how a conditional logistic regression model can be used to handle subject-specific effects. 3. Recognize a generalized linear mixed (logistic) model. 4. Distinguish between random and fixed effects. 5. Contrast the interpretation of an odds ratio obtained from a marginal model with one obtained from a model containing subject-specific effects.
570 16. Other Approaches for Analysis of Correlated Data Presentation I. Overview In this chapter, we provide an introduction to modeling techniques other than GEE for use with dichotomous outcomes in which the responses are correlated. FOCUS Other approaches to modeling outcomes with dichotomous correlated responses Other approaches for correlated In addition to the GEE approach, there are a data: number of alternative approaches that can be applied to model correlated data. These 1. Alternating logistic regressions include (1) the alternating logistic regressions (ALR) algorithm algorithm, which uses odds ratios instead of correlations, (2) conditional logistic regres- 2. Conditional logistic regression sion, and (3) the generalized linear mixed 3. Generalized linear mixed model approach, which allows for random effects in addition to fixed effects. We briefly model describe each of these approaches. This chapter is not intended to provide a thor- ough exposition of these other approaches but rather an overview, along with illustrative exam- ples, of other ways to handle the problem of analyzing correlated dichotomous responses. Some of the concepts that are introduced in this presentation are elaborated in the Practice Exercises at the end of the chapter. Conditional logistic regression has previously been presented in Chap. 11 but is presented here in a some-what different context. The alternating logistic regression and generalized linear mixed model approaches for analyzing correlated dichotomous responses show great promise but at this point have not been fully investigated with regard to numerical estima- tion and possible biases.
Presentation: II. The Alternating Logistic Regressions Algorithm 571 II. The Alternating Logistic Regressions Algorithm Modeling associations: The alternating logistic regressions (ALR) algo- rithm is an analytic approach that can be used GEE approach ALR approach to model correlated data with dichotomous out- comes (Carey et al., 1993; Lipsitz et al., 1991). correlations ðrsÞ odds ratios ðORsÞ This approach is very similar to that of GEE. What distinguishes the two approaches is that ORijk ¼ PðYij ¼ 1; Yik ¼ 1ÞPðYij ¼ 0; Yik ¼ 0Þ with the GEE approach, associations between PðYij ¼ 1; Yik ¼ 0ÞPðYij ¼ 0; Yik ¼ 1Þ pairs of outcome measures are modeled with correlations, whereas with ALR, they are mod- eled with odds ratios. The odds ratio (ORijk) between the jth and kth responses for the ith subject can be expressed as shown on the left. GEE: as and bs estimated by alter- Recall that in a GEE model, the correlation nately updating estimates until parameters (a) are estimated using estimates convergence of the regression parameters (b). The regres- sion parameter estimates are, in turn, updated using estimates of the correlation parameters. The computational process alternately updates the estimates of the alphas and then the betas until convergence is achieved. ALR: as and bs estimated similarly The ALR approach works in a similar manner, BUT except that the alpha parameters are log odds ratio parameters rather than correlation para- ALR : a are log ORs meters. Moreover, for the same data, an odds ðGEE : a are rsÞ ratio between the jth and kth responses that is greater than 1 using an ALR model corre- ALR ORjk > 1 , GEE rjk > 1 sponds to a positive correlation between the ALR ORjk < 1 , GEE rjk < 1 jth and kth responses using a GEE model. Sim- ilarly, an odds ratio less than 1 using an ALR model corresponds to a negative correlation between responses. Same OR can correspond to differ- ent rs OR jk rajk However, the correspondence is not one-to- rbjk one, and examples can be constructed in which the same odds ratio corresponds to dif- ferent correlations (see Practice Exercises 1–3).
