# powerEpiCont: Power Calculation for Cox Proportional Hazards Regression... In powerSurvEpi: Power and Sample Size Calculation for Survival Analysis of Epidemiological Studies

## Description

Power calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies. Some parameters will be estimated based on a pilot data set.

## Usage

 1 powerEpiCont(formula, dat, X1, failureFlag, n, theta, alpha = 0.05) 

## Arguments

 formula a formula object relating the covariate of interest to other covariates to calculate the multiple correlation coefficient. The variables in formula must be in the data frame dat. dat a nPilot by p data frame representing the pilot data set, where nPilot is the number of subjects in the pilot study and the p (>1) columns contains the covariate of interest and other covariates. X1 the covariate of interest. failureFlag a nPilot by 1 vector of indicators indicating if a subject is failure (failureFlag=1) or alive (failureFlag=0). n total number of subjects. theta postulated hazard ratio. alpha type I error rate.

## Details

This is an implementation of the power calculation formula derived by Hsieh and Lavori (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, \boldsymbol{x}_2)=h_0(t)\exp(β_1 x_1+\boldsymbol{β}_2 \boldsymbol{x}_2),

where the covariate X_1 is a nonbinary variable and \boldsymbol{X}_2 is a vector of other covariates.

Suppose we want to check if the hazard ratio of the main effect 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 power required to detect a hazard ratio as small as \exp(β_1)=θ is

power=Φ≤ft(-z_{1-α/2}+√{n[\log(θ)]^2 σ^2 ψ (1-ρ^2)}\right),

where σ^2=Var(X_1), ψ is the proportion of subjects died of the disease of interest, and ρ is the multiple correlation coefficient of the following linear regression:

x_1=b_0+\boldsymbol{b}^T\boldsymbol{x}_2.

That is, ρ^2=R^2, where R^2 is the proportion of variance explained by the regression of X_1 on the vector of covriates \boldsymbol{X}_2.

rho will be estimated from a pilot study.

## Value

 power The power of the test. rho2 square of the correlation between X_1 and X_2. sigma2 variance of the covariate of interest. psi proportion of subjects died of the disease of interest.

## Note

(1) Hsieh and Lavori (2000) assumed one-sided test, while this implementation assumed two-sided test. (2) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

## References

Hsieh F.Y. and Lavori P.W. (2000). Sample-size calculation for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 21:552-560.

powerEpiCont.default
 1 2 3 4 5 6 7 8 9  # generate a toy pilot data set set.seed(123456) X1 <- rnorm(100, mean = 0, sd = 0.3126) X2 <- sample(c(0, 1), 100, replace = TRUE) failureFlag <- sample(c(0, 1), 100, prob = c(0.25, 0.75), replace = TRUE) dat <- data.frame(X1 = X1, X2 = X2, failureFlag = failureFlag) powerEpiCont(formula = X1 ~ X2, dat = dat, X1 = X1, failureFlag = failureFlag, n = 107, theta = exp(1), alpha = 0.05)