1. G89.2247 Lecture 81 Example of Random Regression Model Fitting What if outcome is not normal? Marginal Models and GEE Example of GEE for binary outcome

2. G89.2247 Lecture 82 Psychological Interpretation of Longitudinal Data (Presentation at 2000 meeting of Society for Multivariate Experimental Psychology) Niall Bolger Pat Shrout New York University

3. G89.2247 Lecture 83 Goals of SMEP Presentation Describe a Research Problem as a Case Study  Design addressed advice from 1990  Data available on www.psych.nyu.edu/coupleswww.psych.nyu.edu/couples Focus on interpretation of parameters that arise from application of general random regression methods to this problem Briefly describe new design issues

4. G89.2247 Lecture 84 A Case Study of a Longitudinal Application in Psychology Question: How does social support affect anxiety during a stressful event? Approach: Collect 30+ daily diary reports of support, coping and anxiety levels during acute planned stress event from members of couples  Inquire about support provided by partner  Inquire about support noticed by proband Acute Stressor: NY State Bar Exam

5. G89.2247 Lecture 85 The Bar Exam is Stressful About 30% of examinees will fail Most examinees work full time to prepare for exam in six weeks before the exam Much is at stake  Employment requirement  Self esteem  Social standing and esteem  Investment of time preparing

6. G89.2247 Lecture 86 Diary Reports Show Stress On average, steadily increases to day of exam

7. G89.2247 Lecture 87 Individuals show variations of anxiety buildup

8. G89.2247 Lecture 88 More patterns of Anxiety

9. G89.2247 Lecture 89 A Growth Model Suppose we consider a simple linear growth model for each examinee. Anxiety at day t is represented as some baseline (intercept) plus an increment for the day in the series. Model 1 Question: do persons who get more support over the 30 day period have different slopes?

10. G89.2247 Lecture 810 Interpreting Model 1 We let T be zero for beginning of series b I is expected anxiety at day zero b T is the expected increase in anxiety for each additional day  Assumption that the increase in Anxiety is linear over 30 days is probably not reasonable  Psychological explanation for slope is not trivial Counting days to event? Social norms supporting increase? r t may have autocorrelation structure  Adjacent days have common influences on mood

11. G89.2247 Lecture 811 A Simple Intraindividual Model When support occurs, the trajectory may be affected. Theory says anxiety will be reduced. Data suggests the opposite for visible support. Model 2 Question: On days when support occurs does anxiety vary from expected trajectory?

12. G89.2247 Lecture 812 Interpreting Model 2 We let S be binary, (0,1) b I applies to an unsupported first day b S is the change in anxiety due to support  Causal strength not clear: Support may be provided because concurrent anxiety is increased Gollob and Reichardt issue  Effect of support is limited to one day Residual terms may have autocorrelation

13. G89.2247 Lecture 813 An Autoregressive Model Anxiety tomorrow may be affected by processes other than support and time to exam. A basis for causal inference may be enhanced by adding autoregression term Model 3 Question: Does support today affect anxiety tomorrow, holding constant anxiety today as well as the expected trajectory?

14. G89.2247 Lecture 814 Interpreting Model 3 In our data A=0 is meaningful (no anxiety) b I is the expected change from zero when there is no anxiety today b S is the change in anxiety tomorrow associated with support today, adjusting for anxiety today. All effects are conceivably random over subjects Residual terms may still have autocorrelation

15. G89.2247 Lecture 815 Interpreting the Autoregression Effect, b A Anxiety today may have structural effects on tomorrow that might be mediated by  sleep (may be disrupted, adding to next day stress)  relationships (may be impaired)  preparation (may be disrupted) Anxiety today may be a proxy for additional effects  Poor expectations of achievement  Illness  Chronic stress buildup

16. G89.2247 Lecture 816 Some empirical results based on 68 couples over 30 days The Growth Model

17. G89.2247 Lecture 817 Simple Intraindividual Model on Lagged Support

18. G89.2247 Lecture 818 Autoregressive Model With Lagged Support, Phase and AR errors

19. G89.2247 Lecture 819 Design innovations in ongoing work Respondents are asked to report POMS at waking in addition to bedtime Respondents are randomly assigned to diary, panel and cross-sectional arms POMS is refined to include more response categories Sample is recruited to be more heterogeneous