572 16. Other Approaches for Analysis of Correlated Data ALR: dichotomous outcomes only For many health scientists, an odds ratio mea- GEE: dichotomous and other out- sure, such as that provided with an ALR model, comes are allowed is more familiar and easier to interpret than a correlation measure. However, ALR models can only be used if the outcome is a dichoto- mous variable. In contrast, GEE models are not so restrictive. EXAMPLE The ALR model is illustrated by returning to the Aspirin–Heart Bypass Study example, which GEE vs. ALR was first presented in Chap. 15. Recall that in that study, researchers examined the efficacy of Aspirin–Heart Bypass Study aspirin for prevention of thrombotic graft (Gavaghan et al., 1991) occlusion after coronary bypass grafting in a sample of 214 patients (Gavaghan et al., 1991). Subjects: received up to six coronary Patients were given a variable number of artery bypass grafts bypasses (up to six) and randomly assigned to take either aspirin (ASPIRIN ¼ 1) or a placebo Randomly assigned to treatment (ASPIRIN ¼ 0) every day. One year later, each bypass was checked for occlusion and the out- group: 8 come was coded as blocked (D ¼ 1) or unblocked < 1 if daily aspirin (D ¼ 0). Additional covariates included AGE (in years), GENDER (1 ¼ male, 2 ¼ female), ASPIRIN ¼ : 0 if daily placebo WEIGHT (in kilograms), and HEIGHT (in centimeters). Response (D): occlusion of a bypass graft 1 year later. ( 1 if blocked D¼ 0 if unblocked Additional covariates: Consider the model presented at left, with ASPIRIN, AGE, GENDER, WEIGHT, and AGE (in years) HEIGHT as covariates. GENDER (1 ¼ male, 2 ¼ female) WEIGHT (in kilograms) HEIGHT (in centimeters) Model: logit PðXÞ ¼ b0 þ b1ASPIRIN þ b2AGE þ b3GENDER þ b4WEIGHT þ b5HEIGHT
Presentation: II. The Alternating Logistic Regressions Algorithm 573 EXAMPLE (continued) Output from using the GEE approach is pre- sented on the left. An exchangeable correlation GEE Approach (Exchangeable r) structure is assumed. (This GEE output has previously been presented in Chap. 15) Empirical z Wald Variable Coefficient Std Err p-value The correlation parameter estimate obtained from the working correlation matrix of the INTERCEPT À0.4713 1.6169 0.7707 GEE model is À0.0954, which suggests a nega- 0.1444 0.0001 tive association between reocclusions on the ASPIRIN À1.3302 0.0087 0.3231 same bypass patient. 0.2559 0.0315 AGE À0.0086 0.0066 0.9200 0.0105 0.4448 GENDER À0.5503 WEIGHT À0.0007 HEIGHT 0.0080 Scale 1.0076 Exchangeable Ci (GEE: r = –0.0954) ALR approach (Exchangeable OR) Output obtained from SAS PROC GENMOD using the ALR approach is shown on the left Empirical z wald for comparison. An exchangeable odds ratio Variable Coefficient Std Err p-value structure is assumed. The assumption underly- ing the exchangeable odds ratio structure is INTERCEPT À0.4806 1.6738 0.7740 that the odds ratio between the ith subject’s ASPIRIN À1.3253 0.1444 0.0001 jth and kth responses is the same (for all j and AGE À0.0086 0.0088 0.3311 k, j ¼6 k). The estimated exchangeable odds GENDER À0.5741 0.2572 0.0256 ratio is obtained by exponentiating the coeffi- WEIGHT À0.0003 0.0066 0.9665 cient labeled ALPHA1. HEIGHT 0.0108 0.4761 ALPHA1 0.0077 0.1217 0.0001 À0.4716 expðALPHA1Þ ¼ OcRjkðexchangeableÞ
574 16. Other Approaches for Analysis of Correlated Data EXAMPLE (continued) The regression parameter estimates are very Odds ratios similar for the two models. The odds ratio for OdR ASPIRIN ¼ 1 vs: ASPIRIN ¼ 0: aspirin use on artery reocclusion is estimated as exp(À1.3302) ¼ 0.264 using the GEE model GEE ! expðÀ1:3302Þ ¼ 0:264 and exp(À1.3253) ¼ 0.266 using the ALR ALR ! expðÀ1:3253Þ ¼ 0:266 model. The standard errors for the aspirin S.E. (Aspirin) ¼ 0.1444 (GEE and parameter estimates are the same in both mod- ALR) els (0.1444), although the standard errors for some of the other parameters are slightly larger Measure of association ðOdRjkÞ in the ALR model. OdRjk ¼ expðALPHA1Þ The corresponding measure of association ¼ expðÀ0:4716Þ ¼ 0:62 (the odds ratio) estimate from the ALR model (Negative association: can be found by exponentiating the coeffi- similar to r^ ¼ À0:0954 cient of ALPHA1. This odds ratio estimate is exp(À0.4716) ¼ 0.62. As with the estimated 95% CI for ALPHA1 exchangeable correlation ðr^Þ from the GEE ¼ exp½ðÀ0:4716 Æ 1:96ð0:1217Þ approach, the exchangeable OR estimate, ¼ ð0:49; 0:79Þ which is less than 1, also indicates a negative association between any pair of outcomes P-value ¼ 0.0001 (i.e., reocclusions on the same bypass patient). ) ALPHA1 significant A 95% confidence interval for the OR can be calculated as exp[À0.4716 Æ 1.96(0.1217)], which yields the confidence interval (0.49, 0.79). The P-value for the Wald test is also given in the output at 0.0001, indicating the statistical significance of the ALPHA1 parameter. SE? GEE (r) ALR (ALPHA1) For the GEE model output, an estimated stan- Test? No Yes dard error (SE) or statistical test is not given No Yes for the correlation estimate. This is in contrast to the ALR output, which provides a standard error and statistical test for ALPHA1.
