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

### Description

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

### Usage

1 | ```
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` |
a |

`X1` |
the covariate of interest. |

`failureFlag` |
a |

`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 |

`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

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