1. 1 STA 617 – Chp11 Models for repeated data Analyzing Repeated Categorical Response Data  Repeated categorical responses may come from  repeated measurements over time on each individual  or from a set of measurements that are related because they belong to the same group or cluster (e.g., measurements made on siblings from the same family, measurements made on a set of teeth from the same mouth).  Observations within a cluster are not usually independent of each other, as the response from one child of a family, say, may influence the response from another child, because the two grew up together.  Matched-pairs are the special case of each cluster having two members.

2. 2 STA 617 – Chp11 Models for repeated data  Using repeated measures within a cluster can be an efficient way to estimate the mean response at each measurement time without estimating between-cluster variability.  Many times, one is interested in the marginal distribution of the response at each measurement time, and not substantially interested in the correlation between responses across times.  Estimation methods for marginal modeling include maximum likelihood estimation and generalized estimating equations (GEE).  Maximum likelihood estimation is difficult because the likelihood is written in terms of the I T multinomial joint probabilities for T responses with I categories each, but the model applies to the marginal probabilities.  Lang and Agresti give a method for maximum likelihood fitting of marginal models in Section 11.2.5. Modeling a repeated multinomial response or repeated ordinal response is handled in the same way.

3. 3 STA 617 – Chp11 Models for repeated data Topics  In Section 11.1 we compare marginal distributions in T- way tables. The remaining sections extend models to include explanatory variables.  In Section 11.2 we use ML methods for fitting marginal models.  In Section 11.3 we use generalized estimating equations (GEE), a multivariate version of quasi- likelihood that is computationally simpler than ML.  Section 11.4 covers technical details about the GEE approach.  In the final section we introduce a transitional approach that models observations in terms of previous outcomes.

4. 4 STA 617 – Chp11 Models for repeated data 11.1 COMPARING MARGINAL DISTRIBUTIONS: MULTIPLE RESPONSES  Please review 10.1-10.3.  Example: in treating a chronic condition with some treatment, the primary goal might be to study whether the probability of success increases over the T weeks of a treatment period.  The T success probabilities refer to the T first-order marginal distributions  We want to compare marginal distributions.

5. 5 STA 617 – Chp11 Models for repeated data 11.1.1 Binary Marginal Models and Marginal Homogeneity  T binary responses  Marginal logit model with  All possible outcomes where  Let  the joint distribution of is Mult (n, ( 1,  2, …,  2^T ))

6. 6 STA 617 – Chp11 Models for repeated data Marginal homogeneity  Likelihood  The likelihood-ratio test of marginal homogeneity where sample proportions and is maximized likelihood estimate assuming marginal homogeneity.  asymptotic null chi-squared distribution with DF=T-1

7. 7 STA 617 – Chp11 Models for repeated data 11.1.2 Crossover Drug Comparison Example  each subject used each of three drugs for treatment of a chronic condition at three times.  The response measured the reaction as favorable or unfavorable. (binary)  assume that the drugs have no carryover effects and that the severity of the condition remained stable for each subject throughout the experiment.

8. 8 STA 617 – Chp11 Models for repeated data Test marginal homogeneity  Sample proportions favorable (n=46) [(6+2+16+4)/46=0.61, 28/46=0.61, 16/46=0.35] for drug A, B, C  Clearly, from the sample proportion, A and B are similar, and better than C  The likelihood-ratio test statistic is 5.95 (DF=2). P-value=0.05.

9. 9 STA 617 – Chp11 Models for repeated data SAS

10. 10 STA 617 – Chp11 Models for repeated data simultaneous confidence intervals  The confidence interval for the true difference is (0.00133, 0.520) between B and C

11. 11 STA 617 – Chp11 Models for repeated data CATMOD Suppose the dependent variable A has three levels and is the only response-effect in the MODEL statement.

