Simulations

We perform a simple simulation using the function simulateData. \(1000\) features are generated \(30\) times as normally distributed with mean \(0\) under the null hypothesis and mean under the alternative computed considering the difference in means having the power of the one-sample t-test equals \(0.8\). The proportion of true null hypothesis equals \(\pi = 0.8\).

datas <- simulateData(pi0 = 0.8, m = 1000, n = 30, power = 0.9, rho = 0.5,seed = 123)

pARI then computes the lower bound for the number of true discoveries inside the set containing the first \(200\) features. The user must specify the cluster in the ix set, i.e., from \(1\) to \(200\) in this case. We apply the one-sample t-test for each feature.

out <- pARI(X = datas, ix = c(1:200), test.type = "one_sample", seed = 123)
out$TDP
#>      [,1]
#> [1,] 0.68

Therefore, we can say that at least 68 of features are truly significant inside the ix cluster.

However, the pARI function can analyzed directly the matrix of permuted p-values. We can compute it by signTest function:

out <- signTest(X = datas, B = 1000, rand = F)
P <- cbind(out$pv, out$pv_H0)
pARI(pvalues = P, ix = c(1:200),test.type = "one_sample")$TDP
#>      [,1]
#> [1,] 0.72

The set of features can also be expressed as a vector with length equals the number of features where different values indicate the different sets. For example, we can construct four random clusters as

ix <- sample(c(1:4), size = 1000, replace = T)
out <- pARI(pvalues = P, ix = ix,test.type = "one_sample", clusters = TRUE)$TDP
out
#> [1] 0.1298701 0.1150794 0.1198502 0.1000000

specifying in the pARI function cluster = TRUE. Then, pARI returns the lower bound for the true discovery proportion for each set of features. We can say that we have at least 12.987013 of truly active features in the first cluster.

Gene cluster analysis

Let consider a simple example using Montgomery et al. (2010) and Pickrell et. al (2010) data, i.e., an expression set that combines two studies transcriptome genetics using second generation sequencing HapMap. The data comprises \(52580\) genes and \(129\) samples (i.e., \(60\) unrelated Caucasian individuals of European descent and \(69\) unrelated Nigerian individuals). After pre-processing steps, we perform a two-sample t-test for each gene, and we define the sets of interest the ones computed by the hclust function.

if (!requireNamespace("BiocManager", quietly = TRUE)){
  install.packages("BiocManager")
}
    

if (!requireNamespace("dynamicTreeCut", quietly = TRUE)){
  install.packages("dynamicTreeCut")
}
    

#BiocManager::install(c("Biobase","genefilter", "EnrichmentBrowser"))

library(Biobase)
library(genefilter)
library(dynamicTreeCut)

load(file=url("http://bowtie-bio.sourceforge.net/recount/ExpressionSets/montpick_eset.RData"))

pdata<- pData(montpick.eset)
edata <- as.matrix(exprs(montpick.eset))
fdata <- fData(montpick.eset)

edata <- log2(as.matrix(edata) + 1)
edata <- edata[rowMeans(edata) > 10, ]

my.dist <- dist(edata)
my.tree <- hclust(my.dist, method="ward.D2")

my.clusters <- unname(cutreeDynamic(my.tree, distM=as.matrix(my.dist),minClusterSize=10))

Having the data edata with labels referring to the type of population, we use pARI considering as ix the hierarchical cluster analysis output.

out <-pARI(X = edata,alpha = 0.05, test.type = "two_samples",
           label = as.factor(pdata$population),
           ix = my.clusters, family = "simes", clusters = TRUE)
out$TDP

For each cluster computed by hclust, pARI returns the lower bound for the true discoveries proportion.

Alternatively, you can use as clusters the pathways of genes from the getGenesets function:

pathways <- EnrichmentBrowser::getGenesets(org = "hsa", db = "kegg", gene.id.type = "ENSEMBL")

out <- c()
for(i in seq(pathways)){
  
  ix <- which(rownames(edata) %in% pathways[[i]])
  if(length(ix)!=0){
  out[i] <-pARI(X = edata,alpha = 0.05, test.type = "two_samples",
           label = as.factor(pdata$population),
           ix = ix, family = "simes", clusters = TRUE)$TDP
  }else{
  out[i] <- NA
  }
}

db <- data.frame(TDP = out, size = sapply(seq(pathways), function(x) length(which(rownames(edata) %in% pathways[[x]]))), name = names(pathways))

So, we represent the lower confidence bounds for the proportion of differential expressed genes (TDP) inside each pathway in the following plot:

library(tidyverse)
db <- db %>% dplyr::filter(!is.na(TDP) | TDP!=0) %>% dplyr::filter(size >10) 

db$name = factor(db$name, levels=db[order(db$TDP), "name"])

db %>% dplyr::filter(!is.na(TDP) | TDP!=0) %>% dplyr::filter(size >10) %>% arrange(TDP)%>% ggplot(aes(x = TDP, y = name, size = size)) + geom_point() + 
  xlab(expression(bar(pi)(S[m]))) + ylab("Pathways") + labs(size = expression(paste("|", S, "|"))) + theme_classic()+
  scale_size_continuous(breaks = c(15, 25, 35, 45))

In this case, we filter out the pathways having less than \(10\) genes to simply the plot.

fMRI cluster analysis

pARI is particularly useful in functional Magnetic Resonance Imaging cluster analysis, where it is of interest to select a cluster of voxels and to provide a confidence statement on the percentage of truly activated voxels within that cluster, avoiding the well-known spatial specificity paradox.

We analyzed the Auditory data collected by Pernet et al. (2015), i.e., people listening vocal and non-vocal sounds.

First, let download the data from the fMRIdata package:

if (!requireNamespace("fMRIdata", quietly = TRUE)){
  remotes::install_github("angeella/fMRIdata")
}
library(fMRIdata)
data(Auditory_clusterTH3_2)
data(Auditory_copes)
data(Auditory_mask)

We have three ingredients:

1. The set of copes Auditory_copes as a list of niftiImage objects, one for each subject. The copes represent the \(3\) dimensional (\(91 \times 109 \times 91\)) contrast map. Each element of the array describes the estimated parameter used in the hypotheses. In this case, the copes represent the statistics maps regarding the contrast that describes the difference of neural activation during vocal and non-vocal stimuli for each participant, computed by FSL. The one-sample t-test is computed for each voxel to analyze the hypothesis of zero mean across the subjects, i.e.,

\[ H_0 : \mu_i = 0 \]

where \(\mu_i = \sum_{j = 1}^{J} copes_{ji}/J\), where \(J\) is the total number of subjects.

2. The cluster map Auditory_clusterTH3_2 is used as a set of features in pARIbrain. While our method allows any method for forming clusters, we started from a map computed using Random Field Theory (RFT) with a cluster-forming-threshold equalling \(3.2\).

3. The brain mask Auditory_mask. In this case, we extract it from the group-level analysis by FSL.

Finally, pARIbrain can be used.

auditory_out <- pARIbrain(copes = Auditory_copes, clusters = Auditory_clusterTH3_2, mask = Auditory_mask, alpha = 0.05, silent = TRUE)
auditory_out$out

For each cluster, pARIbrain returns the lower bounds of the proportion of active voxels, the cluster’s size, the coordinates, and the maximum statistical test value inside the cluster.

Finally, you can also produce the True Discovey Proportion brain map using the map_TDP function.