---
title: "Backbone Unit Tests"
author: "Zachary Neal, Michigan State University, zpneal@msu.edu"
output: 
  rmarkdown::html_vignette:
  toc: true
  
bibliography: backbone_bib.bib
link-citations: yes
vignette: >
  %\VignetteIndexEntry{Backbone Unit Tests}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", width = 80)
knitr::opts_knit$set(global.par = TRUE)
```

# Table of Contents {#toc}

1. [Introduction](#introduction)
2. [`backbone_from_projection()`](#projection)
    a. [From matrix](#projection-matrix)
    b. [From igraph](#projection-igraph)
3. [`backbone_from_weighted()`](#weighted)
    a. [From weighted matrix](#weighted-matrix)
    b. [From weighted igraph](#weighted-igraph)
    c. [From weighted projection](#weighted-projection)
    d. [From weighted igraph projection](#weighted-igraph-projection)
4. [`backbone_from_unweighted()`](#unweighted)
    a. [`.escore()`](#unweighted-escore)
    b. [`.normalize()`](#unweighted-normalize)
    c. [`.filter()`](#unweighted-filter)
    d. [Models](#unweighted-models)
5. [Null model functions](#null)
6. [Utility functions](#utility)

# Introduction {#introduction}

The "vignette" is designed solely to display the automated unit tests used to check the `backbone` package during development. To identify any failed unit tests, search in the vignette for the word "failed".

For a general introduction to the backbone package, please see the [Introduction to Backbone](../doc/backbone.html) vignette. Or, for a complete example using empirical data, please see the [Empirical Example of Backbone Extraction](../doc/senate.html) vignette.

```{r}
library(backbone)
library(igraph)
library(tinytest)
```

[back to Table of Contents](#toc)

# `backbone_from_projection()` {#projection}
## Setup {#projection-setup}
```{r}
trace <- function(x){sum(diag(x))}
matcube <- function(x){x%*%x%*%x}
triangle_index <- function(x){(trace(matcube(x)) + trace(matcube(abs(x))))/(2 * trace(matcube(abs(x))))}
```

## From matrix {#projection-matrix}
### SDSM {#sdsm-matrix}
```{r}
B <- rbind(cbind(matrix(rbinom(250,1,.85),10),   #An example block incidence matrix
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10)))

bb <- backbone_from_projection(B, model = "sdsm", return = "everything")  #Extract SDSM matrix, return everything
expect_equal(length(bb),6)  #Returned object contains six elements
expect_true(is(bb$bipartite,"matrix"))
expect_true(is(bb$projection,"matrix"))
expect_true(is(bb$backbone,"matrix"))
expect_true(is(bb$pvalues$upper,"matrix"))
expect_true(is(bb$narrative,"character"))
expect_true(is(bb$call,"call"))

bb <- backbone_from_projection(Matrix::Matrix(B), model = "sdsm", return = "everything")  #Extract SDSM Matrix, return everything
expect_equal(length(bb),6)  #Returned object contains six elements
expect_true(is(bb$bipartite,"Matrix"))
expect_true(is(bb$projection,"Matrix"))
expect_true(is(bb$backbone,"Matrix"))
expect_true(is(bb$pvalues$upper,"matrix"))
expect_true(is(bb$narrative,"character"))
expect_true(is(bb$call,"call"))

bb <- backbone_from_projection(B, model = "sdsm", signed = TRUE)  #Extract SDSM matrix as signed
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### FDSM {#fdsm-matrix}
```{r}
bb <- backbone_from_projection(B, model = "fdsm", signed = TRUE, trials = 250)  #Extract FDSM matrix as signed
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### fixedrow {#fixedrow-matrix}
```{r}
bb <- backbone_from_projection(B, model = "fixedrow", signed = TRUE)  #Extract fixedrow matrix as signed
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### fixedcol {#fixedfill-matrix}
```{r}
bb <- backbone_from_projection(B, model = "fixedcol", signed = TRUE)  #Extract fixedcol matrix as signed
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### fixedfill {#fixedfill-matrix}
```{r}
bb <- backbone_from_projection(B, model = "fixedfill", signed = TRUE)  #Extract fixedfill matrix as signed
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### SDSM-EC {#sdsmec-matrix}
```{r}
B <- as.vector(B)
make_prohibited <- sample(which(B==0), 5, replace = FALSE)  #Pick some missing edges to prohibit
B[make_prohibited] <- 10
make_required <- sample(which(B==1), 5, replace = FALSE)  #Pick some present edges to require
B[make_required] <- 11
B <- matrix(B, 30, 75)  #Reassemble as matrix
bb <- backbone_from_projection(B, model = "sdsm", signed = TRUE)  #Extract SDSM matrix as signed, considering structural values
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

## From igraph {#projection-igraph}
### SDSM {#sdsm-igraph}
```{r}
B <- rbind(cbind(matrix(rbinom(250,1,.85),10),   #An example block incidence matrix
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10)))
B <- graph_from_biadjacency_matrix(B)          #Convert to igraph
V(B)$agent_attrib <- c(c(1:30),rep(NA,75))     #Add agent attribute
V(B)$artifact_attrib <- c(rep(NA,30),c(1:75))  #Add artifact attribute