12. 12 STA 617 – Chp11 Models for repeated data Design Matrix  p_A=alpha+beta1+beta2  P_B=alpha+beta1  P_C=alpha  Alpha=intercept  Beta1=p_B-p_C  Beta2=p_A-p_B

13. 13 STA 617 – Chp11 Models for repeated data Design Matrix  p_A=parameter1  P_B=parameter2  P_C=parameter3 Analysis of Weighted Least Squares Estimates EffectParameterEstimateStandard Error Chi- Square Pr > ChiSq Model10.60870.072071.56<.0001 20.60870.072071.56<.0001 30.34780.070224.53<.0001

14. 14 STA 617 – Chp11 Models for repeated data 11.1.3 Modeling Margins of a Multicategory Response  Saturated model  marginal homogeneity  Test

15. 15 STA 617 – Chp11 Models for repeated data Ordinal response  marginal homogeneity  Test  Model fitting 11.2.5

16. 16 STA 617 – Chp11 Models for repeated data 11.1.4 Wald and Generalized CMH Score Tests of Marginal Homogeneity  Similar with paired data in Chapter 10  SAS

17. 17 STA 617 – Chp11 Models for repeated data 11.2 MARGINAL MODELING: MAXIMUM LIKELIHOOD APPROACH  compared marginal distributions, but accounting for explanatory variables.

18. 18 STA 617 – Chp11 Models for repeated data Longitudinal Mental Depression Example  comparing a new drug with a standard drug  Outcome: mental depression (normal, abnormal)  Stratified randomization by severity of depression (was mild or severe). Four arms n=80, 70, 100, 90  Follow up 1 week, 2 weeks, and 4 weeks

19. 19 STA 617 – Chp11 Models for repeated data  explanatory variables: treatment type and severity of initial diagnosis  T=3  12 marginal distributions result from three repeated observations for each of the four groups.  Let s denote the severity of the initial diagnosis, with s=1 for severe and s=0 for mild.  Let d denote the drug, with d=1 for new and d=0 for standard.  Let t denote the time of measurement. Use score (0, 1, 2), the logs to base 2 of the week (1, 2, 4).

20. 20 STA 617 – Chp11 Models for repeated data Descriptive statistics (sample proportions)  the sample proportion of normal responses after week 1 for subjects with mild initial diagnosis using the standard drug was

