AW Fisher tutorial

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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.

Statistical method

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 $H_0$.

About this tutorial

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:

About the package

How to install the package

To install this package, start R (version "3.6") and enter:

if (!requireNamespace("BiocManager", quietly = TRUE))


How to cite the package


Zhiguang Huo (

Description about the example data -- multi-tissue mouse metabolism transcriptomic data

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

Read in the example data

library(AWFisher) # Include the AWFisher package

# load the data

# Verify gene names match across three tissues
all(rownames(data_mouseMetabolism$brown) == rownames(data_mouseMetabolism$heart))
all(rownames(data_mouseMetabolism$brown) == rownames(data_mouseMetabolism$liver))

dataExp <- data_mouseMetabolism

# Check the dimension of the three studies
sapply(dataExp, dim)

# Check the head of the three studies
sapply(dataExp, function(x) head(x,n=2))

# 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)

Prepare the input p-value matrix -- perform differential expression analysis in each study

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

Perform AW Fisher meta analysis using the multi-tissue mouse metabolism transcriptomic data

res <- AWFisher_pvalue(pmatrix) ## Perform AW Fisehr meta analysis
qvalue <- p.adjust(res$pvalue, "BH") ## Perform BH correction to control for multiple comparison.
sum(qvalue < 0.05) ## Differentially expressed genes with FDR 5%
head(res$weights) ## Show the AW weight of the first few genes

Differential expression pattern (meta-pattern) detection.

Calculate dissimilarity matrix

## 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.

result <- biomarkerCategorization(studies,function_limma,B=100,DEindex=NULL)
sum(result$DEindex) ## print out DE index at FDR 5%
head(result$varibility, n=2) ## print out the head of variability index
print(result$dissimilarity[1:4,1:4]) ## print out the dissimilarity matrix

Apply the tight clustering algorithm to get gene modules with unique meta-pattern

library(tightClust) ## load tightClust package

tightClustResult <- tight.clust(result$dissimilarity, target=4, k.min=15, random.seed=15213)
clusterMembership <- tightClustResult$cluster

Visualize the heatmap of the first meta-pattern module for all three tissues.

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)

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AWFisher documentation built on Nov. 8, 2020, 5:42 p.m.