bb <- backbone_from_projection(B, model = "sdsm", return = "everything")  #Extract SDSM igraph, return everything
expect_equal(length(bb),6)  #Returned object contains six elements
expect_equal(class(bb$bipartite)[1],"igraph")
expect_true(is_bipartite(bb$bipartite))
expect_equal(class(bb$projection)[1],"igraph")
expect_false(is_directed(bb$projection))
expect_equal(class(bb$backbone)[1],"igraph")
expect_false(is_directed(bb$backbone))
expect_equal(class(bb$pvalues$upper)[1],"matrix")
expect_equal(class(bb$narrative)[1],"character")
expect_equal(class(bb$call)[1],"call")

bb <- backbone_from_projection(B, model = "sdsm")                              #Extract SDSM igraph with defaults
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### FDSM {#fdsm-igraph}
```{r}
bb <- backbone_from_projection(B, model = "fdsm", trials = 250)                #Extract FDSM igraph with defaults
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### fixedrow {#fixedrow-igraph}
```{r}
bb <- backbone_from_projection(B, model = "fixedrow")                          #Extract fixedrow igraph with defaults
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### fixedcol {#fixedfill-igraph}
```{r}
bb <- backbone_from_projection(B, model = "fixedrow")                          #Extract fixedcol igraph with defaults
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### fixedfill {#fixedfill-igraph}
```{r}
bb <- backbone_from_projection(B, model = "fixedfill")                         #Extract fixedcol igraph with defaults
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### SDSM-EC {#sdsmec-igraph}
```{r}
E(B)$weight <- NA
E(B)$weight <- sample(c(1,11), length(E(B)$weight), replace = TRUE, prob = c(.9,.1))
bb <- backbone_from_projection(B, model = "sdsm")                              #Extract SDSM igraph with defaults, considering structural values
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

# `backbone_from_weighted()` {#weighted}
## From weighted matrix {#weighted-matrix}
### Disparity
```{r}
W <- matrix(c(0,10,10,10,10,75,0,0,0,0,
              10,0,1,1,1,0,0,0,0,0,
              10,1,0,1,1,0,0,0,0,0,
              10,1,1,0,1,0,0,0,0,0,
              10,1,1,1,0,0,0,0,0,0,
              75,0,0,0,0,0,100,100,100,100,
              0,0,0,0,0,100,0,10,10,10,
              0,0,0,0,0,100,10,0,10,10,
              0,0,0,0,0,100,10,10,0,10,
              0,0,0,0,0,100,10,10,10,0),10)

bb <- backbone_from_weighted(W, model = "disparity", return = "everything")  #Extract disparity backbone, return everything
expect_equal(length(bb),5)  #Returned object contains five elements
expect_true(is(bb$weighted,"matrix"))
expect_true(is(bb$backbone,"matrix"))
expect_true(is(bb$pvalues$upper,"matrix"))
expect_true(is(bb$narrative,"character"))
expect_true(is(bb$call,"call"))

bb <- backbone_from_weighted(Matrix::Matrix(W), model = "disparity", return = "everything")  #Extract disparity backbone, return everything
expect_equal(length(bb),5)  #Returned object contains five elements
expect_true(is(bb$weighted,"Matrix"))
expect_true(is(bb$backbone,"Matrix"))
expect_true(is(bb$pvalues$upper,"matrix"))
expect_true(is(bb$narrative,"character"))
expect_true(is(bb$call,"call"))

