A A R H U S U N I V E R S I T E T

Faculty of Agricultural Sciences

Introduction to analysis of microarray data

David Edwards

The Microarray Study Process

Study Objectives

Class comparison: differential expression

Class prediction: classification

Class discovery: clustering

Differential Expression

How to identify genes whose expression level changes across

conditions in the study?

Analysis Strategy

The study may be to:

Compare two groups (eg treatment vs control)

Compare more than two groups

More than one comparison (eg 2 treatments at 3 timepoints)

As a first approximation, we can think of our approach as:

1. Choose the appropriate analysis method for a single gene

2. Apply to all genes, correcting for multiplicity (eg FDR).

Additive and multiplicative scales

Most statistical models use additive scales and constant variance

Gene expression appears to work more on a multiplicate scale (fold changes rather than expression differences), and the variance in gene expression depends on its absolute value.

Conclusion: transform the data by taking logarithms (conventionally base 2).

Fold Change & Log Ratios

We have transformed our data by taking logarithms! So differences are log-ratios (log fold changes)

log(a/b) = log(a) – log(b)

With two-channel (cDNA) data the numbers we analyze (usually) are the within-spot log-ratios:

M = log(R) – log(G)

To estimate log fold change across replicate slides we compute the average log-ratio across the replicates.

With one-channel (affy) data the numbers we analyze are the logs of the expression measures (eg rma)

To estimate log fold change between two groups of arrays we compute the average log-expression within each group and calculate the difference.

LR = ( Y1i)/n1 – ( Y2i)/n2

Analysis

then for gene 2, ... then for gene 20000.

Some examples of methods

Two-sample t-test Linear regression

yt = y0 + ¯ Z

y0 baseline expression (before treatment)

Z (0=control, 1=treatment) ¯ group effect

ANOVA models Non-parametric tests ....

Multiplicity

Typically a list of p-values is obtained, one per gene.

Now we need to select the ones likely to be differentially expressed.

If we used p<0.05 as criterion this would lead to 1000 (=0.05x20000) genes being selected even though there was no differential expression.

Multiplicity

If select genes using the criterion p < ®/N, where N is total no of genes, (Bonferroni’s correction), this controls the familywise error rate

Pr(any type I error) = Pr(any false selections) < ®

But this is usually too stringent.

False Discovery Rate

FDR= Proportion of false positives within selected genes.

Two uses: If top 100 genes are selected for further study, what

proportion may be expected to be false positive? If we want a proportion of 5% false positives, how many

genes should be selected?

Adjusted p-values can be defined (q-values) such that

selecting genes with qg<® results in FDR<®

LIMMA Package:Linear Models for Microarray Data

arbitrarily complex experiments: linear models, contrasts

empirical Bayes methods for differential expression: t-tests, F-tests, posterior odds

inference methods for duplicate spots, technical replication

analyse log-ratios or log-intensities spot quality weights control of FDR across genes and contrasts stemmed heat diagrams, Venn diagrams pre-processing: background correction, within and

between array normalization

Analysis of differential expression studies

Empirical Bayes Methods in Limma

Problem with ordinary t-tests here:

small estimates of S.D. can arise by chance, giving false positives.

Limma uses an empirical Bayes approach:

the gene variances are given a prior distribution (the sample distribution). Each variance is then updated using the data to obtain posterior distribution, and an an estimate is derived from the posterior distribution.

This shrinks the variances towards the prior mean. This estimate is then substituted in classical t-statistics (the ”degrees of freedom” are adjusted), giving the so-called moderated t-test.

Good evidence that this is more robust than the classical approach.

Given a prior estimate p of the proportion of DE genes, the posterior probability pg that a gene g is DE can be calculated. The B-statistic given by Limma is the log-odds ie log(Og=pg/(1-pg)). This is useful for ranking genes.

Smyth, GK (2004). Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments, Stat. Appl. In Genetics and Mol. Biol., 3, 1.