Sample Size Calculation for Cox Proportional Hazards Regression with Nonbinary Covariates for Epidemiological Studies

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Description

Sample size calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies.

Usage

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ssizeEpiCont(formula, dat, X1, failureFlag, power, 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).

power

postulated power.

theta

postulated hazard ratio.

alpha

type I error rate.

Details

This is an implementation of the sample size 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 total number of subjects required to achieve a sample size of 1-β is

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

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^2, σ^2, and ψ will be estimated from a pilot study.

Value

n

the total number of subjects required.

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 ssize 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.

See Also

ssizeEpiCont.default

Examples

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  # 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)

  ssizeEpiCont(formula = X1 ~ X2, dat = dat, X1 = X1, failureFlag = failureFlag, 
    power = 0.806, theta = exp(1), alpha = 0.05)

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