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1.  Estimation of the 2-sample hazard ratio function using a semiparametric model 
Biostatistics (Oxford, England)  2010;12(2):354-368.
The hazard ratio provides a natural target for assessing a treatment effect with survival data, with the Cox proportional hazards model providing a widely used special case. In general, the hazard ratio is a function of time and provides a visual display of the temporal pattern of the treatment effect. A variety of nonproportional hazards models have been proposed in the literature. However, available methods for flexibly estimating a possibly time-dependent hazard ratio are limited. Here, we investigate a semiparametric model that allows a wide range of time-varying hazard ratio shapes. Point estimates as well as pointwise confidence intervals and simultaneous confidence bands of the hazard ratio function are established under this model. The average hazard ratio function is also studied to assess the cumulative treatment effect. We illustrate corresponding inference procedures using coronary heart disease data from the Women's Health Initiative estrogen plus progestin clinical trial.
doi:10.1093/biostatistics/kxq061
PMCID: PMC3062151  PMID: 20860993
Clinical trial; Empirical process; Gaussian process; Hazard ratio; Simultaneous inference; Survival analysis; Treatment–time interaction
2.  Bias-reduced estimators and confidence intervals for odds ratios in genome-wide association studies 
Biostatistics (Oxford, England)  2008;9(4):621-634.
Genome-wide association studies (GWAS) provide an important approach to identifying common genetic variants that predispose to human disease. A typical GWAS may genotype hundreds of thousands of single nucleotide polymorphisms (SNPs) located throughout the human genome in a set of cases and controls. Logistic regression is often used to test for association between a SNP genotype and case versus control status, with corresponding odds ratios (ORs) typically reported only for those SNPs meeting selection criteria. However, when these estimates are based on the original data used to detect the variant, the results are affected by a selection bias sometimes referred to the “winner's curse” (Capen and others, 1971). The actual genetic association is typically overestimated. We show that such selection bias may be severe in the sense that the conditional expectation of the standard OR estimator may be quite far away from the underlying parameter. Also standard confidence intervals (CIs) may have far from the desired coverage rate for the selected ORs. We propose and evaluate 3 bias-reduced estimators, and also corresponding weighted estimators that combine corrected and uncorrected estimators, to reduce selection bias. Their corresponding CIs are also proposed. We study the performance of these estimators using simulated data sets and show that they reduce the bias and give CI coverage close to the desired level under various scenarios, even for associations having only small statistical power.
doi:10.1093/biostatistics/kxn001
PMCID: PMC2536726  PMID: 18310059
Bias-reduced estimator; Genome-wide association study; Odds ratio; Selection adjusted confidence interval; Selection bias

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