bb <- backbone_from_weighted(W, model = "disparity")  #Extract disparity backbone
expect_true(is(bb,"matrix"))                          #Returns as matrix
bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
expect_true(is_tree(bb))                      #Backbone is a tree
```

### LANS
```{r}
bb <- backbone_from_weighted(W, model = "lans")       #Extract lans backbone
expect_true(is(bb,"matrix"))                          #Returns as matrix
bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
expect_true(is_tree(bb))                      #Backbone is a tree
```

### MLF
```{r}
bb <- backbone_from_weighted(W, model = "mlf")        #Extract mlf backbone
expect_true(is(bb,"matrix"))                          #Returns as matrix
bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
expect_true(is_tree(bb))                      #Backbone is a tree
```

### Global
```{r}
bb <- backbone_from_weighted(W, model = "global")     #Extract global backbone (unsigned)
expect_true(is(bb,"matrix"))                          #Returns as matrix
expect_true(table(bb)[1]==58 & table(bb)[2]==42)      #Contains 58 0s and 42 1s
bb <- backbone_from_weighted(W, model = "global", parameter = c(10,74))     #Extract global backbone (signed)
expect_true(table(bb)[1]==12 & table(bb)[2]==78 & table(bb)[3]==10)      #Contains 12 -1s, 78 0s, and 10 1s
```

## From weighted igraph {#weighted-igraph}
### Disparity
```{r}
W <- graph_from_adjacency_matrix(W, mode = "undirected", weighted = TRUE)

bb <- backbone_from_weighted(W, model = "disparity")  #Extract disparity backbone
expect_true(is(bb,"igraph"))                          #Returns as igraph
expect_true(is_tree(bb))                      #Backbone is a tree
```

### LANS
```{r}
bb <- backbone_from_weighted(W, model = "lans")       #Extract lans backbone
expect_true(is(bb,"igraph"))                          #Returns as igraph
expect_true(is_tree(bb))                      #Backbone is a tree
```

### MLF
```{r}
bb <- backbone_from_weighted(W, model = "mlf")        #Extract mlf backbone
expect_true(is(bb,"igraph"))                          #Returns as igraph
expect_true(is_tree(bb))                      #Backbone is a tree
```

### Global
```{r}
bb <- backbone_from_weighted(W, model = "global")     #Extract global backbone (unsigned)
expect_true(is(bb,"igraph"))                          #Returns as matrix
bb <- as_adjacency_matrix(bb, sparse = FALSE)   #Get matrix
expect_true(table(bb)[1]==58 & table(bb)[2]==42)      #Contains 58 0s and 42 1s
bb <- backbone_from_weighted(W, model = "global", parameter = c(10,74))     #Extract global backbone (signed)
bb <- as_adjacency_matrix(bb, sparse = FALSE, attr = "sign")   #Get matrix
expect_true(table(bb)[1]==12 & table(bb)[2]==78 & table(bb)[3]==10)      #Contains 12 -1s, 78 0s, and 10 1s
```

## From weighted projection {#weighted-projection}
### Disparity
```{r}
W <- rbind(cbind(matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10)))
W <- W%*%t(W)
diag(W) <- 0

bb <- backbone_from_weighted(W, model = "disparity", signed = TRUE, alpha = 0.5)  #Extract signed disparity matrix
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### LANS
```{r}
bb <- backbone_from_weighted(W, model = "lans", signed = TRUE, alpha = 0.5)  #Extract signed lans matrix
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### MLF
```{r}
bb <- backbone_from_weighted(W, model = "mlf", signed = TRUE, alpha = 0.5)  #Extract signed mlf matrix
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

### Global
```{r}
upper <- mean(W) + sd(W)             #Use mean + sd as positive edge threshold
lower <- mean(W) - sd(W)             #Use mean - sd as negative edge threshold
bb <- backbone_from_weighted(W, model = "global", parameter = c(lower, upper))  #Extract signed global matrix
expect_true(is(bb,"matrix"))         #Returns as matrix
expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
expect_true(any(bb %in% c(-1)))      #Contains some negative edges
expect_true(any(bb %in% c(0)))       #Contains some missing edges
expect_true(any(bb %in% c(1)))       #Contains some positive edges
triangle_index(bb)
expect_true(triangle_index(bb)>.8)   #Is nearly balanced
```

