Sample-Size Calculations for the Cox Proportional Hazards Regression Model with Nonbinary Covariates

Abstract

This paper derives a formula to calculate the number of deaths required for a proportional hazards regression model with a nonbinary covariate. The method does not require assumptions about the distributions of survival time and predictor variables other than proportional hazards. Simulations show that the censored observations do not contribute to the power of the test in the proportional hazards model, a fact that is well known for a binary covariate. This paper also provides a variance inflation factor together with simulations for adjustment of sample size when additional covariates are included in the model. Control Clin Trials 2000;21:552–560

Introduction

In survival analysis, the Cox proportional hazards (PH) regression model assumes that the hazard function λ(t) for the survival time T given the predictors X₁, X₂, … , X_k has the following regression formulation:

$$\text{log}\text{λ}\text{t|X}\text{λ}{}_0\text{t}\text{=}\text{θ}{}_1\text{X}{}_1\text{+}\text{θ}{}_2\text{X}{}_2\text{+ … +}\text{θ}{}_{\mathrm{k}}\text{X}{}_{\mathrm{k}}$$

where λ₀(t) is the baseline hazard. The survival analysis allows the response, the survival time variable t, to be censored. In the use of this model, one often wishes to test the effect of a specific predictor, X₁, possibly in the presence of other predictors or covariates, on the response variable. The null hypothesis on the parameter $\text{θ}{}_1\text{is H}{}_0\text{:}\text{θ}{}_1\text{,}\text{θ}{}_2\text{, … ,}\text{θ}{}_{\mathrm{k}}\text{=}\text{0,}\text{θ}{}_2\text{, … ,}\text{θ}{}_{\mathrm{k}}$ tested against alternative [θ* θ₂, … , θ_k]. In the proportional hazards model, θ₁ represents the predicted change in log hazards at one unit change in X₁ when covariates X₂ to X_k are held constant.

When comparing two groups in a univariate model, the group indicator X₁ is binary, and θ₁ = logΔ is the log hazard ratio of the two groups. Cox proposed testing H₀ with the Rao-type statistic, also known as the score statistic [1]. When there is only one binary covariate X₁, the score test is the same as the Mantel-Haenszel test and the log-rank test if there are no ties in survival times. It is known that the power of the log-rank test depends on the sample size only through the number of deaths. This simplifies the sample-size formula. For comparing two groups, Schoenfeld derived the following formula:

$$\text{D =}\text{Z}{}_{\mathrm{<mtext>1-</mtext><mtext>\alpha </mtext>}}\text{+ Z}{}_{\mathrm{<mtext>1-</mtext><mtext>\beta </mtext>}}{}^2\text{P}\text{1−P}\text{logΔ}{}^2{}^{\mathrm{-1}}$$

where D is the total number of deaths, P is the proportion of the sample assigned to the first treatment group, and Z_1−α and Z_1−β are standard normal deviates at the desired one-sided significance level α and power 1 − β, respectively [2]. Formula (1) was designed for a randomized comparison of two groups using survival analysis with covariates, although it applies to nonrandomized comparisons as well. In addition to formula (1), there are other sample-size formulas, such as Freedman's and Lakatos' formulas, proposed for the log-rank test to compare two survival distributions 3, 4. Zhen and Murphy derived a formula for a nonbinary covariate assuming exponential survival time [5]. In this paper we derive a sample-size formula for a nonbinary covariate X₁ without assuming exponential survival time by generalizing formula (1). However, in our experience, a nonbinary covariate occurs most often in a nonexperimental context such as an epidemiologic study. In this context, one must often adjust for other confounding covariates to appropriately model the effect of X₁, the covariate of greatest interest.

Schoenfeld extended the multivariate model power calculation for binary X₁ to the case that additional covariates X₂ … X_k are included [2]. His argument depends on the assumption that X₁ is independent of X₂ … X_k, as would occur if X₁ were randomly assigned in a controlled experiment. We show that Schoenfeld's argument also works when X₁ is nonbinary and is independent of X₂ … X_k. This could be relevant in a randomized study if, for example, X₁ records dose levels to which the study subjects are randomized. In epidemiologic studies, X₁ is often a measure of a risk factor, such as numbers of cigarettes smoked per day, of interest to the investigators, and X₂ … X_k are possible confounders such as age and sex. By definition, covariates X₂ … X_k are correlated with the main factor of interest, X₁, and formula (1) doesn't apply. We describe a method for adjusting sample sizes to preserve power when X₁ is correlated with X₂ … X_k.