Last updated: 2017-03-07

Code version: ce12916

Purpose

My purpose here is just to outline some thoughts about the LDAK model. Disclaimer: these are very preliminary.

A simple block model for LD (to help thinking)

Consider a model relating phenotype \(Y\) (\(n\) by 1) to SNPs \(X\) (\(n\) by \(p\)), \[Y=X\beta + E.\]

I find it useful to think about really extreme and simple situations. So suppose your \(X\) variables consist of blocks of completely correlated SNPs, where here completely correlated means population correlation = 1. Suppose also that the blocks are independent.

Let \(p_j\) denote the number of SNPs in block \(j\).

So the first \(p_1\) SNPs are completely correlated, next \(p_2\) are completely correlated etc. And the first \(p_1\) are independent of the next \(p_2\) etc.

Let \(g_j\) denote the genotypes (an \(n\) vector) of any SNP in block \(j\). (They are all the same by assumption).

Classic polygenic model

I will consider the “classic polygenic model” to be that the \(\beta_j\) are iid and normal,
with \[\beta_j \sim N(0,\sigma_b^2)\].

Then \(Y \sim N(0, \sigma_b^2 XX' + \sigma^2 I)\), or \[Y \sim N(0, \sigma_b^2 \sum_{j} p_j g_j g_j' + \sigma^2 I)\].

That is, in the covariance matrix each block is weighted by the number of SNPs \(p_j\).

LDAK model

As I understand it (check this!) the LDAK model would instead assume that \(\beta_j\) is somehow weighted by the number of SNPs it is in LD with: \[\beta_j \sim N(0,(1/p_j)\sigma_b^2)\]

so \[Y \sim N(0, \sigma_b^2 \sum_{j} g_j g_j' + \sigma^2 I)\].

Marginal effects and LD-score regression

Let \(\hat{\beta}_j\) denote the estimated marginal effect of \(\beta_j\) from a univariate regression of \(Y\) on \(X_j\). What is the expected \(\hat{\beta}_j\)? We can compute this from RSS work.

I haven’t yet done the algebra, but it seems that:

  • Under the classic polygenic model it would scale roughly linearly with \(p_j\). [fill in details]

  • Under the LDAK model, it would be constant among \(j\).

The LD score regression model attempts to distinguish between polygenicity and confounding by assuming that:

  1. the “polygenic” effects scale with the number of SNPs you are in LD with. That is, the classic polygenic model.

  2. the “confounding” effects are constant, independent of the number of SNPs you are in LD with. That is, the LDAK model.

In other words, from the point of view of LD score regression, the LDAK model is going to capture confounding, not polygenicity.

Speculation

Consider fitting a combined model, with two variance components \[Y \sim N(0, \sigma_1^2 K_1 + \sigma_2^2 K_2 + \sigma^2 I)\] where \(K_1 = \sum_{j} p_j g_j g_j'\) and \(K_2 = \sum_j g_j g_j'\). I speculate that the estimate of \(\sigma_1\) corresponds to the slope of the LD regression model and that the estimate of \(\sigma_2\) corresponds to the intercept.

If correct, this would give a “full data” model that corresponds to the LD score regression model.

Can you formalize this?

Comments

  • There seems to be no good reason to assume that the true (causal) effect of a SNP will depend on how many other SNPs it is in LD with. (Maybe one could make convoluted arguments based on selection, but it seems to be not the most natural starting point.) That is, to me the LDAK assumption seems unnatural as a model for polygenicity.

  • If the above is correct then there seems to be something of a contradiction in the results from LD-score regression papers and LDAK. Specifically the LD score regression papers seems to suggest that in most diseases the inflation of test statistics is due to polygenicity and not confounding. Whereas, if the interpretation of the LDAK model as capturing confounding is correct, then the LDAK papers suggest that it is mostly due to confounding (although the LDAK authors do not interpret it this way I think.)

Session information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] backports_1.0.5 magrittr_1.5    rprojroot_1.2   tools_3.3.2    
 [5] htmltools_0.3.5 yaml_2.1.14     Rcpp_0.12.9     stringi_1.1.2  
 [9] rmarkdown_1.3   knitr_1.15.1    git2r_0.18.0    stringr_1.2.0  
[13] digest_0.6.12   evaluate_0.10

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