Presentation: III. Conditional Logistic Regression 575 Key difference: GEE vs. ALR This points out a key difference in the GEE and ALR approaches. With the GEE approach, the GEE: rjk are typically nuisance correlation parameters are typically consid- parameters ered to be nuisance parameters, with the para- meters of interest being the regression ALR: ORjk are parameters of coefficients (e.g., ASPIRIN). In contrast, with interest the ALR approach, the association between different responses is also considered to be of ALR: allows inferences about both interest. Thus, the ALR approach allows statis- ^a and b^s tical inferences to be assessed from both the alpha parameter and the beta parameters (regression coefficients). III. Conditional Logistic Regression EXAMPLE Another approach that is applicable for certain types of correlated data is a matched analysis. Heartburn Relief Study This method can be applied to the Heartburn (“subject” as matching factor) Relief Study example, with “subject” used as the matching factor. This example was pre- 40 subjects received: sented in detail in Chap. 15. Recall that the dataset contained 40 subjects, each receiving Active treatment (“exposed”) an active or standard treatment for the relief of Standard treatment heartburn. In this framework, within each matched stratum (i.e., subject), there is an (“unexposed”) exposed observation (the active treatment) and an unexposed observation (the standard CLR model treatment). A conditional logistic regression (CLR) model, as discussed in Chap. 11, can 39 then be formulated to perform a matched anal- ysis. The model is shown on the left. logit PðXÞ ¼ b0 þ b1RX þ ~ giVi; This model differs from the GEE model for the i¼1 same data, also shown on the left, in that the conditional model contains 39 dummy variables where for subject i besides RX. Each of the parameters (gi) for the (1 otherwise 39 dummy variables represents the (fixed) effects for each of 39 subjects on the outcome. Vi ¼ The 40th subject acts as the reference group since all of the dummy variables have a value 0 of zero for the 40th subject (see Chap. 11). GEE model logit PðXÞ ¼ b0 þ b1RX CLR vs. GEE # # 39 Vi no Vi (dummy variables)
576 16. Other Approaches for Analysis of Correlated Data CLR approach ) When using the CLR approach for modeling responses assumed independent P(X), the responses from a specific subject are assumed to be independent. This may seem Subject-specific gi allows for conditioning by surprising since throughout this chapter we subject have viewed two or more responses on the same subject as likely to be correlated. Never- fixed effect theless, when dummy variables are used for each subject, each subject has his/her own Responses can be independent if subject-specific fixed effect included in the conditioned by subject model. The addition of these subject-specific fixed effects can account for correlation that may exist between responses from the same subject in a GEE model. In other words, responses can be independent if conditioned by subject. However, this is not always the case. For example, if the actual underlying correlation structure is autoregressive, condi- tioning by subject would not account for the within-subject autocorrelation. EXAMPLE (continued) Returning to the Heartburn Relief Study data, the output obtained from running the condi- Model 1: conditional logistic tional logistic regression is presented on the left. regression Std. Wald Variable Coefficient error P-value RX 0.4055 0.5271 0.4417 No b0 or gi estimates in CLR model With a conditional logistic regression, parame- (cancel out in conditional ter estimates are not obtained for the intercept likelihood) or the dummy variables representing the matched factor (i.e., subject). These para- Odds ratio and 95% CI meters cancel out in the expression for the OdR ¼ expð0:4055Þ ¼ 1:50 conditional likelihood. However, this is not a 95% CI ¼ ð0:534; 4:214Þ problem because the parameter of interest is the coefficient of the treatment variable (RX). The odds ratio estimate for the effect of treatment for relieving heartburn is exp (0.4055) ¼ 1.50, with a 95% confidence interval of (0.534, 4.214).
