# Sample Size Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

### Description

Sample size calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

### Usage

1 | ```
ssizeEpiInt(X1, X2, failureFlag, power, theta, alpha = 0.05)
``` |

### Arguments

`X1` |
a |

`X2` |
a |

`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 Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemoilogical studies:

*h(t|x_1, x_2)=h_0(t)\exp(β_1 x_1+β_2 x_2 + γ (x_1 x_2)),*

where both covariates *X_1* and *X_2* are binary variables.

Suppose we want to check if
the hazard ratio of the interaction effect *X_1 X_2=1* to *X_1 X_2=0* is equal to *1*
or is equal to *\exp(γ)=θ*.
Given the type I error rate *α* for a two-sided test, the total
number of subjects required to achieve the desired power *1-β* is:

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

where *ψ* is the proportion of subjects died of
the disease of interest, and

*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)*, and

*G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1},*

and
*p0=Pr(X_1=1 | X_2=0)=myc/(mya+myc)*,
*p1=Pr(X_1=1 | X_2=1)=myd/(myb+myd)*,
*p=Pr(X_1=1)=(myc+myd)/n*,
*q=Pr(X_2=1)=(myb+myd)/n*,
*n=mya+myb+myc+myd*.

*p_{00}=Pr(X_1=0,\mbox{and}, X_2=0)*,
*p_{01}=Pr(X_1=0,\mbox{and}, X_2=1)*,
*p_{10}=Pr(X_1=1,\mbox{and}, X_2=0)*,
*p_{11}=Pr(X_1=1,\mbox{and}, X_2=1)*.

*p_{00}*, *p_{01}*, *p_{10}*, *p_{11}*, and *ψ* will be
estimated from the pilot data.

### Value

`n` |
the total number of subjects required. |

`p` |
estimated |

`q` |
estimated |

`p0` |
estimated |

`p1` |
estimated |

`rho2` |
square of the estimated |

`G` |
a factor adjusting the sample size. The sample size needed to
detect an effect of a prognostic factor with given error probabilities has
to be multiplied by the factor |

`mya` |
estimated number of subjects taking values |

`myb` |
estimated number of subjects taking values |

`myc` |
estimated number of subjects taking values |

`myd` |
estimated number of subjects taking values |

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

### References

Schmoor C., Sauerbrei W., and Schumacher M. (2000).
Sample size considerations for the evaluation of prognostic factors in survival analysis.
*Statistics in Medicine*. 19:441-452.

### See Also

`ssizeEpiInt.default0`

, `ssizeEpiInt2`

### Examples

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