31 Biostatistics 540 FINAL REVIEW

32 Correlated Data Models Three formulations of regression models for correlated outcomes lead to models with distinct interpretations of parameter estimates: – Marginal (or direct) models – random effects (or mixed) models – transition models

33 Practical matters Use of the xtgee procedure in Stata – Differences between models with/without “robust” specification Use of regress, poisson and logistic procedures in Stata – Differences between models with/without “cluster” specification – Differences between models with/without “robust” specification

34 Marginal Models – Modeling assumptions – Interpretation of parameters – Linear and nonlinear models What are they?

35 Marginal Models We directly (and separately) specify – the mean model g(E[y ij ]) = b 0 +b 1 x 1ij +b 2 x 2ij +…+b p x pij – g(.) is a link function (e.g., logit, log, identity, etc.) – a working variance (and correlation) structure E.g., independence, exchangeable, AR(1), etc. We do not specify distributions of the data – Specify only the means and variances – Valid estimates of means and regression coefficients can be obtained by ignoring correlation – Statistical inference ignoring correlation is not valid

36 Random Effects Models Modeling assumptions Model Specification (of random effects) – Y | b ~ Normal, b ~ Normal – Y | b ~ Poisson, b ~ Normal (or b ~ Gamma) – Y | b ~ Binomial (Bernoulli), b ~ Normal Use xtreg or xtmixed for Normal, xtpois for Poisson, xtlogit for binomial Subject specific interpretation of regression parameters – Same as population average (PA) estimates for linear model – Same PA estimates for predictors in Poisson model – Subject specific interpretation for logistic model

37 Random Effects Models Random (or mixed) models Use “conditional” means and variances – by conditioning on the random effects – Typically one assumes “conditional” independence of the outcomes given the random effects – All modeling assumptions need to be correct for valid inference (and checked!) Useful for studying – Sources of variation (between versus within) – Magnitude of the variances – Predicting unobserved random variables (BLUPs)

38 Transition Models Transition models A type of “conditional” model – One conditions upon some function of the outcome history – Markov models are a special class of transition models – Models can be fitted using likelihood methods or GEE Useful for – Estimating transition probabilities – In binary response models, incidence probabilities

39 Missing Data Missing data types – Missing by design, incomplete data, unbalanced data – Unintended missing data Missing data mechanisms – MCAR – Missing completely at random – MAR – Missing at random – NI – Non-ignorable missingness Problems with missing data – bias, loss of efficiency, power

40 Missing Data Accomodating missing data – Understanding missing data mechanism(s) – Complete case analysis, simple imputation methods – Modeling the missingness (IPW, MI, joint modeling) – Sensitivity analysis methods

41 Questions?