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The focus of this article is on a class of regression models that are widely used for the analysis of cluster-correlated data. Linear mixed models are a natural extension of classical linear regression models that allow for the incorporation of random effects to account for the correlation among repeated measures on the same individual or cluster. Linear mixed models are used extensively for the analysis of longitudinal data as well as cluster-correlated data arising from other types of study designs. This article describes a general linear mixed model for clustered continuous data, outlining the model development and its underlying assumptions. Many of the conceptual ideas that are fundamental to understanding mixed models are reinforced using graphical representations of the linear mixed model equations. Finally, an example, based on data from a longitudinal study of growth in body fat in girls before and after menarche, is used to illustrate the application of linear mixed models.

The focus of this article is on regression models for a binary response. In most fields of application, logistic regression has become the standard method for relating a binary response to a set of covariates. In this article the main features of logistic regression are described and some aspects of interpretation of logistic regression are illustrated with an example. The logistic regression model is first introduced for the simple case where there is only a single predictor variable. This model is compared and contrasted with the classical linear regression model. Later the generalizations to more than one predictor variable are considered. A major emphasis of this article is placed on how logistic regression is used in practice and how the logistic regression coefficients should be interpreted. An example, based on data from the US General Social Survey, is used to illustrate and reinforce the main concepts. Finally, three important extensions of standard logistic regression are briefly reviewed: conditional logistic regression, exact logistic regression, and logistic regression for clustered data.

Alzheimer's disease (AD) is a genetically complex and heterogeneous disorder. To date four genes have been established to either cause early-onset autosomal-dominant AD (APP, PSEN1, and PSEN2(1-4)) or to increase susceptibility for late-onset AD (APOE5). However, the heritability of late-onset AD is as high as 80%, (6) and much of the phenotypic variance remains unexplained to date. We performed a genome-wide association (GWA) analysis using 484,522 single-nucleotide polymorphisms (SNPs) on a large (1,376 samples from 410 families) sample of AD families of self-reported European descent. We identified five SNPs showing either significant or marginally significant genome-wide association with a multivariate phenotype combining affection status and onset age. One of these signals (p = 5.7 x 10(-14)) was elicited by SNP rs4420638 and probably reflects APOE-epsilon4, which maps 11 kb proximal (r2 = 0.78). The other four signals were tested in three additional independent AD family samples composed of nearly 2700 individuals from almost 900 families. Two of these SNPs showed significant association in the replication samples (combined p values 0.007 and 0.00002). The SNP (rs11159647, on chromosome 14q31) with the strongest association signal also showed evidence of association with the same allele in GWA data generated in an independent sample of approximately 1,400 AD cases and controls (p = 0.04). Although the precise identity of the underlying locus(i) remains elusive, our study provides compelling evidence for the existence of at least one previously undescribed AD gene that, like APOE-epsilon4, primarily acts as a modifier of onset age.

Traditional epidemiological study concepts such as case-control or cohort designs can be used in the design of genetic association studies, giving them a prominent role in genetic association analysis. A different class of designs based on related individuals, typically families, uses the concept of Mendelian transmission to achieve design-independent randomization, which permits the testing of linkage and association. Family-based designs require specialized analytic methods but they have distinct advantages: They are robust to confounding and variance inflation, which can arise in standard designs in the presence of population substructure; they test for both linkage and association; and they offer a natural solution to the multiple comparison problem. This chapter focuses on family-based designs. We describe some basic study designs as well as general approaches to analysis for qualitative, quantitative, and complex traits. Finally, we review available software.

Researchers frequently use multiple informants to predict a single outcome and compare the marginal relationships of each informant with response; a common application is diagnostic testing where the goal is to determine which diagnostic test best predicts disease. We review generalized estimating equations (GEE) for marginal regression models using continuous multiple source predictors with a continuous outcome and introduce a new maximum likelihood (ML) approach. ML and GEE yield the same regression coefficient estimates when (1) allowing different regression coefficients for each informant report, (2) assuming equal variance for the two multiple informant reports and constraining the marginal regression coefficients to be equal and (3) including non-multiple informant covariates with cases 1 or 2. With the ML technique, likelihood ratio tests (LRTs) can be formed to easily compare regression models and a broader array of models can be fit. Using asymptotic relative efficiency (ARE), we show that a constrained model assuming equal variance is more efficient than an unconstrained model. We apply the methods to a study investigating the effect of vigorous exercise on body mass index (BMI) with measures of exercise collected on two informants: children and their mothers.

PUBLISHED, Several independent studies and meta-analyses aimed at identifying genomic regions linked to bipolar disorder (BP) have failed to find clear and consistent evidence of linkage regions. Our hypothesis is that combining the original genotype data provides benefits of increased power and control over sources of heterogeneity that outweigh the difficulty and potential pitfalls of the implementation. We conducted a combined analysis using the original genotype data from 11 BP genomewide linkage scans comprising 5,179 individuals from 1,067 families. Heterogeneity among studies was minimized in our analyses by using uniform methods of analysis and a common, standardized marker map and was assessed using novel methods developed for meta-analysis of genome scans. To date, this collaboration is the largest and most comprehensive analysis of linkage samples involving a psychiatric disorder. We demonstrate that combining original genome-scan data is a powerful approach for the elucidation of linkage regions underlying complex disease. Our results establish genomewide significant linkage to BP on chromosomes 6q and 8q, which provides solid information to guide future gene-finding efforts that rely on fine-mapping and association approaches.

Publisher Summary This chapter focuses on the joint discrete-time proportional hazards model and investigates the extent of efficiency gains for various scenarios of censoring, failure-time and auxiliary data Y . The discrete-time model is a natural choice when survival data are grouped. This occurs, for example, when survival status is noted at a sequence of scheduled follow-up visits. When survival time is grouped, but not coarsely grouped, the discrete-time models may be reasonable approximations to continuous-time models. The numbers of parameters in the joint model are large but do not depend on sample size, so that the asymptotic relative efficiency (ARE) is a valid means of comparing the joint and standard models. The chapter outlines the standard and joint models. The ARE, which is computed as the amount of censoring and probability law for ( Y | T, Z ), is varied. In investigation of ARE presented in the chapter, survival parameters and the shape for censoring are fixed. The chapter presents efficiency gains for repeated categorical measurements. This includes the case of baseline binary and polychotomous measurement and the case of two binary measurements. Recurrence time as auxiliary data is investigated and the alternative specifications for the censoring shape and survival distribution are discussed.

We describe a linear regression model for a spatially correlated dependent variable when a covariate is also spatially correlated and measured with error. Maximum likelihood estimates and likelihood-based confidence intervals for the regression parameters of the linear model are obtained and the method is extended to logistic regression analysis of grouped binomial data by methods analogous to empirical logistic regression. When the independent variable is spatially correlated, the measurement error variance and the other parameters of the regression model can be estimated without additional assumptions or data. The methods are used to characterize the association between outdoor concentrations of volatile organic compounds and respiratory health of school-children attending 73 elementary schools in Kanawha County, West Virginia. Results are compared to those from two-stage estimation procedures in which the dependent variable is regressed on the expectation of the true covariate conditional on the observed covariate values.