20. G89.2247 Lecture 820 Longitudinal Models when Outcome or Residual is Not Normal Both estimation (ML, REML) and inference (s.e. estimates) in PROC MIXED assume normal residuals  When violated we might have Mispecified regression model Inefficient estimates Misleading inference Normal theory makes computations more convenient  ML and REML have nice forms that depend on means and covariances (first two moments)  Linear models usually work well with normal data

21. G89.2247 Lecture 821 Mixed Models for Non-normal Outcomes Modeling non-normal outcomes  Binary outcomes: logistic, probit regression  Count outcomes: Poisson regression  Ordinal outcomes: Multivariate probit, multinomial logistic Alternative models work best for large n If number of time points is small, then level 1 (within subject) models may be difficult to estimate Special software is needed in any case  PROC NLMIXED (SAS)  MIXOR, MIXREG (Hedeker and Gibbons)

22. G89.2247 Lecture 822 An Alternative Analysis: Marginal Models (GEE) If one is mainly interested in the fixed effects (population averages) then consider marginal models  Random effects are considered nuisance parameters  Model is specified only for population (fixed) effects  Residuals are correlated because of individual effects  Estimates and inference take into account correlated residuals Marginal Models ignore ZU in the mixed model

23. G89.2247 Lecture 823 Example: Marginal Model From PROC MIXED 1 8 PROC MIXED NOCLPRINT COVTEST METHOD=REML; 19 CLASS id time; 20 MODEL anx=group week group*week /s; 21 REPEATED time /TYPE=UN SUBJECT=ID R RCORR; 22 TITLE2 'Fixed (Marginal Model): ASSUMES RESIDUALS HAVE GENERAL CORR PATTERN';

24. G89.2247 Lecture 824 Marginal effects: Taking Correlations Among R.Measures into Account PROC MIXED (All fixed, R estimated to be Unstructured) Solution for Fixed Effects Effect Estimate S. Error DF t Value Pr > |t| Intercept 1.1718 0.07454 133 15.72 <.0001 group -0.6221 0.10580 133 -5.88 <.0001 week 0.2733 0.02379 133 11.49 <.0001 group*week -0.2944 0.03377 133 -8.72 <.0001

25. G89.2247 Lecture 825 General Linear Models through PROC GENMOD: A similar algorithm to the one used by PROC MIXED is available in GENMOD  We can specify a structure for Var(Y|B)=V  V is estimated and used to give weighted estimates of B  GENMOD uses Generalized Score Estimation for V

26. G89.2247 Lecture 826 GENMOD OUTPUT

27. G89.2247 Lecture 827 The GEE Method of GENMOD can be used with Non-Normal Data Suppose we have a model h(Y) = X'B  where h() is a function that describes how Y is related to X'B E.g. If Y is binary (0,1) and P=Prob(Y=1), then h(Y) might be a logistic function h(Y) = ln[P/(1-P)] h(Y) is called a LINK function Marginal models describe the relation between Y and X at the population level  Instead of describing the average of random subjects’ models, it models the average response pattern.

28. G89.2247 Lecture 828 Example of GENMOD Analysis of Binary Outcome Modeling daily provision of practical support  Is level of Anxious and Depressed Mood at waking related to the provision of practical support on that day?  Is there a tendency for partners to report more practical support over the course of a diary study? We analyze binary reports of practical support provision over 28 days by 87 persons in our Grad Couple Study (comparison group in exercises)  POMS Anxiety and Depression measured at waking  Diary day indicates course of study

29. G89.2247 Lecture 829 PROC GENMOD SETUP IRCPRP is binary received practical support  The MODEL statement says that support will be modeled with a logistic link function, and that day, AM-anxiety and AM-depression are predictors  The variance structure is “exchangeable”, which is the same as the sphericity or compound symmetry structure of MIXED.  POMS Anxiety and Depression are on 0-4 scale.

30. G89.2247 Lecture 830 PROC GENMOD OUTPUT Logistic results say that odds of support at day zero for zero anxiety and depression is exp(-.60)=.55 (corresponding to p=.35) For each point increase of Anxiety, odds of support goes up by a factor of exp(.168)=1.18. Persons with values of 4 on Anxiety would have about 2 times the chance of support as persons at 0.