Presentation: III. Conditional Logistic Regression 577 EXAMPLE (continued) The estimated odds ratios and the standard errors for the parameter estimate for RX are Model comparison shown at left for the conditional logistic regres- sion (CLR) model, as well as for the GEE and Model OR sb^ standard logistic regression (SLR) discussed in Chap. 15. The odds ratio estimate for the CLR CLR 1.50 0.5271 model is somewhat larger than the estimate GEE 1.35 0.3868 obtained at 1.35 using the GEE approach. The SLR 1.35 0.4486 standard error for the RX coefficient estimate in the CLR model is also larger than what was obtained in either the GEE model using empir- ical standard errors or in the standard logistic regression, which uses model-based standard errors. Estimation of An important distinction between the CLR and predictors GEE analytic approaches concerns the treat- ment of the predictor (independent) variables Within- Between- in the analysis. A matched analysis (CLR) relies on within-subject variability (i.e., variability subject subject within the matched strata) for the estimation of its parameters. A correlated (GEE) analysis Analysis variability variability takes into account both within-subject varia- bility and between-subject variability. In fact, Matched (CLR) p p if there is no within-subject variability for an Correlated (GEE) p independent variable (e.g., a time-independent variable), then its coefficient cannot be esti- No within-subject variability for an mated using a conditional logistic regression. independent variable In that situation, the parameter cancels out in + the expression for the conditional likelihood. This is what occurs to the intercept as well as parameter will not be estimated to the coefficients of the matching factor using CLR dummy variables when CLR is used. EXAMPLE To illustrate the consequences of only includ- ing independent variables with no within- CLR with time-independent cluster variability in a CLR, we return to the predictors (Aspirin–Heart Bypass Aspirin–Heart Bypass Study discussed in the Study) previous section. Recall that patients were given a variable number of artery bypasses in Subjects: 214 patients received up to 6 a single operation and randomly assigned to coronary bypass grafts. either aspirin or placebo therapy. One year later, angiograms were performed to check Treatment: each bypass for reocclusion. 8 <> 1 if daily aspirin ASPIRIN ¼ >: 8 0 if daily placebo >< 1 if graft blocked D ¼ >: 0 if graft unblocked
578 16. Other Approaches for Analysis of Correlated Data EXAMPLE (continued) Besides ASPIRIN, additional covariates include AGE, GENDER, WEIGHT, and logit PðXÞ ¼ b0 þ b1ASPIRIN þ b2AGE HEIGHT. We restate the model from the previ- þ b3GENDER ous section at left, which also includes 213 þ b4WEIGHT dummy variables for the 214 study subjects. 213 The output from running a conditional logistic regression is presented on the left. Notice that þ b5HEIGHT þ ~ giVi all of the coefficient estimates are zero with their standard errors missing. This indicates i¼1 that the model did not execute. The problem occurred because none of the independent CLR model variables changed their values within any clus- ter (subject). In this situation, all of the predic- Standard Wald tor variables are said to be concordant in all the Variable Coefficient Error p-value matching strata and uninformative with respect to a matched analysis. Thus, the condi- AGE 0 .. tional logistic regression, in effect, discards all GENDER 0 .. of the data. WEIGHT 0 .. HEIGHT 0 .. ASPIRIN 0 .. All strata concordant ) model will not run Within-subject variability for one If at least one variable in the model does vary or more independent variable within a cluster (e.g., a time-dependent variable), then the model will run. However, estimated + coefficients will be obtained only for those vari- ables that have within-cluster variability. Model will run Parameters estimated for only those variables Matched analysis: An advantage of using a matched analysis with subject as the matching factor is the ability to Advantage: control of control for potential confounding factors that confounding factors can be difficult or impossible to measure. When the study subject is the matched vari- Disadvantage: cannot separate able, as in the Heartburn Relief example, effects of time-independent there is an implicit control of fixed genetic factors and environmental factors that comprise each subject. On the other hand, as the Aspirin– Heart bypass example illustrates, a disadvan- tage of this approach is that we cannot model the separate effects of fixed time-independent factors. In this analysis, we cannot examine the separate effects of aspirin use, gender, and height using a matched analysis, because the values for these variables do not vary for a given subject.
Presentation: III. Conditional Logistic Regression 579 Heartburn Relief Model: With the conditional logistic regression (Subject modeled as fixed effect) approach, subject is modeled as a fixed effect with the gamma parameters (g), as shown on the left for the Heartburn Relief example. logit PðXÞ ¼ b0 þ b1RX 39 þ ~ giVi; i¼1 where for subject i (1 otherwise Vi ¼ 0 Alternative approach: An alternative approach is to model subject as Subject modeled as random effect a random effect. What if study is replicated? To illustrate this concept, suppose we attempted to replicate the heartburn relief study using a Different sample different sample of 40 subjects. We might expect ) different subjects the estimate for b1, the coefficient for RX, to change due to sampling variability. However, b1 unchanged (fixed effect) the true value of b1 would remain unchanged g different (i.e., b1 is a fixed effect). In contrast, because there are different subjects in the replicated Parameters themselves may be study, the parameters representing subject random (not just their estimates) (i.e., the gammas) would therefore also be dif- ferent. This leads to an additional source of variability that is not considered in the CLR, in that some of the parameters themselves (and not just their estimates) are random. In the next section, we present an approach for modeling subject as a random effect, which takes into account that the subjects represent a random sample from a larger population.
