An Introduction to Association Analysis

Abstract

This chapter focuses on techniques commonly used in GWAS studies to estimate single SNP marker associations in samples of unrelated individuals; when the phenotype is discrete (disease/no disease) then case–control methods, conditional and unconditional logistic regression, are typically utilized. Maximum likelihood estimation for generalized linear models is reviewed, and the score, Wald, and likelihood ratio tests are defined and discussed. The analysis of data from nuclear family-based designs is also briefly introduced. Issues regarding confounding, measurement error, effect mediation, and interactions are described. Control for multiple comparisons is reviewed with an emphasis placed on the behavior of the Bonferroni criteria for multiple correlated tests. The effects on statistical estimation and inference of the loss of independence between outcomes are characterized for a specific model of loss of independence, which is relevant to the presence of hidden population structure or relatedness. These last results build on a basic theme described in Chap. 2 and are then carried forward in Chap. 4.

Appendix

Proof of Equation (3.26)

We have the OLS estimate of σ ² equal to $\frac{1}{N-r}Y\prime \left(I-\mathbf{X}{\left(\mathbf{X}\hbox{'}\mathbf{X}\right)}^{-1}\mathbf{X}\prime \right)Y$. In order to simply the notation re-write this as

$$ \frac{1}{N-r}Y\prime \left(I-\mathbf{P}\right)Y, $$

where P = X(X′X)^− 1 X′. Note that PP = P and (I − P)(I − P) = (I − P) i.e. both P and I − P are idempotent with trace equal to r and N − r respectively. Now we take the expected value of the estimate. We have

$$ \begin{array}{l}E\left({\widehat{\sigma}}^2\right)=E\left\{\frac{1}{N-r}Y\prime \left(\mathbf{I}-\mathbf{P}\right)Y\right\}=\frac{1}{N-r}\mathrm{tr}\left\{\left(\mathbf{I}-\mathbf{P}\right)E\right( YY\prime \left)\right\}\\ {}=\frac{1}{N-r}\mathrm{tr}\left\{\left(\mathbf{I}-\mathbf{P}\right)\right[\mathrm{Var}(Y)+E(Y)E\left(Y\prime \right)\left]\right\}.\\ {}\end{array} $$

Note that

$$ \left(\mathbf{I}-\mathbf{P}\right)E(Y)E\left(Y\prime \right)=\mathbf{X}\beta \beta \prime \mathbf{X}\prime -\mathbf{X}\beta \beta \prime \mathbf{X}\prime \mathbf{X}{\left(\mathbf{X}\prime \mathbf{X}\right)}^{-1}\mathbf{X}=0. $$

Thus the above expression simplifies to

$$ \frac{1}{N-r}\mathrm{tr}\left\{\left(\mathbf{I}-\mathbf{P}\right)\left[\mathrm{Var}(Y)\right]\right\}. $$

Since it assumed that Var(Y) = σ ² I + γ ² K, the above expression is equal to

$$ \begin{array}{l}\frac{1}{N-r}\mathrm{tr}\left\{\left(\mathbf{I}-\mathbf{P}\right)\left[{\sigma}^2\mathbf{I}+{\gamma}^2\mathbf{K}\right]\right\}=\frac{\sigma^2}{N-r}\mathrm{tr}\left\{\right(\mathbf{I}-\mathbf{P}\left)\right\}+\frac{\gamma^2}{N-r}\mathrm{tr}\left\{\right(\mathbf{I}-\mathbf{P}\left)\mathbf{K}\right\}\\ {}={\sigma}^2+\frac{\gamma^2}{N-r}\mathrm{tr}\left\{\left(\mathbf{K}-\mathbf{P}\mathbf{K}\right)\right\}={\sigma}^2+\frac{\gamma^2}{N-r}\mathrm{tr}\left\{\right(\mathbf{K}-\mathbf{X}{\left(\mathbf{X}\hbox{'}\mathbf{X}\right)}^{-1}\mathbf{X}\mathbf{K}\left)\right\}\\ {}={\sigma}^2+\frac{\gamma^2}{N-r}\left[\mathrm{tr}\left\{\mathbf{K}\right\}-\mathrm{tr}\left\{{\left(\mathbf{X}\hbox{'}\mathbf{X}\right)}^{-1}\mathbf{X}\prime \mathbf{KX}\right\}\right]\end{array} $$