estimate_densities_pval: Estimate parameters under the alternative density
In kjgleason/primo: Package in R for Integrative Multi-Omics association analysis

Description Usage Arguments Details Value

Estimate the parameters of the alternative density function using a mixture of chi-squared distributions of -2*log(P)-values.

1	estimate_densities_pval(pvals, alt_prop, method_moments = FALSE)

`pvals`	vector of P-values.
`alt_prop`	proportion of P-values from the alternative density.
`method_moments`	logical, denoting whether to estimate scale and degree of freedom parameters using Method of Moments (`TRUE`) or optimization (`FALSE`).

The function estimates densities under the null and under the alternative given a vector of P-values from test statistics and the marginal probability of coming from the alternative distribution. Under the null hypothesis, -2*log() transformed P-values follow a chi-squared distribution with 2 degrees of freedom. Under the alternative, the function assumes that the -2*log() transformed P-values follow a scaled chi-squared distribution with unknown scale parameter and unknown degrees of freedom.

When method_moments=TRUE, the two unknown parameters are estimated by using the first and second moments of the transformed P-values given the known proportion of alternatives (specified in alt_prop). The function solves a theoretical formula of the first and second moments as functions of the scale parameter and degrees of freedom. Note that the alternative density estimation may fail if alt_prop is small and method_moments=TRUE.

When method_moments=FALSE, the two unknown parameters are estimated by minimizing the differences between the P-values given the parameters and the nominal P-values based on each statistic's rank, using optimization algorithms.

A list with the following elements:

`chi_mix`	vector of -2log(P)*-values.
`scaler`	estimated scaling factor for the alternative distribution.
`df_alt`	estimated degrees of freedom for the alternative distribution.