Meta-analysis aims to combine summary statistics (e.g., effect sizes, p-values) from multiple clinical or genomic studies in order to enhance statistical power. Another appealing feature of meta-analysis is that batch effect (non-biological differences between studies because of sample platforms and experimental protocols) can be avoided, because the summary statistics are usually considered as standardized. The adaptively weighted Fisher's method (AW-Fisher) is an effective approach to combine \(p\)-values from \(K\) independent studies and to provide better biological interpretability by characterizing which studies contribute to the meta-analysis.
Denote \(\theta_k\) is the effect size of study \(k\), \(1\le k \le K\)).
The AW-Fisher's method targets on biomarkers differentially expressed in one or more studies.
The null hypothesis \(H_0\) and the alternative hypothesis are listed below.
\[H_0: \vec{\boldsymbol{\theta}}\in \bigcap \{ \theta_k=0 \}\]
\[H_A: \vec{\boldsymbol{\theta}}\in \bigcup \{ \theta_k \ne 0 \},\]
Define \(T(\vec{\textbf{P}}; \vec{\textbf{w}} ) = -2 \sum_{k=1}^K w_k \log P_k\), where \(\vec{\textbf{w}} = (w_1, \ldots, w_K) \in {\{ 0,1 \} }^K\) is the AW weight associated with \(K\) studies and \(\vec{\textbf{P}} = (P_1, \ldots, P_K) \in {(0,1)}^K\) is the random variable of input \(p\)-value vector for \(K\) studies. The AW-Fisher's method will find the optimal weight \(\vec{\textbf{w}}^*\), and calculate the test statistics and AW-Fisher p-value based on \(\vec{\textbf{w}}^*\).
Collectively, the AW-Fisher's method will provide knowledge about which study contributes to the meta-analysis result via \(\vec{\textbf{w}}^*\), and also generate p-value for rejecting the null hypothesis \(H_0\).
This is a tutorial for the usage of the AWFisher package. A real data example of the multiple-tissue mouse metabolism data is used. The major contents of this tutorial includes:
To install this package, start R (version “3.6”) and enter:
if (!requireNamespace("BiocManager", quietly = TRUE))
install.packages("BiocManager")
BiocManager::install("AWFisher")
Huo, Z., Tang, S., Park, Y. and Tseng, G., 2020. P-value evaluation, variability index and biomarker categorization for adaptively weighted Fisher’s meta-analysis method in omics applications. Bioinformatics, 36(2), pp.524-532.
The manuscript can be found here: https://www.ncbi.nlm.nih.gov/pubmed/31359040
Zhiguang Huo (zhuo@ufl.edu)
The purpose of the multi-tissue mouse metabolism transcriptomic data is to study how the gene expression changes with respect to the energy deficiency using mouse models. Very long-chain acyl-CoA dehydrogenase (VLCAD) deficiency was found to be associated with energy metabolism disorder in children. Two genotypes of the mouse model - wild type (VLCAD +/+) and VLCAD-deficient (VLCAD -/-) - were studied for three types of tissues (brown fat, liver, heart) with 3 to 4 mice in each genotype group. The sample size information is available in the table below. A total of 6,883 genes are available in this example dataset.
| Tissue | Wild Type | VLCAD-deficent |
|---|---|---|
| Brown Fat | 4 | 4 |
| Heart | 3 | 4 |
| Skeleton | 4 | 4 |
library(AWFisher) # Include the AWFisher package
# load the data
data(data_mouseMetabolism)
# Verify gene names match across three tissues
all(rownames(data_mouseMetabolism$brown) == rownames(data_mouseMetabolism$heart))
#> [1] TRUE
all(rownames(data_mouseMetabolism$brown) == rownames(data_mouseMetabolism$liver))
#> [1] TRUE
dataExp <- data_mouseMetabolism
# Check the dimension of the three studies
sapply(dataExp, dim)
#> brown heart liver
#> [1,] 6883 6883 6883
#> [2,] 8 7 8
# Check the head of the three studies
sapply(dataExp, function(x) head(x,n=2))