580 16. Other Approaches for Analysis of Correlated Data IV. The Generalized Linear Mixed Model Approach Mixed models: The generalized linear mixed model (GLMM) provides another approach that can be used for Random effects correlated dichotomous outcomes. GLMM is a Fixed effects generalization of the linear mixed model. Cluster effect is random Mixed models refer to the mixing of random and fixed effects. With this approach, the clus- variable ter variable is considered a random effect. This means that the cluster effect is a random vari- able following a specified distribution (typi- cally a normal distribution). Mixed logistic model (MLM): A special case of the GLMM is the mixed logis- tic model (MLM). This type of model combines Special case of GLMM some of the features of the GEE approach and Combines GEE and CLR features some of the features of the conditional logistic regression approach. As with the GEE GEE CLR approach, the user specifies the logit link func- tion and a structure (Ci) for modeling response User specifies Subject-specific correlation. As with the conditional logistic g(m) and Ci effects regression approach, subject-specific effects are directly included in the model. However, GLMM: subject-specific effects random here these subject-specific effects are treated as random rather than fixed effects. The model is commonly stated in terms of the ith subject’s mean response (mi). EXAMPLE We again use the heartburn data to illustrate the model (shown on the left) and state it in Heartburn Relief Study terms of the ith subject’s mean response, which logit mi = b0 + b1RXi + b0i in this case is the ith subject’s probability of heartburn relief. The coefficient b1 is called a b1 = fixed effect fixed effect, whereas b0i is called a random b0i = random effect, effect. The random effect (b0i) in this model is where b0i is a random variable assumed to follow a normal distribution with mean 0 and variance sb0 2. Subject-specific ran- ~ N(0, sb02) dom effects are designed to account for the subject-to-subject variation, which may be due to unexplained genetic or environmental factors that are otherwise unaccounted for in the model. More generally, random effects are often used when levels of a variable are selected at random from a large population of possible levels.
Presentation: IV. The Generalized Linear Mixed Model Approach 581 For each subject: With this model, each subject has his/her own logit of baseline risk = (b0 + b0i) baseline risk, the logit of which is the intercept plus the random effect (b0 þ b0i). The sum b0i = subject-specific intercept (b0 þ b0i) is typically called the subject-specific No random effect for RX intercept. The amount of variation in the + baseline risk is determined by the variance ðsb0 2Þ of b0i. RX effect is same for each subject i.e., exp(b1) In addition to the intercept, we could have added another random effect allowing the treatment (RX) effect to also vary by subject (see Practice Exercises 4–9). By not adding this additional random effect, there is an assump- tion that the odds ratio for the effect of treat- ment is the same for each subject, exp(b1). Mixed logistic model (MLM) The output obtained from running the MLM on the heartburn data is presented on the left. Variable Coefficient Standard Wald This model was run using SAS’s GLIMMIX Error p-value procedure. (See the Computer Appendix for INTERCEPT À0.2285 details and an example of program coding.) RX 0.3445 0.3583 0.5274 0.4425 0.4410 Odds ratio and 95% CI: The odds ratio estimate for the effect of treat- OdR ¼ expð0:3445Þ ¼ 1:41 ment for relieving heartburn is exp(0.3445) ¼ 95% CI ¼ ð0:593; 3:360Þ 1.41. The 95% confidence interval is (0.593, 3.360). Model comparison The odds ratio estimate using this model is slightly larger than the estimate obtained Model OdR sb^ (1.35) using the GEE approach, but somewhat smaller than the estimate obtained (1.50) using MLM 1.41 0.4425 the conditional logistic regression approach. GEE 1.35 0.3868 The standard error at 0.4425 is also larger CLR 1.50 0.5271 than what was obtained in the GEE model (0.3868), but smaller than in the conditional logistic regression (0.5271).
