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Detecting Differentially Expressed Genes. Pengyu Hong 09/13/2005. Background (Microarray). Extract RNA. Cells. Background. Extract RNA. Cells. Background. Extract RNA. Cells. Background. Extract RNA. Cells. Background. Extract RNA. Cells. 10 4 + genes. Background. Extract RNA.

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background microarray
Background (Microarray)

Extract RNA



Extract RNA



Extract RNA



Extract RNA



Extract RNA


104+ genes


Extract RNA


104+ genes


Extract RNA


104+ genes


biological variability

technical variability


Biological sample

  • RNA extraction (total RNA or mRNA)
  • Amplification (in vitro transcription)
  • Label samples
  • Hybridization
  • Washing and staining
  • Microarrays are highly noisy
  • Use replicated experiments to make inferences about differential expression for the population from which the biological samples originate




Calculate Gene Expression Index


An Example

5 normal sample and 9 myeloma (MM) samples 12558 genes (rows)

genes of interest
Genes of Interest
  • Statistical significance: that the observed differential expression is unlikely to be due to chance.
  • Scientific significance: that the observed level of differential expression is of sufficient magnitude to be of biological relevance.

Parametric Test: t-test

Statistical significance in the two group problem

Group 1 (N samples): X1, X2, … XN

Group 2 (M samples): Y1, Y2, … YM


Xi ~ Normal (μ1, σ2)

Yj ~ Normal (μ2, σ2)

Null hypothesis: Group 1 is the “same” to Group 2

(i.e., μ1= μ2)


Parametric Test: t-test

Statistical significance in the two group problem

Xi ~ Normal (μ1, σ2)

Yj ~ Normal (μ2, σ2)

Null hypothesis:μ1= μ2

Test null hypothesis with test statistics:


Xi ~ Normal (μ1, σ12)

σ1 σ2

If variances are unequal

Yj ~ Normal (μ2, σ22)

(1) When N+M > 30, this is approximately normal

(2) When 1 >> 2, this is approximately t(df = N–1)

(3) In general, Welch approximation: t’ ~ t(df’), where

wilcoxon rank sum test
Wilcoxon rank sum test

Consider row 7 of MM study

16 253 633 1008 708 36 72 28 14 33 19 49 58 23

13 4 3 1 2 8 5 10 14 9 12 7 6 11


rank sum = 23

This test is more appropriate than the t-tests when the underlying distribution is far from normal. (But it requires large group sizes)

p value
  • p-value = P(|T|>|t|) is calculated based on the distribution of T under the null hypothesis.
  • p-value is a function of the test statistics and can be viewed as a random variable.
    • e.g. p-value = 2(1 - F(|t*|), F = cdf of t(N+M – 2).
  • A small p-value represents evidence against the null hypothesis  differentially expressed in our case.
permutation test
Permutation test
  • A non-parametric way of computation p-value for any test statistics.
    • In the MM-study, each gene has (14 choose 5) = 2002 different test values obtainable from permuting the group labels.
  • Under the null hypothesis that the distribution for the two groups are identical, all these test values are equally probable. What is the probability of getting a test value at least as extreme as the observed one? This is the permutation p-value.
permutation technique
Permutation technique

Compute TS0

Compute TS1

Compute TS2

Compute TS3

The set of TSi form the empirical distribution of the test statistic TS

scientific significance
Scientific Significance
  • Fold change FC =
  • May not be high when statistical significance is high.
  • Not an appropriate measure if the dispersion is not taken into consideration.

Conservative fold change

Conservative fold change (CFC) =

Max (25th percentile of sample 1 / 75th percentile of sample 2,

25th percentile of sample 2 / 75th percentile of sample 1)


Sample 1: Normal (100, 1)

Sample 2: Normal (103, 1)

CFC = 1.0164






gene selection and ranking
Gene Selection and Ranking
  • A high threshold of statistical significance  Select genes with p-values smaller than a threshold
  • The selected genes are ordered according to their scientific significance (i.e. ranked by fold-changes)
the false positive rate fpr
The False Positive Rate (FPR)
  • If we select genes with p-value < 0.01, then the probability of making a positive call when the gene is in fact not differential is less than 0.01. Thus selection by p-value controls the FPR.
  • However, if we have 12,000 genes in a microarray, then a FPR = 0.01 still allows up to 120 false positives. To make sensible decision, we must take multiple comparisons into consideration.
dealing with multiple comparison
Dealing with Multiple Comparison
  • Bonferroni inequality: To control the family-wise error rate for testing m hypotheses at level α, we need to control the FPR for each individual test at α/m
  • Then P(false rejection at least one hypothesis) < α

or P(no false rejection) > 1- α

  • This is appropriate for some applications (e.g. testing a new drug versus several existing ones), but is too conservative for our task of gene selection.