G89 2247 Lecture 8 Example of Random Regression Model Fitting What if outcome is not normal Marginal Models and GEE Example of GEE for binary outcome G89 2247 Lecture 8 1 Psychological Interpretation of Longitudinal Data Presentation at 2000 meeting of Society for Multivariate Experimental Psychology Niall Bolger Pat Shrout New York University G89 2247 Lecture 8 2 Goals of SMEP Presentation Describe a Research Problem as a Case Study Design addressed advice from 1990 Data available on www 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 G89 2247 Lecture 8 3 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 G89 2247 Lecture 8 4 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 G89 2247 Lecture 8 5 Diary Reports Show Stress On average steadily increases to day of exam Anxiety over time 3 00 2 50 POMS 2 00 Mean 1 50 Var 1 00 0 50 0 00 0 5 10 15 20 25 30 35 Days G89 2247 Lecture 8 6 Individuals show variations of anxiety buildup Individual patterns of change 4 50 4 00 3 50 POMS 3 00 2 50 2 00 1 50 1 00 0 50 0 00 0 5 10 15 20 25 30 35 Days of Exam Preparation G89 2247 Lecture 8 7 More patterns of Anxiety Patterns of anxiety 4 50 4 00 POMS Anxiety 3 50 3 00 2 50 2 00 b 1 50 1 00 0 50 0 00 0 5 10 15 20 25 30 35 Days of Exam Preparation G89 2247 Lecture 8 8 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 At bI bT Tt rt Model 1 Question do persons who get more support over the 30 day period have different slopes G89 2247 Lecture 8 9 Interpreting Model 1 We let T be zero for beginning of series bI is expected anxiety at day zero bT 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 rt may have autocorrelation structure Adjacent days have common influences on mood G89 2247 Lecture 8 10 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 At bI bT Tt bS St rt Model 2 Question On days when support occurs does anxiety vary from expected trajectory G89 2247 Lecture 8 11 Interpreting Model 2 We let S be binary 0 1 bI applies to an unsupported first day bS 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 G89 2247 Lecture 8 12 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 At 1 bI bT Tt bS St bA At rt Model 3 Question Does support today affect anxiety tomorrow holding constant anxiety today as well as the expected trajectory G89 2247 Lecture 8 13 Interpreting Model 3 In our data A 0 is meaningful no anxiety bI is the expected change from zero when there is no anxiety today bS 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 G89 2247 Lecture 8 14 Interpreting the Autoregression Effect bA 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 G89 2247 Lecture 8 15 Some empirical results based on 68 couples over 30 days The Growth Model B Intercept 0 9943 Diaryday 0 0341 MeanSupport 0 1229 Mean Diaryday 0 0016 AR 1 0 458 Random Var Intercept 0 32317 Diaryday 0 00039 Residual 0 551 AIC 2262 G89 2247 Lecture 8 se 0 1751 0 0074 0 2742 0 0116 16 Simple Intraindividual Model on Lagged Support Intercept Diary Day LagRecSup AR 1 Random Intercept Diary Day LagRecSup Residual AIC B 1 0503 0 0351 0 0225 0 4558 Var 0 31510 0 00039 0 00828 0 548 2261 G89 2247 Lecture 8 se 0 0856 0 0036 0 0383 17 Autoregressive Model With Lagged Support Phase and AR errors B INTERCEPT LagANX DIARYDAY PHASE LagSupp PHASE LagSupp AR Random Effects INTERCEPT LagANX DIARYDAY LagSupp Residual AIC 1 2268 0 1133 0 0330 0 0864 0 0135 0 2076 0 530 Var 0 36855 0 00955 0 00045 0 00165 0 591 2251 G89 2247 Lecture 8 se 0 0969 0 0247 0 0046 0 0913 0 0382 0 0898 18 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 G89 2247 Lecture 8 19 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 G89 2247 Lecture 8 20 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 G89 2247 Lecture 8 21 An Alternative