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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.

We conducted a combined genome-wide association (GWAS) analysis of 7,481 individuals affected with bipolar disorder and 9,250 control individuals within the Psychiatric Genomewide Association Study Consortium Bipolar Disorder group (PGC-BD). We performed a replication study in which we tested 34 independent SNPs in 4,493 independent bipolar disorder cases and 42,542 independent controls and found strong evidence for replication. In the replication sample, 18 of 34 SNPs had P value < 0.05, and 31 of 34 SNPs had signals with the same direction of effect (P = 3.8 × 10−7). In the combined analysis of all 63,766 subjects (11,974 cases and 51,792 controls), genome-wide significant evidence for association was confirmed for CACNA1C and found for a novel gene ODZ4. In a combined analysis of non-overlapping schizophrenia and bipolar GWAS samples we observed strong evidence for association with SNPs in CACNA1C and in the region of NEK4/ITIH1,3,4. Pathway analysis identified a pathway comprised of subunits of calcium channels enriched in the bipolar disorder association intervals. The strength of the replication data implies that increasing samples sizes in bipolar disorder will confirm many additional loci.

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