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.

1 | ```
numDEpi.default(power, theta, p, rho2, alpha = 0.05)
``` |

`power` |
the postulated power. |

`theta` |
postulated hazard ratio |

`p` |
proportion of subjects taking the value one for the covariate of interest. |

`rho2` |
square of the correlation between the covariate of interest and the other covariate. |

`alpha` |
type I error rate. |

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)*.

The number of deaths required to achieve the desired power with given type I error rate.

(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 cause-specific hazard for a particular failure,
while in Schoenfeld (1983), the hazard ratio *θ* measures
the difference of effect on subdistribution hazard.

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.

1 2 3 4 5 6 7 8 9 10 | ```
# Example at the end of Section 5.2 of Latouche et al. (2004)
# for a cohort study.
D <- numDEpi.default(power = 0.8, theta = 2, p = 0.39,
rho2 = 0.132^2, alpha = 0.05)
# proportion of subjects died of the disease of interest.
psi <- 0.505
# total number of subjects required to achieve the desired power
ceiling(D / psi)
``` |

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