SNP Regression Model

For each SNP with coded genotype $g \in {0,1,2}$ and phenotype vector $\mathbf{y}$, the basic linear model is:

\[y_i = \mu + \beta g_i + \mathbf{x}_i^\top \boldsymbol{\gamma} + \varepsilon_i, \quad \varepsilon_i \sim \mathcal{N}(0, \sigma^2)\]

where $\beta$ is the additive SNP effect, $\mathbf{x}_i$ contains covariates (age, sex, principal components), and $\boldsymbol{\gamma}$ are their coefficients. The genome-wide significance threshold is $p < 5 \times 10^{-8}$, correcting for approximately one million independent tests.

Mixed-Model Correction for Population Structure

Population stratification inflates test statistics. The linear mixed model (LMM) adds a polygenic random effect:

\[\mathbf{y} = \mu + \mathbf{g}\beta + \mathbf{X}\boldsymbol{\gamma} + \mathbf{u} + \boldsymbol{\varepsilon}\] \[\mathbf{u} \sim \mathcal{N}(\mathbf{0},\, \sigma_g^2 \mathbf{K}), \quad \boldsymbol{\varepsilon} \sim \mathcal{N}(\mathbf{0},\, \sigma_e^2 \mathbf{I})\]

where $\mathbf{K}$ is the genomic relationship matrix (GRM). Software such as BOLT-LMM and SAIGE implement efficient approximations for biobank-scale data.

LD Score Regression

LD score regression disentangles genuine polygenicity from confounding. The expected $\chi^2$ statistic for SNP $j$ is:

\[\mathbb{E}[\chi^2_j] = \frac{Nh^2}{M} \ell_j + Na + 1\]

where $\ell_j = \sum_k r^2_{jk}$ is the LD score, $h^2$ is SNP heritability, $M$ is the number of SNPs, and $a$ captures confounding. The intercept $Na + 1$ reveals inflation not due to polygenicity.

Visualisation

Plot	Purpose
Manhattan plot	Genome-wide $-\log_{10}(p)$ by chromosomal position
QQ plot	Observed vs. expected $p$-values; slope (genomic $\lambda$) measures inflation
Regional LocusZoom	LD colouring around index SNP for fine-mapping