---
title: "Custom Network Comparison Tests"
bibliography: ../inst/REFERENCES.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Custom Network Comparison Tests}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Introdution
A key feature of `nct` is the ability to test anything hypothesis, as opposed
to the set of defaults provided in the `R` package **NetworkComparisonTest** 
[@van2017comparing]. 


# Correlation
This first example compares the correlation between edges

```{r}
library(GGMncv)

set.seed(1)

main <- gen_net(p = 10)

y1 <- MASS::mvrnorm(n = 500, 
                    mu = rep(0, 10), 
                    Sigma = main$cors)

y2 <- MASS::mvrnorm(n = 500, 
                    mu = rep(0, 10), 
                    Sigma = main$cors)

Correlation <- function(x, y){
  cor(x[upper.tri(x)], y[upper.tri(x)])
}

compare_ggms <- nct(y1, y2, 
                    FUN = Correlation, 
                    progress = FALSE)

compare_ggms
```
By default, the $p$-value corresponds to the proportion of the null distribution 
greater than the observed. For some
test statistics, this might not make sense so they are not provided. 

A fail safe is to visualize the
results and compute the $p$-value manually.

```{r}
hist(compare_ggms$Correlation_perm, 
     main = "null dist: correlation")
abline(v = compare_ggms$Correlation_obs)
```

Because a higher correlation is a good thing, 1 minus the proportion 
of the null distribution greater than the observed make sense.

```{r}
1 - mean(compare_ggms$Correlation_perm > compare_ggms$Correlation_obs)
```

# Predictability

Next is predictability, or $R^2$, for the first node in the networks. The
first step is defining a function.

```{r}
# define function
r2 <- function(x, y){
  diag(x) <- 1
  diag(y) <- 1
  
  # network 1
  inv1 <- solve(corpcor::pcor2cor(x))
  beta1 <- -(inv1[1,-1] / inv1[1,1])
  r21 <- cor(y1[,1], y1[,-1] %*% beta1)^2
  
  # network 2
  inv2 <- solve(corpcor::pcor2cor(y))
  beta2 <- -(inv2[1,-1] / inv2[1,1])
  r22 <- cor(y2[,1], y2[,-1] %*% beta2)^2
  
  return(as.numeric(r21 - r22))
}
```

Next use `nct`
```{r}
compare_ggms <- nct(y1, y2, 
                    progress = FALSE,
                    FUN = r2)

hist(compare_ggms$r2_perm,  
     main = "null dist: R2 Difference")
abline(v = compare_ggms$r2_obs)
```

Note anything can be tested, say, expected influence. As such, this is not limited
to only one value returned by the custom function (as in the above examples), 
but can be any number. There is an example in the documentation.

Further, although not provided here, both tests could have been done at the same time
by passing a list to `FUN`, i.e., 
`list_fun <- list(r2 = r2, Correlation = Correlation) `.

# References
