---
title: "Empirical comparison of P-value combination methods in mvMAPIT"
output: rmarkdown::html_vignette
description: >
  Empirical comparison of P-value combination methods in mvMAPIT.
vignette: >
  %\VignetteIndexEntry{Empirical comparison of P-value combination methods in mvMAPIT}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#",
  fig.width=7,
  fig.height=5
)
library(knitr)
```

```{r load_dependencies, message=FALSE}
library(mvMAPIT)
library(GGally)
library(tidyr)
library(dplyr)
```

In Stamp et al. (2023)[^4] we discuss the meta analysis of the pariwise comparison $P$-values that mvMAPIT gives.
We provide three methods to compute a per-variant combined $P$-value:

  1. Fisher's Method[^1]
  2. The harmonic mean $P$[^2]
  3. The Cauchy combination test[^3]

The Fisher's method requires independent $P$-values. The other two tests handle arbitrary
covariances between the $P$-values. Here we show how these three methods compare empirically
when applied to the same $P$-values.

## Generate data

Draw random uniform $P$-values from [0, 0.05]. 

```{r generate_data, eval = TRUE}
n_variants <- 10000
n_combine <- 3
pvalues <- tidyr::tibble(
   id = rep(as.character(c(1:n_variants)), each = n_combine),
   trait = rep(as.character(c(1:n_combine)), n_variants),
   p =  runif(n_variants * n_combine, min = 0, max = 1)
)
```

## Combine $P$-Values with all methods

Use the provided methods to compute the meta analysis $P$-values.

```{r combine, eval = TRUE}
cauchy <- cauchy_combined(pvalues) %>%
  rename(p_cauchy = p) %>%
  select(-trait)
fisher <- fishers_combined(pvalues) %>%
  rename(p_fisher = p) %>%
  select(-trait)
harmonic <- harmonic_combined(pvalues) %>%
  rename(p_harmonic = p) %>%
  select(-trait)
min_max <- pvalues %>%
  group_by(id) %>%
  summarise(p_min = min(p), p_max = max(p))

combined_wide <- fisher %>%
  left_join(harmonic) %>%
  left_join(cauchy) %>%
  left_join(min_max) %>%
  select(-id)
```

# Plot the result

The figure shows the data distribution on the diagonal and paired 2D historgam plots for all combinations of $P$-values. The brighter yellows correspond to higher counts, the green is in the middle of the scale, and the darker blues correspond to the low values of the histogram.

```{r plot, eval = TRUE}
my_bin <- function(data, mapping) {
  ggplot(data = data, mapping = mapping) +
  geom_bin2d() +
  scale_fill_continuous(type = "viridis")
}
ggpairs(combined_wide, columns = 1:5,
        lower = list(continuous = my_bin)) +
  theme_bw() +
  theme(axis.text.x = element_text(angle=90, hjust=1))
```



# References
[^1]: Fisher, R.A. (1925). Statistical Methods for Research Workers. Oliver and Boyd (Edinburgh). ISBN 0-05-002170-2.
[^2]: Wilson, D.J., 2019. The harmonic mean p-value for combining dependent tests. Proceedings of the National Academy of Sciences, 116(4), pp.1195-1200. <https://doi.org/10.1073/pnas.1814092116>
[^3]: Liu, Y. and Xie, J., 2020. Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures. Journal of the American Statistical Association, 115(529), pp.393-402. <https://doi.org/10.1080/01621459.2018.1554485>
[^4]: J. Stamp, A. DenAdel, D. Weinreich, L. Crawford (2023). Leveraging the
    Genetic Correlation between Traits Improves the Detection of Epistasis in
    Genome-wide Association Studies. G3 Genes|Genomes|Genetics, 13(8), jkad118; doi:
    <https://doi.org/10.1093/g3journal/jkad118>