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 = ( Y 1i )/n 1 – ( Y 2i )/n 2

Analysis

then for gene 2,... then for gene

Some examples of methods  Two-sample t-test  Linear regression y t = y 0 + ¯ Z y 0 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 q g < ® 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 p g that a gene g is DE can be calculated. The B-statistic given by Limma is the log-odds ie log(O g =p g /(1- p g )). 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.