## From weighted igraph projection {#weighted-igraph-projection}
### Disparity
```{r}
W <- rbind(cbind(matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10),
                 matrix(rbinom(250,1,.15),10)),
           cbind(matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.15),10),
                 matrix(rbinom(250,1,.85),10)))
W <- graph_from_biadjacency_matrix(W)
W <- bipartite_projection(W, which = "false")
V(W)$agent_attrib <- c(c(1:30))     #Add agent attribute

bb <- backbone_from_weighted(W, model = "disparity", return = "everything")  #Extract disparity igraph, return everything
expect_equal(length(bb),5)  #Returned object contains five elements
expect_equal(class(bb$weighted)[1],"igraph")
expect_equal(class(bb$backbone)[1],"igraph")
expect_equal(class(bb$pvalues$upper)[1],"matrix")
expect_equal(class(bb$narrative)[1],"character")
expect_equal(class(bb$call)[1],"call")

bb <- backbone_from_weighted(W, model = "disparity", alpha = 0.25)            #Extract unweighted disparity igraph
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### LANS
```{r}
bb <- backbone_from_weighted(W, model = "lans", alpha = 0.25)                 #Extract unweighted lans igraph
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### MLF
```{r}
bb <- backbone_from_weighted(W, model = "mlf", alpha = 0.25)                  #Extract unweighted mlf igraph
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

### Global
```{r}
threshold <- mean(E(W)$weight) + sd(E(W)$weight)              #Use mean + sd as edge threshold
bb <- backbone_from_weighted(W, model = "global", parameter = threshold)      #Extract unweighted global igraph
expect_true(is(bb,"igraph"))                                                  #Returns as igraph
expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
```

# `backbone_from_unweighted()` {#unweighted}
## `.escore()` {#unweighted-escore}
```{r}
A <- matrix(sample(c(0:1), 100, replace = TRUE),10,10)  #A binary, square, symmetric matrix
diag(A) <- 0
A <- pmax(A, t(A))

test <- backbone:::.escore(A, "random")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "betweenness")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0))  #All values are 0 or larger
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "triangles")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test%%1==0))  #All values are integers
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "jaccard")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "dice")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "quadrangles")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test%%1==0))  #All values are integers
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "quadrilateral")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "degree")
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test%%1==0))  #All values are integers
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "meetmin")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "geometric")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.escore(A, "hypergeometric")
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
```

## `.normalize()` {#unweighted-normalize}
```{r}
A1 <- matrix(sample(c(0,0,0,1,2,3), 100, replace = TRUE),10,10)  #A weighted, square matrix
diag(A1) <- 0
A2 <- pmax(A1, t(A1))  #A weighted, square, symmetric matrix

test <- backbone:::.normalize(A1, "rank")
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test%%1==0))  #All values are integers
expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
test <- backbone:::.normalize(A2, "rank")
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test%%1==0))  #All values are integers
expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result

test <- backbone:::.normalize(A1, "embeddedness")
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
test <- backbone:::.normalize(A2, "embeddedness")
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
```

## `.filter()` {#unweighted-filter}
```{r}
A1 <- matrix(sample(c(0:10), 2500, replace = TRUE),50,50)  #A weighted, square matrix
diag(A1) <- 0
A2 <- A1; A2[upper.tri(A2)] <- t(A1)[upper.tri(A1)]  #Symmetrize using lower triangle

test <- backbone:::.filter(A1, "threshold", 2)
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
expect_true(all(test[A1 <= 2] == 0))  #If edge is below threshold, it is missing in result
test <- backbone:::.filter(A2, "threshold", 2)
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
expect_true(all(test[A2 <= 2] == 0))  #If edge is below threshold, it is missing in result

test <- backbone:::.filter(A1, "proportion", .5)
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
sum(test!=0) / sum(A1!=0)
expect_true((sum(test!=0) / sum(A1!=0)) > 0.3 & (sum(test!=0) / sum(A1!=0)) < 0.7)  #Should keep 30-70% of original edges on average
test <- backbone:::.filter(A2, "proportion", .5)
expect_true(isSymmetric(test))  #Output is symmetric
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
expect_true((sum(test!=0) / sum(A2!=0)) > 0.3 & (sum(test!=0) / sum(A2!=0)) < 0.7)  #Should keep 30-70% of original edges on average

test <- backbone:::.filter(A1, "degree", .5)
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
test <- backbone:::.filter(A2, "degree", .5)
expect_true(all(diag(test)==0))  #Diagonal contains 0s
expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
```

## Models {#unweighted-models}
```{r}
#skeleton (no particular structure expected in backbone)
U <- igraph::sample_sbm(60, matrix(c(.75,.25,.25,.25,.75,.25,.25,.25,.75),3,3), c(20,20,20))  #Unweighted graph with three hidden communities

test <- backbone_from_unweighted(U, model = "skeleton", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
test2 <- backbone_from_unweighted(U, model = "skeleton", parameter = .3, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification

#gspar
test <- backbone_from_unweighted(U, model = "gspar", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "gspar", parameter = .3, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification

#lspar
test <- backbone_from_unweighted(U, model = "lspar", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "lspar", parameter = .3, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification

#simmelian
test <- backbone_from_unweighted(U, model = "simmelian", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "simmelian", parameter = .7, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#jaccard
test <- backbone_from_unweighted(U, model = "jaccard", parameter = .3, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "jaccard", parameter = .5, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#meetmin
test <- backbone_from_unweighted(U, model = "meetmin", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "meetmin", parameter = .7, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#geometric
test <- backbone_from_unweighted(U, model = "geometric", parameter = .3, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "geometric", parameter = .5, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#hyper
test <- backbone_from_unweighted(U, model = "hyper", parameter = .6, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "hyper", parameter = .8, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#quadrilateral
test <- backbone_from_unweighted(U, model = "quadrilateral", parameter = .3, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
test2 <- backbone_from_unweighted(U, model = "quadrilateral", parameter = .5, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification

#degree
U <- igraph::sample_pa(n = 60, m = 3, directed = FALSE)  #A dense, scale-free network
test <- backbone_from_unweighted(U, model = "degree", parameter = .5, return = "everything")
expect_true(length(test)==4)  #Returned object has four elements
expect_true(is(test$narrative,"character"))  #Narrative element is character class
expect_true(is(test$call,"call"))  #Call element is call class
expect_true(all.equal(U,test$original))  #Original element matches starting graph
expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
expect_true(which.max(igraph::degree(U)) == which.max(igraph::degree(test$backbone)))  #Backbone preserves highest-degree node
expect_true(cor(igraph::degree(U),igraph::degree(test$backbone)) > 0.75)  #Backbone preserves degree distribution
test2 <- backbone_from_unweighted(U, model = "degree", parameter = .2, return = "everything")
expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification
```

# Null model functions {#null}
## `.sdsm()` {#sdsm}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.sdsm(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.sdsm_ec()` {#sdsm_ec}
```{r}
M <- rbind(c(10,0,1),c(0,11,0),c(1,0,1))
test <- backbone:::.sdsm_ec(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.fixedrow()` {#fixedrow}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.fixedrow(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.fixedcol()` {#fixedcol}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.fixedcol(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.fixedfill()` {#fixedfill}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.fixedfill(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.fdsm()` {#fdsm}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.fdsm(M, signed = TRUE, missing_as_zero = TRUE, alpha = 0.05, mtc = "none", trials = 1000)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.disparity()` {#disparity}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.disparity(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.lans()` {#lans}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.lans(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

## `.mlf()` {#mlf}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- backbone:::.mlf(M, signed = TRUE, missing_as_zero = TRUE)
test$upper <- round(test$upper,3)
test$lower <- round(test$lower,3)
expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
```

# Utility functions {#utility}
## `bicm()` {#bicm}
```{r}
M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
test <- round(bicm(M),3)
expect_equal(test, rbind(c(.216,.216,.568),c(.216,.216,.568),c(.568,.568,.863)))  #BiCM probabilities
```

## `fastball()` {#fastball}
```{r}
M <- matrix(rbinom(100*1000,1,0.5),100,1000)
test <- fastball(M)
expect_equal(rowSums(test), rowSums(M))  #Row sums match
expect_equal(colSums(test), colSums(M))  #Column sums match
```

## `.retain()` {#retain}
```{r}
upper <- rbind(c(.01,.02,.03),  #Unsigned
               c(.05,.06,.07),
               c(0.5,0.6,0.7))
p <- list(upper = upper)
test <- backbone:::.retain(p, alpha = 0.05, mtc = "none")
expect_equal(test, rbind(c(0,1,1),
                         c(0,0,0),
                         c(0,0,0)))

upper <- rbind(c(.01,.02,.03),  #Signed
               c(.05,.06,.07),
               c(0.5,0.6,0.7))
lower <- rbind(c(0.5,0.6,0.7),
               c(.05,.06,.07),
               c(.01,.02,.03))
p <- list(lower = lower, upper = upper)
test <- backbone:::.retain(p, alpha = 0.1, mtc = "none")  #Higher alpha because this is a two-tailed test
expect_equal(test, rbind(c(0,1,1),
                         c(0,0,0),
                         c(-1,-1,0)))
```