21. 21 STA 617 – Chp11 Models for repeated data data depress; input case diagnose treat time outcome ; * outcome=1 is normal; datalines; 1 0 0 0 1 1 0 0 1 1 1 0 0 2 1 2 0 0 0 1 2 0 0 1 1 2 0 0 2 1 3 0 0 0 1 3 0 0 1 1 3 0 0 2 1 4 0 0 0 1 4 0 0 1 1 4 0 0 2 1 5 0 0 0 1 5 0 0 1 1 5 0 0 2 1 6 0 0 0 1 6 0 0 1 1 6 0 0 2 1 7 0 0 0 1 7 0 0 1 1 7 0 0 2 1 8 0 0 0 1 8 0 0 1 1 8 0 0 2 1 9 0 0 0 1 9 0 0 1 1 9 0 0 2 1 10 0 0 0 1 10 0 0 1 1 10 0 0 2 1 11 0 0 0 1 11 0 0 1 1 11 0 0 2 1 12 0 0 0 1 12 0 0 1 1 12 0 0 2 1 13 0 0 0 1 13 0 0 1 1 13 0 0 2 1 14 0 0 0 1 14 0 0 1 1 14 0 0 2 1 15 0 0 0 1 15 0 0 1 1 15 0 0 2 1 16 0 0 0 1 16 0 0 1 1 16 0 0 2 1 17 0 0 0 1 17 0 0 1 1 17 0 0 2 0 18 0 0 0 1 18 0 0 1 1 18 0 0 2 0 19 0 0 0 1 19 0 0 1 1 19 0 0 2 0 20 0 0 0 1 20 0 0 1 1 20 0 0 2 0 21 0 0 0 1 21 0 0 1 1 21 0 0 2 0 22 0 0 0 1 22 0 0 1 1 22 0 0 2 0 23 0 0 0 1 23 0 0 1 1 23 0 0 2 0 24 0 0 0 1 24 0 0 1 1 24 0 0 2 0 25 0 0 0 1 25 0 0 1 1 25 0 0 2 0 26 0 0 0 1 26 0 0 1 1 26 0 0 2 0 27 0 0 0 1 27 0 0 1 1 27 0 0 2 0 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1 1 2 1 307 1 1 0 0 307 1 1 1 0 307 1 1 2 1 308 1 1 0 0 308 1 1 1 0 308 1 1 2 1 309 1 1 0 0 309 1 1 1 0 309 1 1 2 1 310 1 1 0 0 310 1 1 1 0 310 1 1 2 1 311 1 1 0 0 311 1 1 1 0 311 1 1 2 1 312 1 1 0 0 312 1 1 1 0 312 1 1 2 1 313 1 1 0 0 313 1 1 1 0 313 1 1 2 1 314 1 1 0 0 314 1 1 1 0 314 1 1 2 1 315 1 1 0 0 315 1 1 1 0 315 1 1 2 1 316 1 1 0 0 316 1 1 1 0 316 1 1 2 1 317 1 1 0 0 317 1 1 1 0 317 1 1 2 1 318 1 1 0 0 318 1 1 1 0 318 1 1 2 1 319 1 1 0 0 319 1 1 1 0 319 1 1 2 1 320 1 1 0 0 320 1 1 1 0 320 1 1 2 1 321 1 1 0 0 321 1 1 1 0 321 1 1 2 1 322 1 1 0 0 322 1 1 1 0 322 1 1 2 1 323 1 1 0 0 323 1 1 1 0 323 1 1 2 1 324 1 1 0 0 324 1 1 1 0 324 1 1 2 1 325 1 1 0 0 325 1 1 1 0 325 1 1 2 1 326 1 1 0 0 326 1 1 1 0 326 1 1 2 1 327 1 1 0 0 327 1 1 1 0 327 1 1 2 1 328 1 1 0 0 328 1 1 1 0 328 1 1 2 1 329 1 1 0 0 329 1 1 1 0 329 1 1 2 1 330 1 1 0 0 330 1 1 1 0 330 1 1 2 0 331 1 1 0 0 331 1 1 1 0 331 1 1 2 0 332 1 1 0 0 332 1 1 1 0 332 1 1 2 0 333 1 1 0 0 333 1 1 1 0 333 1 1 2 0 334 1 1 0 0 334 1 1 1 0 334 1 1 2 0 335 1 1 0 0 335 1 1 1 0 335 1 1 2 0 ; proc sort; by diagnose treat time; proc means n mean std; class diagnose treat time; var outcome; run;

22. 22 STA 617 – Chp11 Models for repeated data  The sample proportion of normal responses  increased over time for each group;  increased at a faster rate for the new drug than the standard, for each fixed initial diagnosis;  and was higher for the mild than the severe initial diagnosis, for each treatment at each occasion.  The company would hope to show that patients have a significantly higher rate of improvement with the new drug.

23. 23 STA 617 – Chp11 Models for repeated data Modeling  The marginal logit model 1 (main effects model)  Time (t) is continuous  The natural sampling assumption is multinomial for the eight cells in the 2 3 cross-classification of the three responses  A check of model fit compares the 32 cell counts in Table 11.2 to their ML fitted values. Since the model describes 12 marginal logits using four parameters, residual df=8. The deviance G 2 =34.6.  Lack of fit, since model assumes a common rate of improvement (should be higher for new drug)

24. 24 STA 617 – Chp11 Models for repeated data Model 2

25. 25 STA 617 – Chp11 Models for repeated data  For each drug-time combination, the estimated odds of a normal response when the initial diagnosis was severe equal exp(-1.29)=0.27 times the estimated odds when the initial diagnosis was mild.  The estimate indicates an insignificant difference between the drugs after 1 week.  At time t, the estimated odds of normal response with the new drug are exp(-0.06+1.01 t) times the estimated odds for the standard drug, for each initial diagnosis level.  Conclusion: severity of initial diagnosis, drug treatment, and time all have substantial effects on the probability of a normal response.