582 16. Other Approaches for Analysis of Correlated Data Typical model for random Y: The modeling of any response variable typi- cally contains a fixed and random component. Fixed component (fixed effects) The random component, often called the error Random component (error) term, accounts for the variation in the response variables that the fixed predictors fail to explain. Random effects model: A model containing a random effect adds Fixed component (fixed effects) another layer to the random part of the Random components (random model. With a random effects model, there are at least two random components in the model: effects) 1. Random effects: b 1. The first random component is the variation explained by the random effects. VarðbÞ ¼ G For the heartburn data set, the random effect is designed to account for random 2. Residual variation: « subject-to-subject variation Varð«Þ ¼ R (heterogeneity). The variance–covariance matrix of this random component (b) is Random components layered: called the G matrix. 2. The second random component is the Yij = 1 + eij residual error variation. This is the 1+ exp variation unexplained by the rest of the p + model (i.e., unexplained by fixed or +Σ random effects). For a given subject, this is –b0 h=1 bhXhij boi the difference of the observed and expected response. The variance–covariance matrix random residual of this random component is called the R effects variation matrix. For mixed logistic models, the layering of these random components is tricky. This layering can be illustrated by presenting the model (see left side) in terms of the random effect for the ith subject (b0i) and the residual varia- tion (eij) for the jth response of the ith subject (Yij).
Presentation: IV. The Generalized Linear Mixed Model Approach 583 eij ¼ Yij À PðYij ¼ 1jXÞ; The residual variation (eij) accounts for the dif- ference between the observed value of Y and where the mean response, P(Y ¼ 1| X), for a given sub- PðYij ¼ 1jXÞ ¼ m ject. The random effect (b0i), on the other hand, allows for randomness in the modeling of ¼ \"1 p # the mean response [i.e., P(Y ¼ 1| X)], which is 1 þ exp Àðb0 þ ~ bhXhij þ b0iÞ modeled using both fixed (bs) and random (bs) effects. h¼1 Model GLM GEE For GLM and GEE models, the outcome Y is modeled as the sum of the mean and the resid- R Yi ¼ mi þ eij Yij ¼ mij þ eij ual variation [Y ¼ m þ eij, where the mean (m) G Independent Correlated is fixed] determined by the subject’s pattern of covariates. For GEE, the residual variation is — — modeled with a correlation structure, whereas for GLM, the residual variation (the R matrix) is modeled as independent. Neither GLM nor GEE models contain a G matrix, as they do not contain any random effects (b). GLMM: Yij ¼ g|fflÀfflffl1fflfflðfflfflXfflffl{;zbfflffl;fflfflbfflfflffl0fflffli}Þ þ eij In contrast, for GLMMs, the mean also con- tains a random component (b0i). With GLMM, mij the user can specify a covariance structure for the G matrix (for the random effects), the R User specifies covariance struc- matrix (for the residual variation), or both. tures for R, G, or both Even if the G and R matrices are modeled to contain zero correlations separately, the com- bination of both matrices in the model gener- ally forms a correlated random structure for the response variable. GEE: correlation structure speci- Another difference between a GEE model and a fied mixed model (e.g., MLM) is that a correlation structure is specified with a GEE model, GLMM: covariance structure spe- whereas a covariance structure is specified cified with a mixed model. A covariance structure con- tains parameters for both the variance and Covariance structure contains covariance, whereas a correlation structure con- parameters for both the variance tains parameters for just the correlation. Thus, and covariance with a covariance structure, there are additional variance parameters and relationships among those parameters (e.g., variance heterogeneity) to consider (see Practice Exercises 7–9).
584 16. Other Approaches for Analysis of Correlated Data Covariance Þ unique correlation If a covariance structure is specified, then the but correlation Þ unique covariance correlation structure can be ascertained. The reverse is not true, however, since a correlation matrix does not in itself determine a unique covariance matrix. For ith subject: For a given subject, the dimensions of the R matrix depend on how many observations (ni) R matrix dimensions depend the subject contributes, whereas the dimen- on number of observations for sions of the G matrix depend on the number subject i (ni) of random effects (e.g., q) included in the model. For the heartburn data example, in G matrix dimensions depend on which there are two observations per subject number of random effects (q) (ni ¼ 2), the R matrix is a 2 Â 2 matrix modeled with zero correlation. The dimensions of G are Heartburn data: 1 Â 1 (q ¼ 1) since there is only one random effect (b0i), so there is no covariance structure R ¼ 2 Â 2 matrix to consider for G in this model. Nevertheless, the combination of the G and R matrix in this G ¼ 1 Â 1 matrix (only one model provides a way to account for correla- random effect) tion between responses from the same subject. CLR vs. MLM: subject-specific We can compare the modeling of subject- effects specific fixed effects with subject-specific ran- dom effects by examining the conditional logis- CLR: logit mi ¼ b0 þ b1RX þ gi, tic model (CLR) and the mixed logistic model where gi is a fixed effect (MLM) in terms of the ith subject’s response. Using the heartburn data, these models can be MLM: logit mi ¼ b0 þ b1RX þ b0i, expressed as shown on the left. where b0i is a random effect Fixed effect gi: impacts modeling of m The fixed effect, gi, impacts the modeling of the mean response. The random effect, b0i, is a Random effect b0i: used to random variable with an expected value of characterize the variance zero. Therefore, b0i does not directly contribute to the modeling of the mean response; rather, it is used to characterize the variance of the mean response.
