Kung‐Yee Liang

This paper proposes an extension of generalized linear models to the analysis of longitudinal data. We introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence. The estimating equations are derived without specifying the joint distribution of a subject's observations yet they reduce to the score equations for multivariate Gaussian outcomes. Asymptotic theory is presented for the general class of estimators. Specific cases in which we assume independence, m-dependence and exchangeable correlation structures from each subject are discussed. Efficiency of the proposed estimators in two simple situations is considered. The approach is closely related to quasi-likelihood.

Abstract The first edition of Analysis for Longitudinal Data has become a classic. Describing the statistical models and methods for the analysis of longitudinal data, it covers both the underlying statistical theory of each method, and its application to a range of examples from the agricultural and biomedical sciences. The main topics discussed are design issues, exploratory methods of analysis, linear models for continuous data, general linear models for discrete data, and models and methods for handling data and missing values. Under each heading, worked examples are presented in parallel with the methodological development, and sufficient detail is given to enable the reader to reproduce the author's results using the data-sets as an appendix. This new edition of Analysis for Longitudinal Data provides a thorough and expanded revision of this important text. It includes two new chapters; the first discusses fully parametric models for discrete repeated measures data, and the second explores statistical models for time-dependent predictors.

Abstract Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect. The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate Gaussian distribution from a sample of size 1. In this representation the Gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator. Thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the Gaussian random variable onto the region of admissible values for the mean. A variety of examples is provided for which the limiting distributions of likelihood ratio statistics are mixtures of chi-squared distributions. One example is provided with a nuisance parameter on the boundary for which the asymptotic distribution is not a mixture of chi-squared distributions.

CONTEXT: Substantial resources are being devoted to identify candidate genes for complex mental and behavioral disorders through inclusion of environmental exposures following the report of an interaction between the serotonin transporter linked polymorphic region (5-HTTLPR) and stressful life events on an increased risk of major depression. OBJECTIVE: To conduct a meta-analysis of the interaction between the serotonin transporter gene and stressful life events on depression using both published data and individual-level original data. DATA SOURCES: Search of PubMed, EMBASE, and PsycINFO databases through March 2009 yielded 26 studies of which 14 met criteria for the meta-analysis. STUDY SELECTION: Criteria for studies for the meta-analyses included published data on the association between 5-HTTLPR genotype (SS, SL, or LL), number of stressful life events (0, 1, 2, > or = 3) or equivalent, and a categorical measure of depression defined by the Diagnostic and Statistical Manual of Mental Disorders (Fourth Edition) or the International Statistical Classification of Diseases, 10th Revision (ICD-10) or use of a cut point to define depression from standardized rating scales. To maximize our ability to use a common framework for variable definition, we also requested original data from all studies published prior to 2008 that met inclusion criteria. Of the 14 studies included in the meta-analysis, 10 were also included in a second sex-specific meta-analysis of original individual-level data. DATA EXTRACTION: Logistic regression was used to estimate the effects of the number of short alleles at 5-HTTLPR, the number of stressful life events, and their interaction on depression. Odds ratios (ORs) and 95% confidence intervals (CIs) were calculated separately for each study and then weighted averages of the individual estimates were obtained using random-effects meta-analysis. Both sex-combined and sex-specific meta-analyses were conducted. Of a total of 14,250 participants, 1769 were classified as having depression; 12,481 as not having depression. RESULTS: In the meta-analysis of published data, the number of stressful life events was significantly associated with depression (OR, 1.41; 95% CI,1.25-1.57). No association was found between 5-HTTLPR genotype and depression in any of the individual studies nor in the weighted average (OR, 1.05; 95% CI, 0.98-1.13) and no interaction effect between genotype and stressful life events on depression was observed (OR, 1.01; 95% CI, 0.94-1.10). Comparable results were found in the sex-specific meta-analysis of individual-level data. CONCLUSION: This meta-analysis yielded no evidence that the serotonin transporter genotype alone or in interaction with stressful life events is associated with an elevated risk of depression in men alone, women alone, or in both sexes combined.