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

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

`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 |

`n` |
total number of subjects. |

`theta` |
postulated hazard ratio. |

`alpha` |
type I error rate. |

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.

`power` |
The power of the test. |

`rho2` |
square of the correlation between |

`sigma2` |
variance of the covariate of interest. |

`psi` |
proportion of subjects died of the disease of interest. |

(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`

.

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.

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)
``` |

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