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Longitudinal Data

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Three possible ways to analyze repeated measures dataExample of Binary Outcome: Sex, Drugs and TeenagersDataTransition Model for Teenage Sex and Drug-UseTransition ModelsSexual Activity and drug/alcohol use among teenagers revistedTransition Model – teenage sexResults using xtgee in STATARandom Effects ModelsRandom Effects Model for Teenage Sex and Drug-UseMotivation for This ApproachMotivation for This ApproachSome available software for random effects modelsRandom effects using xtlogit in STATAMarginal Models (GEE)Marginal Models (GEE)Parameter Interpretation in a GEE modelParameter Interpretation in a GEE model, cont.Marginal Models (GEE)Examples of Correlation ModelsStructure for R0Correlation Models (contd): Uniform correlation (compound symmetry or exchangeable)Correlation Models (contd):Time-Decaying Correlations (Auto-regressive)Examples of var-cov. modelsThe GEE AlgorithmStandard Errors of CoefficientsGEE Marginal Model for Teenage Sex and Drug-UseSexual Activity and drug/alcohol use among teenagersResults using xtgee in STATAxtgee OptionsModel 2 – same marginal model, different working correlation.Results of Model 2 using STATAEstimated Working CorrelationModel 3 – adjusting for day of weekResults of Model 3 using STATAModel for drug/alcohol use vs. day of weekResults of drug/alcohol use Model using STATAContinuous Outcome Example (Linear Model): Respiratory FunctionComparison of Standard ErrorsLongitudinal DataSpring 2006Chapter 5Approaches to Repeated MeasuresChapter 6Marginal (GEE) ModelsInstructorsAlan HubbardNick JewellGSIEd BeinThree possible ways to analyze repeated measures data• Transition Models• Random Effects Models• Marginal Models (GEE)Example of Binary Outcome: Sex, Drugs and Teenagers! A longitudinal study of the effects of drug-use on sexual activity.! Let Xijindicate whether or not subject ireported drug-use (1=yes, 0=no) on day j.! Let Yijdenote whether subject had sex (1=yes, 0=no), i.e., Yijis a binary outcome and thus its expectation can be modeled via the logit transform.Dataeid today drgalcoh sx24hrs1. 10122 03 Jun 98 yes no2. 10123 04 Jun 98 no no3. 10123 05 Jun 98 no no4. 10123 06 Jun 98 yes no5. 10123 07 Jun 98 no no6. 10123 08 Jun 98 no no7. 10123 09 Jun 98 no no8. 10123 12 Jun 98 no no9. 10123 14 Jun 98 yes no10. 10123 16 Jun 98 no no11. 10123 17 Jun 98 no no12. 10123 18 Jun 98 no yes13. 10123 19 Jun 98 no no14. 10123 20 Jun 98 no no15. 10123 21 Jun 98 no no16. 10123 23 Jun 98 no no17. 10123 25 Jun 98 no yes18. 10123 28 Jun 98 no no19. 10123 29 Jun 98 no yes20. 10123 01 Jul 98 no yes21. 10123 02 Jul 98 no no22. 10123 03 Jul 98 no no23. 10123 04 Jul 98 no no24. 10123 05 Jul 98 no no25. 10124 04 Jun 98 no no26. 10124 07 Jun 98 no no27. 10124 08 Jun 98 no noTransition Model for Teenage Sex and Drug-Use " For time-sequenced repeated measures, build the joint distribution by specifying a sequence of distributions that are conditioned on previous measurements on the individual. These are called transition (Markov) models." For the study of teenage sex:where Yi1is outcome at time ti1,Yi2at ti2, ..., and ti1< ti2<...< tini.)1(10121)],...,,,|1([logit−−−=++==jiijTMTMiijijijijijYxYYYxXYPζββTransition Models! exp(ζ) = odds ratio (OR) of among subjects who did versus did not have sex during the prior day, keeping drug status fixed. ! exp(β1TM) = OR of drug use vs. not for either subjects who reported having sex or did not have sex the previous day.! use generalized linear models (glm) software (e.g., linear, logistic, poisson regression).! Relevant for nice, time-structured data.Sexual Activity and drug/alcohol use among teenagers revistedMain Variablessex24hrs - sex in last 24 hrs. (0=no, 1=yes) drgalcoh - drug or alcohol use in last 24 hrs.tues-sun - dummy variables designating day of weekTransition Model – teenage sex)1(10121)],...,,,|1([logit−−−=++==jiijTMTMiijijijijijYxYYYxXYPζββXij= 0 if drug/alcohol use is no, 1 if yesYij= 0 if no sex in last 24 hours, 1 if yesYi(j-1)= 0 if no sex the day before, 1 if yesResults using xtgee in STATA.sort eid today.by eid: gen sxyest = sx24hrs[_n-1].by eid: replace sxyest = . if _n==1.logistic sx24hrs drgalcoh sxyest. Logit estimates Number of obs = 1607LR chi2(2) = 55.39Prob > chi2 = 0.0000Log likelihood = -942.60915 Pseudo R2 = 0.0285------------------------------------------------------------------------------sx24hrs | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------exp(β1TM)drgalcoh 1.63798 .1986677 4.07 0.000 1.291421 2.07754exp(ξ1TM) sxyest 2.051903 .2478562 5.95 0.000 1.619338 2.600018------------------------------------------------------------------------------Random Effects Models! Uses a random effect to model the relative similarity of observations made on same statistical unit (e.g., person)! Assumes Yijand Yik, j≠k are independent given some realized value of a random effect (βi0) that appears in the conditional distribution of Yijgiven βi0(random effects models).! The model assumes these random effects are randomly drawn from a known distribution.Random Effects Model for Teenage Sex and Drug-Use! Assume that the repeated observations for the ith teenager are independent of one another given βi0and Xij.! Must assume parametric distribution for the βi0, usually βi0~N(0,τ2).! exp(β1RE) is odds ratio for having sex infection when subject i reports drug-use relative to when same subject does not report drug-use.ijREiREijijiijijijiijijijiijxxXYPxXYPxXYPit100000),|0(),|1(log)],|1([ββββββ++========logMotivation for This Approach" Natural for modeling heterogeneity across individuals in their regression coefficients." This heterogeneity can be represented by a probability distribution" Most useful when object is to make


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