Statistical formula
🎯 Summary of equations
| Quantity | Definition | |
|---|
| Genotype properties | | |
| MAF estimate for a SNP $j$ | $\hat p_j = \frac {\bar{x}_j} {2}$ | |
| Expected MAF sampling variance | ${SE}^2_{\hat p_j} = \frac {\hat p_j(1 - \hat p_j)} {2N}$ | |
| Expected variance of a SNP | $var(X_j) = 2\hat p_j (1 - \hat p_j)$ | |
| LD matrix | $L = \frac {X^TX}{N}$ | |
| GRM | $A = \frac {XX^T}{M}$ | |
| LD score of a SNP | $l_j = \frac {1}{N^2}X^T_jXX^TX_j$ | |
| SNP effect estimates | | |
| OLS effect estimate for model with one SNP($\hat \beta_{GWAS}$) | $\hat \beta_{j, GWAS} = \frac{X^T_jy}{X^T_jX_j} = \frac{cov(X_j, y)}{var(X_j)}$ | |
| Mixed linear mode association (MLMA) estimate for one SNP | $\beta _{j,MLMA} = \frac {X^T_jV^{-1}y}{X^T_jV{-1}X_j}$ | |
| OLS effect estimate for model with all SNPs ($\hat \beta_{OLS}$) | $\hat \beta_{OLS} = (X^TX)^{-1}X^Ty$ | |
| BLUP effect estimate | $\hat \beta_{BLUP} = (X^TX + \lambda I)^{-1}X^Ty$ | |
| Precision of SNP effect estimates | | |
| Excepted sampling variance of $\hat \beta^*_{j,GWAS}$ | $SE^2_{\hat \beta^_j} = var({\hat \beta^_j} | {\hat \beta*_j}) \approx \frac{1}{N \times var(X_j)}$ |
✅ 1. Genotype Properties
1.1 MAF estimate for SNP $j$:
\[\hat p_j = \frac {\bar{x}_j} {2}\]
- Meaning: the estimated frequency of the MAF for SNP $j$, assuming that genotypes are coded as 0/1/2.
- R script:
p_hat_j = mean(x[, j]) / 2
1.2 Expected MAF sampling variance:
\[{SE}^2_{\hat p_j} = \frac {\hat p_j(1 - \hat p_j)} {2N}\]
- Meaning: the estimated variance of MAF under sample size $N$.
- R script:
se2_p_hat_j = p_hat_j*(1 - p_hat_j) / (2 * N)
1.3 Expected variance of a SNP:
\[var(X_j) = 2\hat p_j (1 - \hat p_j)\]
- Meaning: the genotype variance of SNP $j$, theoretically, consistent with the variance of the variable encoded as 0/1/2.
- If we set $p$ as MAF and $q$ as another allele frequency, then $var(X_j) = 2\hat p_j * \hat q_j$
- R script:
var_x_j = 2 * p_hat_j * (1 - p_hat_j)
1.4 LD matrix:
\[L = \frac {X^TX}{N}\]
- Meaning: the normalized SNP covariance matrix (LD matrix).
- R script:
