Time-dependent covariates in the cox proportional-hazards regression model

The Cox proportional-hazards regression model has achieved widespread use in the analysis of time-to-event data with censoring and covariates. The covariates may change their values over time. This article discusses the use of such time-dependent covariates, which offer additional opportunities but must be used with caution. The interrelationships between the outcome and variable over time can lead to bias unless the relationships are well understood. The form of a time-dependent covariate is much more complex than in Cox models with fixed (non–time-dependent) covariates. It involves constructing a function of time. Further, the model does not have some of the properties of the fixed-covariate model; it cannot usually be used to predict the survival (time-to-event) curve over time. The estimated probability of an event over time is not related to the hazard function in the usual fashion. An appendix summarizes the mathematics of time-dependent covariates

Evaluation of population impact of candidate polymorphisms for coronary heart disease in the Framingham Heart Study Offspring Cohort

In order to evaluate the population impact of putative causal genetic variants over the life course of disease, we extended the static estimation of population-attributable risk fraction and developed a novel tool to evaluate how the population impact changes over time using the Framingham Heart Study Offspring Cohort data provided to the Genetic Analysis Workshop 16, Problem 2. A set of population-attributable risk fractions based on survival functions were estimated under the proportional hazards models. The development of this novel measure of population impact creates a more comprehensive estimate of population impact over the life course of disease, which may help us to better understand genetic susceptibility at the population level

Checking the Cox model with cumulative sums of martingale-based residuals

This paper presents a new class of graphical and numerical methods for checking the adequacy of the Cox regression model. The procedures are derived from cumulative sums of martingale-based residuals over follow-up time and/or covariate values. The distributions of these stochastic processes under the assumed model can be approximated by zero-mean Gaussian processes. Each observed process can then be compared, both visually and analytically, with a number of simulated realizations from the approximate null distribution. These comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are. Special attention is given to checking the functional form of a covariate, the form of the link function, and the validity of the proportional hazards assumption. An omnibus test, consistent against any model misspecification, is also studied. The proposed techniques are illustrated with two real data sets

Longitudinal age-dependent effect on systolic blood pressure

Age-dependent genetic effects on susceptibility to hypertension have been documented. We present a novel variance-component method for the estimation of age-dependent genetic effects on longitudinal systolic blood pressure using 57,827 Affymetrix single-nucleotide polymorphisms (SNPs) on chromosomes 17-22 genotyped in 2,475 members of the Offspring Cohort of the Framingham Heart Study. We used the likelihood-ratio test statistic to test the main genetic effect, genotype-by-age interaction, and simultaneously, main genetic effect and genotype-by-age interactions (2 degrees of freedom (df) test) for each SNP. Applying Bonferroni correction, three SNPs were significantly associated with longitudinal blood pressure in the analysis of main genetic effects or in combined 2-df analyses. For the associations detected using the simultaneous 2-df test, neither main effects nor genotype-by-age interaction p-values reached genome-wide statistical significance. The value of the 2-df test for screening genetic interaction effects could not be established in this study

Assessing the sensitivity of regression results to unmeasured confounders in observational studies

This paper presents a general approach for assessing the sensitivity of the point and interval estimates of the primary exposure effect in an observational study to the residual confounding effects of unmeasured variables after adjusting for measured covariates. The proposed method assumes that the true exposure effect can be represented in a regression model that includes the exposure indicator as well as the measured and unmeasured confounders. One can use the corresponding reduced model that omits the unmeasured confounder to make statistical inferences about the true exposure effect by specifying the distributions of the unmeasured confounder in the exposed and unexposed groups along with the effects of the unmeasured confounder on the outcome variable. Under certain conditions, there exists a simple algebraic relationship between the true exposure effect in the full model and the apparent exposure effect in the reduced model. One can then estimate the true exposure effect by making a simple adjustment to the point and interval estimates of the apparent exposure effect obtained from standard software or published reports. The proposed method handles both binary response and censored survival time data, accommodates any study design, and allows the unmeasured confounder to be discrete or normally distributed. We describe applications to two major medical studies

Large-scale genome-wide association analysis of bipolar disorder identifies a new susceptibility locus near ODZ4