#> $brown
#> b.wt b.wt.1 b.wt.2 b.wt.3 b.LCAD b.LCAD.1 b.LCAD.2
#> Copg1 8.086841 8.047482 8.140015 8.010229 8.206645 8.151032 7.956093
#> Atp6v0d1 9.807054 9.637094 10.044481 9.825333 9.868880 9.667059 9.541244
#> b.LCAD.3
#> Copg1 8.086200
#> Atp6v0d1 9.581028
#>
#> $heart
#> h.wt h.wt.1 h.wt.2 h.LCAD h.LCAD.1 h.LCAD.2 h.LCAD.3
#> Copg1 7.859429 7.955171 8.045601 8.145281 8.016827 7.961778 7.964703
#> Atp6v0d1 9.479398 9.499617 9.571348 9.469063 9.516679 9.437079 9.559526
#>
#> $liver
#> l.wt l.wt.1 l.wt.2 l.wt.3 l.LCAD l.LCAD.1 l.LCAD.2
#> Copg1 8.501327 8.698994 8.095882 8.519093 8.539002 8.305171 8.588183
#> Atp6v0d1 9.969806 9.975494 10.000650 10.161694 10.051711 10.084761 9.989209
#> l.LCAD.3
#> Copg1 8.554201
#> Atp6v0d1 10.035293
# Before performing differential expression analysis for each of these three tissues.
# Create an empty matrix to store p-value.
# Each row represents a gene and each column represent a study/tissue.
pmatrix <- matrix(0,nrow=nrow(dataExp[[1]]),ncol=length(dataExp))
rownames(pmatrix) <- rownames(dataExp[[1]])
colnames(pmatrix) <- names(dataExp)
library(limma) # Include the limma package to perform differential expression analyses for the microarray data
for(s in 1:length(dataExp)){
adata <- dataExp[[s]]
ControlLabel = grep('wt',colnames(adata))
caseLabel = grep('LCAD',colnames(adata))
label <- rep(NA, ncol(adata))
label[ControlLabel] = 0
label[caseLabel] = 1
design = model.matrix(~label) # design matrix
fit <- lmFit(adata,design) # fit limma model
fit <- eBayes(fit)
pmatrix[,s] <- fit$p.value[,2]
}
head(pmatrix, n=2) ## look at the head of the p-value matrix
#> brown heart liver
#> Copg1 0.7148393 0.3554053 0.7586203
#> Atp6v0d1 0.1584368 0.7154922 0.8502931
res <- AWFisher_pvalue(pmatrix) ## Perform AW Fisehr meta analysis
qvalue <- p.adjust(res$pvalue, "BH") ## Perform BH correction to control for multiple comparison.
sum(qvalue < 0.05) ## Differentially expressed genes with FDR 5%
#> [1] 755
head(res$weights) ## Show the AW weight of the first few genes
#> [,1] [,2] [,3]
#> [1,] 0 1 0
#> [2,] 1 0 0
#> [3,] 1 0 1
#> [4,] 1 1 0
#> [5,] 1 1 1
#> [6,] 0 0 1
## prepare the data to feed function biomarkerCategorization
studies <- NULL
for(s in 1:length(dataExp)){
adata <- dataExp[[s]]
ControlLabel = grep('wt',colnames(adata))
caseLabel = grep('LCAD',colnames(adata))
label <- rep(NA, ncol(adata))
label[ControlLabel] = 0
label[caseLabel] = 1
studies[[s]] <- list(data=adata, label=label)
}
## See help file about about how to use function biomarkerCategorization.
## Set B = 1,000 (at least) for real data application
## You may need to wrap up a function (i.e., function_limma)
## to perform differential expression analysis for each study.
set.seed(15213)
result <- biomarkerCategorization(studies,function_limma,B=100,DEindex=NULL)
#> generate DE index since it is NULL
#> based on AW fdr 0.05
#> calculating permutated score, b = 1,2,..., B (= 100) [one "." per sample]:
#> .................................................. 50
#> .................................................. 100
#>
#> calculating variability index
sum(result$DEindex) ## print out DE index at FDR 5%
#> [1] 755
head(result$varibility, n=2) ## print out the head of variability index
#> [,1] [,2] [,3]
#> [1,] 0.7884 0.9424 0.9600
#> [2,] 0.6864 0.6400 0.7296
print(result$dissimilarity[1:4,1:4]) ## print out the dissimilarity matrix
#> Psph Trappc4 Atg5 Cox18
#> Psph 1.00 0.00 0.01 0
#> Trappc4 0.00 1.00 0.75 0
#> Atg5 0.01 0.75 1.00 0
#> Cox18 0.00 0.00 0.00 1
library(tightClust) ## load tightClust package
tightClustResult <- tight.clust(result$dissimilarity, target=4, k.min=15, random.seed=15213)
#> Number of points: 755 Dimension: 755
#>
#> Looking for tight cluster 1 ...
#> k = 15
#> k = 16
#> 1 tight cluster(s) found!
#> Cluster size: 85 Remaining number of points: 670
#>
#> Looking for tight cluster 2 ...
#> k = 14
#> k = 15
#> 2 tight cluster(s) found!
#> Cluster size: 66 Remaining number of points: 604
#>
#> Looking for tight cluster 3 ...
#> k = 13
#> k = 14
#> 3 tight cluster(s) found!
#> Cluster size: 65 Remaining number of points: 539
#>
#> Looking for tight cluster 4 ...
#> k = 12
#> k = 13
#> 4 tight cluster(s) found!
#> Cluster size: 64 Remaining number of points: 475
clusterMembership <- tightClustResult$cluster
for(s in 1:length(dataExp)){
adata <- dataExp[[s]]
aname <- names(dataExp)[s]
bdata <- adata[qvalue<0.05, ][tightClustResult$cluster == 1 ,]
cdata <- as.matrix(bdata)
ddata <- t(scale(t(cdata))) # standardize the data such that for each gene, the mean is 0 and sd is 1.
ColSideColors <- rep("black", ncol(adata))
ColSideColors[grep('LCAD',colnames(adata))] <- "red"
B <- 16
redGreenColor <- rgb(c(rep(0, B), (0:B)/B), c((B:0)/16, rep(0, B)), rep(0, 2*B+1))
heatmap(ddata,Rowv=NA,ColSideColors=ColSideColors,col= redGreenColor ,scale='none',Colv=NA, main=aname)
}
sessionInfo()
#> R version 4.2.1 (2022-06-23)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Ventura 13.0
#>
#> Matrix products: default
#> BLAS: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
#>
#> locale:
#> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] tightClust_1.1 limma_3.54.0 AWFisher_1.12.0
#>
#> loaded via a namespace (and not attached):
#> [1] edgeR_3.40.0 compiler_4.2.1 magrittr_2.0.3 tools_4.2.1
#> [5] Rcpp_1.0.9 stringi_1.7.8 highr_0.9 grid_4.2.1
#> [9] locfit_1.5-9.6 knitr_1.39 stringr_1.4.0 xfun_0.31
#> [13] lattice_0.20-45 evaluate_0.15