26. 26 STA 617 – Chp11 Models for repeated data 11.2.2 Modeling a Repeated Multinomial Response  At observation t, the marginal response distribution has I-1 logits.  nominal responses, baseline-category logit models describe the odds of each outcome relative to a baseline.  For ordinal responses, one might use cumulative logit models.  checking for interaction is crucial.

27. 27 STA 617 – Chp11 Models for repeated data 11.2.3 Insomnia Example  randomized, double-blind clinical trial comparing an active hypnotic drug with a placebo in patients who have insomnia problems.  response is the patient’s reported time in minutes to fall asleep after going to bed.

28. 28 STA 617 – Chp11 Models for repeated data Proportional odds model  Sample marginal distributions proc sort; by treat time; proc freq; tables treat*time*outcome /nocol NOFREQ NOPERCENT; run;

29. 29 STA 617 – Chp11 Models for repeated data ML model fitting  G2=8.0 (df=6)  shows evidence of interaction  At the initial observation, the estimated odds that time to falling asleep for the active treatment is below any fixed level equal exp(0.046)=1.04 times the estimated odds for the placebo treatment;  at the follow-up observation, the effect is exp(0.046+0.662)=2.03.  In other words, initially the two groups had similar distributions, but at the follow-up those with the active treatment tended to fall asleep more quickly.  Follow-up with placebo or treatment, both tended to fall sleep more quickly (exp(1.07)=2.9)

30. 30 STA 617 – Chp11 Models for repeated data 11.2.4 Comparisons That Control for Initial Response  Model assumption: the marginal distributions for initial response are identical for the treatment groups.  This is true if random assignment of subjects to the groups (one of the principles in experimental design: randomization, other two: replication, blocking)  If the initial marginal distributions are not identical, however, the difference between follow-up and initial marginal distributions may differ between treatment groups, even though their conditional distributions for follow-up response are identical.  In such cases, although marginal models can be useful, they may not tell the entire story. It may be more informative to construct models that compare the follow-up responses while controlling for the initial response.

31. 31 STA 617 – Chp11 Models for repeated data transitional model  Let Y 2 denote the follow-up response, for treatment x with initial response y 1.

32. 32 STA 617 – Chp11 Models for repeated data 11.2.5 ML Fitting of Marginal Logit Models*  For T observations on an I-category response, at each setting of predictors the likelihood refers to I T multinomial joint probabilities, but the model applies to T sets of marginal multinomial parameters  The marginal multinomial variates are not independent.  Marginal logit models have the generalized loglinear model form where denote the complete set of multinomial joint probabilities for all settings of predictors.

33. 33 STA 617 – Chp11 Models for repeated data Example, model (11.1)  the model of marginal homogeneity (T=2)

34. 34 STA 617 – Chp11 Models for repeated data likelihood  The likelihood function for a marginal logit model is the product of the multinomial mass functions from the various predictor settings.  Usually, no continuous predictor is allowed if U denote a full column rank matrix such that the space spanned by the columns of U is the orthogonal complement of the space spanned by the columns of X.  maximizing the likelihood incorporates these model constraints as well as identifiability constraints

35. 35 STA 617 – Chp11 Models for repeated data ML  Joseph Lang ( jblang@stat.uiowa.edu) has R and S-Plus functions for ML fitting of marginal models through the generalized loglinear model (11.8), using the constraint approach with Lagrange multipliers. http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.2.0.htm http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.2.0.htm  The program MAREG (Kastner et al. 1997) provides GEE fitting and ML fitting of marginal models with the Fitzmaurice and Laird (1993) approach, allowing multicategory responses.

36. 36 STA 617 – Chp11 Models for repeated data Generalized Estimating Equation (GEE)