Presentation: IV. The Generalized Linear Mixed Model Approach 585 GEE vs. MLM A GEE model can also be expressed in terms of GEE model: logit m = b0 + b1RX the ith subject’s mean response (mi), as shown at left using the heartburn example. The GEE No subject-specific model contrasts with the MLM, and the condi- random effects (b0i) tional logistic regression, since the GEE model does not contain subject-specific effects (fixed Within-subject correlation specified or random). With the GEE approach, the in R matrix within-subject correlation is handled by MLM model: logit mi = b0 + b1RX + b0i the specification of a correlation structure for the R matrix. However, the mean response is not directly modeled as a function of the indi- vidual subjects. Subject-specific random effects Marginal model ) E(Y| X) not A GEE model represents a type of model called a conditioned on cluster-specific marginal model. With a marginal model, the information mean response E(Y| X) is not directly conditioned on any variables containing infor- (e.g., not allowed as X mation on the within-cluster correlation. For example, the predictors (X) in a marginal Earlier values of Y model cannot be earlier values of the response Subject-specific effects) from the same subject or subject-specific effects. Marginal models (examples): Other examples of marginal models include the ALR model, described earlier in the chapter, GEE and the standard logistic regression with one ALR observation for each subject. In fact, any model SLR using data in which there is one observation per subject is a marginal model because in that situation, there is no information available about within-subject correlation. Heartburn Relief Study Returning to the Heartburn Relief Study exam- ple, the parameter of interest is the coefficient b1 ¼ parameter of interest of the RX variable, b1, not the subject-specific BUT effect, b0i. The research question for this study is whether the active treatment provides interpretation of exp (b1) depends greater relief for heartburn than the standard on type of model treatment. The interpretation of the odds ratio exp(b1) depends, in part, on the type of model that is run.
586 16. Other Approaches for Analysis of Correlated Data Heartburn Relief Study: The odds ratio for a marginal model is the ratio of the odds of heartburn for RX ¼ 1 vs. RX ¼ 0 GEE: marginal model among the underlying population. In other expðb^1Þ is population OdR words, the OR is a population average. The odds ratio for a model with a subject-specific MLM: effect, as in the mixed logistic model, is the expðb^1Þ is OdR for an individual ratio of the odds of heartburn for RX ¼ 1 vs. RX ¼ 0 for an individual. What is an individual OR? What is meant by an odds ratio for an individ- ual? We can conceptualize each subject as hav- Each subject has separate prob- ing a probability of heartburn relief given the abilities active treatment and having a separate proba- bility of heartburn relief given the standard PðX ¼ 1jRX ¼ 1Þ treatment. These probabilities depend on the PðX ¼ 1jRX ¼ 0Þ fixed treatment effect as well as the subject- specific random effect. With this conceptualiza- + tion, the odds ratio that compares the active vs. OR compares RX ¼ 1 vs. RX ¼ 0 standard treatment represents a parameter for an individual that characterizes an individual rather than a population (see Practice Exercises 10–15). The mixed logistic model supplies a structure that gives the investigator the ability to estimate an odds ratio for an individual, while simulta- neously accounting for within-subject and between-subject variation. Goal OR The choice of whether a population averaged or individual level odds ratio is preferable Population ) marginal depends, in part, on the goal of the study. If inferences the goal is to make inferences about a popula- ) individual tion, then a marginal effect is preferred. If the Individual goal is to make inferences on the individual, inferences then an individual level effect is preferred.
