Author: David Gerard

# Fast LD Estimation

#### David Gerard

# Abstract

We demonstrate our new moment-based linkage disequilibrium (LD) estimators, as implemented in the `ldfast()` function. These methods are described in detail in Gerard (2021). The following code chunks are only evaluated if `{updog}` is installed with a version of at least 2.0.2.

# Analysis

We will load the packages and data that we will need

    library(updog)
    library(ldsep)
    library(corrplot)
    #> corrplot 0.95 loaded
    data("uit", package = "ldsep")
    class(uit)
    #> [1] "multidog"

We need the posterior probabilities for each genotype for each individual at each SNP. In `{updog}`, this corresponds to the `Pr_n` variables in its output, where `n` varies from 0 to the ploidy. We can use `updog::format_multidog()` to extract these values into an array with the proper dimensions.

    ploidy <- 4
    gp <- format_multidog(x = uit, varname = paste0("Pr_", 0:ploidy))
    class(gp)
    #> [1] "array"
    dim(gp)
    #> [1] 10 84  5

The first dimension indexes the SNPs, the second dimension indexes the individuals, and the third dimension indexes the possible dosages. So `gp[i, j, k]` contains the posterior probability that individual `j` has dosage `k-1` at SNP `i`.

You can use other genotyping programs to obtain these posterior probabilities, but they have to be formatted as an array with the dimensions in the correct order (SNPs by individuals by dosages).

However, in order for these estimates to perform well, you need to use posterior genotype probabilities that have been calculated using adaptive priors, i.e. empirical/hierarchical Bayes approaches. There are many approaches that do this, such as `{updog}`, `{polyRAD}`, `{fitPoly}`, or `{SuperMASSA}`. Note that GATK uses a uniform prior, so would be inappropriate for use in `ldfast()`.

Let’s use our genotype posterior probabilities (`gp`) to estimate LD.

    ldout <- ldfast(gp = gp, type = "r2")

`ldout` contains:

- `ldmat`: The estimated squared correlation coefficients, adjusted for genotype uncertainty.

      corrplot(corr = ldout$ldmat,
               method = "color",
               type = "upper",
               diag = FALSE,
               tl.pos = "n")

- `semat`: The estimated standard errors for the squared correlation coefficient estimates.

      corrplot(corr = ldout$semat,
               method = "color",
               type = "upper",
               diag = FALSE,
               tl.pos = "n",
               is.corr = FALSE)

- `rr`: The estimated “reliability ratios”, the multiplicative effects applied to the naive LD estimates using just the posterior mean genotypes.

      graphics::hist(x = ldout$rr,
                     main = "Reliability Ratios",
                     xlab = "Reliability Ratios")

The slowest part of `ldfast()` is using hierarchical shrinkage on the log of the reliability ratios (`shrinkrr = TRUE`), but I wouldn’t disable this as some reliability ratios have high variance and are greatly improved through shrinkage.

Other LD estimators are available: `D`, `Dprime`, `r`, and `z`. See the help file of `ldfast()` for details.

# References

- Gerard, David (2021). “Scalable Bias-corrected Linkage Disequilibrium Estimation Under Genotype Uncertainty.” Heredity, 127(4), 357–362. doi:10.1038/s41437-021-00462-5.