Most fine-mapping methods assume summary statistics from marginal association studies are normally distributed, with covariance determined by LD 1
\[\begin{align*} \hat{ {\bf z} } \sim N({\bf z}, R) \end{align*}\]
Statistical property of OLS– what if the marginal effects are coming from somewhere else?
Method | Notes | Summary stats |
---|---|---|
Generalized IBSS | “correct” model, hueristic algorithm | No |
Logistic + RSS | ad-hoc, actually used (5,6) | Yes |
Linear + RSS | mis-specified model, correct algorithm | Yes |
Logistic + RSS
Logistic GIBSS
An under-appreciated source of “LD mismatch”?
\(n = 500\), \(b_0 = -1\), \(b = 0, 1, 2, 3\) 1
Figure 1: Wakefield’s ABF can be order of magnitude off when the \(z\)-score is large
!()[resources/abf_biased.png]
!()[resources/abf_eq.png]
Simulation: one causal variant in the locus that explains \(1\%\) of heritability of liability. \(h^2 = 0.1, 0.2, 0.5, 0.9\)
\[\begin{align*} y \sim Bin(1, \sigma(\psi)) \\ \psi = b_0 + b x + \epsilon \\ \epsilon \sim N(0, \sigma^2) \end{align*}\]
95% C.I. for different \(h^2\)
\[ \begin{align*} y_i &\sim Bin\left(1, \sigma \left(b_0 + \sum_{j=1}^q b_j x_{ij} + \delta\right)\right)\\ b &\sim N(0, \sigma^2) \\ \delta &\sim N(0, \nu - q \sigma^2)\\ \end{align*} \]
Value | Description |
---|---|
\(X\) | Standardized genotypes |
\(\sigma^2\) | Variance of standardized effects, i.e. \(b \sim N(0, \sigma^2)\) |
\(q\) | Number of causal variants in locus |
\(\rho\) | Fraction of variance of genetic component in-locus |
\(k\) | Fraction of cases (determines \(b_0\)) |
\(q\sigma^2\) | (Expected) variance of genetic component in-locus |
\(\nu\) | \(q \sigma^2/\rho\), (expected) variance of genetic component |
\(h^2\) | \(\nu / (\nu + \pi^2/3)\), (expected) heritability of liability 1 |
(5), Alzheimers meta analysis combining linear and logistic association studies (6) logistic-mixed model SAIGE + SuSiE
A few options:
Limiting BF
Idea put a normal prior on all covariates \(\begin{bmatrix} \alpha \\ \beta \end{bmatrix} \sim N(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} I_{p-1} \tau_0^{-1} & 0 \\ 0 & \tau_1^{-1} \end{bmatrix}\) and compute Laplace approximation to the BF. Take \(\tau_0 \rightarrow 0+\).
Q: How variable is the scaling factor? Can we get away with just using the univariate BF?
Gauss-Hermite quadrature
\[ I = \int f(x) e^{-x^2} dx \approx \sum_{i=1}^n w_i f(x_i) \]
\((x_i)_{i=1}^n\) are the roots of the Hermite polynomial \(H_n(x)\), \(w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 H_{n-1}^2 (x_i)}\)
\[ I = \int f(x) dx = \int \left[\frac{f(x)}{q(x)} \right] q(x) dx, \;\; q(x) = N(x | \mu, \sigma^2)\; \text{s.t.}\; \frac{f}{q} \approx 1 \]
(note: change of variable + scaling factor to apply the \(n\) point Hermite quadrature rule)