Presentation: IV. The Generalized Linear Mixed Model Approach 587 Parameter estimation for MLM in There are various methods that can be used for SAS: parameter estimation with mixed logistic mod- els. The parameter estimates, obtained for the GLIMMIX Heartburn Relief data from the SAS procedure GLIMMIX use an approach termed penalized Penalized quasi-likelihood quasi-likelihood equations (Breslow and Clay- equations ton, 1993; Wolfinger and O’Connell, 1993). Alternatively, the SAS procedure NLMIXED User specifies G and R can also be used to run a mixed logistic model. NLMIXED fits nonlinear mixed models NLMIXED by maximizing an approximation to the likeli- hood integrated over the random effects. Maximized approximation to Unlike GLIMMIX, NLMIXED does not allow likelihood integrated over the user to specify a correlation structure for random effects the G and R matrices (SAS Institute, 2000). User does not specify G and R Instead, NLMIXED allows the user to specify User specifies variance the individual variance components within the G matrix, but assumes that the R matrix has an components of G matrix and independent covariance structure (i.e. 0s on assumes an independent R the off-diagonals of the R matrix). matrix (i.e., R ¼ s2I) Mixed models are flexible: Mixed models offer great flexibility by allowing the investigator to layer random components, Layer random components model clusters nested within clusters (i.e., per- Handle nested clusters form hierarchical modeling), and control for Control for subject effects subject-specific effects. The use of mixed linear models is widespread in a variety of disciplines because of this flexibility. Performance of mixed logistic Despite the appeal of mixed logistic models, models not fully evaluated their performance, particularly in terms of numerical accuracy, has not yet been ade- quately evaluated. In contrast, the GEE approach has been thoroughly investigated, and this is the reason for our emphasis on that approach in the earlier chapters on corre- lated data (Chaps. 14 and 15).
588 16. Other Approaches for Analysis of Correlated Data V. SUMMARY The presentation is now complete. Several alternate approaches for the analysis of cor- üChapter 16. Other Approaches related data were examined and compared to for Analysis of Cor- the GEE approach. The approaches discussed related Data included alternating logistic regressions, con- ditional logistic regression, and the gene- ralized linear mixed (logistic) model. The choice of which approach to implement for the primary analysis can be difficult and should be determined, in part, by the research hypothesis. It may be of interest to use several different approaches for comparison. If the results are different, it can be informative to investigate why they are different. If they are similar, it may be reassuring to know the results are robust to different methods of analysis. Computer Appendix We suggest that you review the material cov- ered here by reading the detailed outline that follows. A Computer Appendix is presented in the fol- lowing section. This appendix provides details on performing the analyses discussed in the various chapters using SAS, SPSS, and Stata statistical software.
Detailed Outline 589 Detailed I. Overview (page 570) Outline A. Other approaches for analysis of correlated data: i. Alternating logistic regressions (ALR) algorithm ii. Conditional logistic regression iii. Generalized linear mixed model (GLMM) II. Alternating logistic regressions algorithm (pages 571–575) A. Similar to GEE except that i. Associations between pairs of responses are modeled with odds ratios instead of correlations: ORijk ¼ PðYij ¼ 1; Yik ¼ 1ÞPðYij ¼ 0; Yik ¼ 0Þ : PðYij ¼ 1; Yik ¼ 0ÞPðYij ¼ 0; Yik ¼ 1Þ ii. Associations between responses may also be of interest, and not considered nuisance parameters. III. Conditional logistic regression (pages 575–579) A. May be applied in a design where each subject can be viewed as a stratum (e.g., has an exposed and an unexposed observation). B. Subject-specific fixed effects are included in the model through the use of dummy variables [Example: Heartburn Relief Study (n ¼ 40)]: 39 logit PðXÞ ¼ b0 þ b1RX þ ~ giVi; i¼1 where Vi ¼ 1 for subject i and Vi ¼ 0 otherwise. C. In the output, there are no parameter estimates for the intercept or the dummy variables representing the matched factor, as these parameters cancel out in the conditional likelihood. D. An important distinction between CLR and GEE is that a matched analysis (CLR) relies on the within-subject variability in the estimation of the parameters, whereas a correlated analysis (GEE) relies on both the within- subject variability and the between-subject variability. IV. The generalized linear mixed model approach (pages 579–587) A. A generalization of the linear mixed model. B. As with the GEE approach, the user can specify the logit link function and apply a variety of covariance structures to the model.
590 16. Other Approaches for Analysis of Correlated Data C. As with the conditional logistic regression approach, subject-specific effects are included in the model: logit mi ¼ PðD ¼ 1jRXÞ ¼ b0 þ b1RXi þ b0i; where bi is a random variable from a normal distribution with mean ¼ 0 and variance ¼ sb0 2. D. Comparing the conditional logistic model and the mixed logistic model: i. The conditional logistic model: logit mi ¼ b0 þ b1RX þ gi; where gi is a fixed effect ii. The mixed logistic model: logit mi ¼b0 þ b1RX þ b0i; where b0i is a random effect: E. Interpretation of the odds ratio: i. Marginal model: population average OR ii. Subject-specific effects model: individual OR. V. Summary (page 588)
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