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---
title: Shiny for Genetic Analysis Package (gap) Designs
output:
  html_document:
    mathjax:  default
    fig_caption:  true
    toc: true
    section_numbering: true
bibliography: shinygap.bib
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Shiny for Genetic Analysis Package (gap) Designs}
  %\VignetteEncoding{UTF-8}
---

# Appendix: Theory for gap designs

## Family-based and population-based designs

See the R/gap package vignette jss or @zhao07.

## Case-cohort design

### Power

Following @cai04, we have
$$\Phi\left(Z_\alpha+\tilde{n}^\frac{1}{2}\theta\sqrt{\frac{p_1p_2p_D}{q+(1-q)p_D}}\right)$$

where $\alpha$ is the significance level, $\theta$ is the log-hazard ratio for
two groups, $p_j, j = 1, 2$, are the proportion of the two groups
in the population ($p_1 + p_2 = 1$), $\tilde{n}$ is the total number of subjects in the subcohort, $p_D$ is the proportion of the failures in
the full cohort, and $q$ is the sampling fraction of the subcohort.

### Sample size

$$\tilde{n}=\frac{nBp_D}{n-B(1-p_D)}$$ where $B=\frac{Z_{1-\alpha}+Z_\beta}{\theta^2p_1p_2p_D}$ and $n$ is the whole cohort size.

# References

@article{cai04,
   author = {Cai, J. and Zeng, D.},
   title = {Sample size/power calculation for case–cohort studies},
   journal = {Biometrics},
   volume = {60},
   number = {4},
   pages = {1015-1024},
   ISSN = {1471-0056},
   DOI = {10.1111/j.0006-341X.2004.00257.x},
   year = {2004},
   pmid = {15606422},
   type = {Journal Article}
}

@article{zhao07,
   Author = {Zhao, J. H.},
   Title = {gap: genetic analysis package},
   Journal = {Journal of Statistical Software},
   Volume = {23},
   Number = {8},
   Pages = {1-18},
   Year = {2007}
}

csl: biomed-central.csl