# numDEpi: Calculate Number of Deaths Required for Cox Proportional... In powerSurvEpi: Power and Sample Size Calculation for Survival Analysis of Epidemiological Studies

## Description

Calculate number of deaths required for Cox proportional hazards regression with two covariates for epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates. Some parameters will be estimated based on a pilot data set.

## Usage

 1 numDEpi(X1, X2, power, theta, alpha = 0.05) 

## Arguments

 X1 a nPilot by 1 vector, where nPilot is the number of subjects in the pilot data set. This vector records the values of the covariate of interest for the nPilot subjects in the pilot study. X1 should be binary and take only two possible values: zero and one. X2 a nPilot by 1 vector, where nPilot is the number of subjects in the pilot study. This vector records the values of the second covariate for the nPilot subjects in the pilot study. X2 can be binary or non-binary. power the postulated power. theta postulated hazard ratio alpha type I error rate.

## Details

This is an implementation of the calculation of the number of required deaths derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(β_1 x_1+β_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 should be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(β_1)=θ. Given the type I error rate α for a two-sided test, the total number of deaths required to achieve a power of 1-β is

D=\frac{≤ft(z_{1-α/2}+z_{1-β}\right)^2}{ [\log(θ)]^2 p (1-p) (1-ρ^2), }

where

ρ=corr(X_1, X_2)=(p_1-p_0)\times √{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

p and rho will be estimated from a pilot data set.

## Value

 D the number of deaths required to achieve the desired power with given type I error rate. p proportion of subjects taking X_1=1. rho2 square of the correlation between X_1 and X_2.

## Note

(1) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

(2) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio θ measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio θ measures the difference of effect on the cause-specific hazard.

## References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

numDEpi.default
  1 2 3 4 5 6 7 8 9 10 11 12  # generate a toy pilot data set X1 <- c(rep(1, 39), rep(0, 61)) set.seed(123456) X2 <- sample(c(0, 1), 100, replace = TRUE) res <- numDEpi(X1, X2, power = 0.8, theta = 2, alpha = 0.05) print(res) # proportion of subjects died of the disease of interest. psi <- 0.505 # total number of subjects required to achieve the desired power ceiling(res\$D / psi)