We conducted a combined genome-wide association study (GWAS) of 7,481 individuals with bipolar disorder (cases) and 9,250 controls as part of the Psychiatric GWAS Consortium. Our replication study tested 34 SNPs in 4,496 independent cases with bipolar disorder and 42,422 independent controls and found that 18 of 34 SNPs had P less than 0.05, with 31 of 34 SNPs having signals with the same direction of effect (P = 3.8 × 10−7). An analysis of all 11,974 bipolar disorder cases and 51,792 controls confirmed genome-wide significant evidence of association for CACNA1C and identified a new intronic variant in ODZ4. We identified a pathway comprised of subunits of calcium channels enriched in bipolar disorder association intervals. Finally, a combined GWAS analysis of schizophrenia and bipolar disorder yielded strong association evidence for SNPs in CACNA1C and in the region of NEK4-ITIH1-ITIH3-ITIH4. Our replication results imply that increasing sample sizes in bipolar disorder will confirm many additional loci

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox‐type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time‐transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co‐workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson‐type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple‐infection data taken from a clinical study on chronic granulomatous disease is also provided

Genetic relationship between five psychiatric disorders estimated from genome-wide SNPs

Most psychiatric disorders are moderately to highly heritable. The degree to which genetic variation is unique to individual disorders or shared across disorders is unclear. To examine shared genetic etiology, we use genome-wide genotype data from the Psychiatric Genomics Consortium (PGC) for cases and controls in schizophrenia, bipolar disorder, major depressive disorder, autism spectrum disorders (ASD) and attention-deficit/hyperactivity disorder (ADHD). We apply univariate and bivariate methods for the estimation of genetic variation within and covariation between disorders. SNPs explained 17–29% of the variance in liability. The genetic correlation calculated using common SNPs was high between schizophrenia and bipolar disorder (0.68 ± 0.04 s.e.), moderate between schizophrenia and major depressive disorder (0.43 ± 0.06 s.e.), bipolar disorder and major depressive disorder (0.47 ± 0.06 s.e.), and ADHD and major depressive disorder (0.32 ± 0.07 s.e.), low between schizophrenia and ASD (0.16 ± 0.06 s.e.) and non-significant for other pairs of disorders as well as between psychiatric disorders and the negative control of Crohn's disease. This empirical evidence of shared genetic etiology for psychiatric disorders can inform nosology and encourages the investigation of common pathophysiologies for related disorders

Genome-wide association study identifies five new schizophrenia loci

We examined the role of common genetic variation in schizophrenia in a genome-wide association study of substantial size: a stage 1 discovery sample of 21,856 individuals of European ancestry and a stage 2 replication sample of 29,839 independent subjects. The combined stage 1 and 2 analysis yielded genome-wide significant associations with schizophrenia for seven loci, five of which are new (1p21.3, 2q32.3, 8p23.2, 8q21.3 and 10q24.32-q24.33) and two of which have been previously implicated (6p21.32-p22.1 and 18q21.2). The strongest new finding (P = 1.6 × 10−11) was with rs1625579 within an intron of a putative primary transcript for MIR137 (microRNA 137), a known regulator of neuronal development. Four other schizophrenia loci achieving genome-wide significance contain predicted targets of MIR137, suggesting MIR137-mediated dysregulation as a previously unknown etiologic mechanism in schizophrenia. In a joint analysis with a bipolar disorder sample (16,374 affected individuals and 14,044 controls), three loci reached genome-wide significance: CACNA1C (rs4765905, P = 7.0 × 10−9), ANK3 (rs10994359, P = 2.5 × 10−8) and the ITIH3-ITIH4 region (rs2239547, P = 7.8 × 10−9)

PreMeta: a tool to facilitate meta-analysis of rare-variant associations

Abstract Background Meta-analysis is essential to the discovery of rare variants that influence complex diseases and traits. Four major software packages, namely MASS, MetaSKAT, RAREMETAL, and seqMeta, have been developed to perform meta-analysis of rare-variant associations. These packages first generate summary statistics for each study and then perform the meta-analysis by combining the summary statistics. Because of incompatible file formats and non-equivalent summary statistics, the output files from the study-level analysis of one package cannot be directly used to perform meta-analysis in another package. Results We developed a computationally efficient software program, PreMeta, to resolve the non-compatibility of the four software packages and to facilitate meta-analysis of large-scale sequencing studies in a consortium setting. PreMeta reformats the output files of study-level summary statistics generated by the four packages (text files produced by MASS and RAREMETAL, binary files produced by MetaSKAT, and R data files produced by seqMeta) and translates the summary statistics from one form to another, such that the summary statistics from any package can be used to perform meta-analysis in any other package. With this tool, consortium members are not required to use the same software for study-level analyses. In addition, PreMeta checks for allele mismatches, corrects summary statistics, and allows the rescaled inverse normal transformation to be performed at the meta-analysis stage by rescaling summary statistics. Conclusions PreMeta processes summary statistics from the four packages to make them compatible and avoids the need to redo study-level analyses. PreMeta documentation and executable are available at: http://dlin.web.unc.edu/software/premeta