Detecting Differentially Expressed Genes

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Detecting Differentially Expressed Genes
Pengyu Hong 09/13/2005
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Background (Microarray)
Extract RNA
Cells
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Background
Extract RNA
Cells
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Background
Extract RNA
Cells
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Background
Extract RNA
Cells
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Background
Extract RNA
Cells
104 genes
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Background
Extract RNA
Cells
104 genes
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Background
Extract RNA
Cells
104 genes
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Background
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

Scanning
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Background
Normalization
Calculate Gene Expression Index
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An Example
5 normal sample and 9 myeloma (MM) samples 12558
genes (rows)
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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.

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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
Assume
Xi Normal (µ1, s2)
Yj Normal (µ2, s2)
Null hypothesis Group 1 is the same to Group 2
(i.e., µ1 µ2)
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Parametric Test t-test
Statistical significance in the two group problem
Yj Normal (µ2, s2)
Xi Normal (µ1, s2)
Null hypothesis µ1 µ2
Test null hypothesis with test statistics
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Xi Normal (µ1, s12)
s1 ? s2
If variances are unequal
Yj Normal (µ2, s22)
(1) When NM gt 30, this is approximately normal
(2) When ?1 gtgt ?2, this is approximately t(df
N1)
(3) In general, Welch approximation t t(df),
where

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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)

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P-value

• p-value P(Tgtt) 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(NM
  2).
• A small p-value represents evidence against the
  null hypothesis ? differentially expressed in our
  case.

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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.

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Permutation technique
Compute TS0
Compute TS1
Compute TS2
Compute TS3
The set of TSi form the empirical distribution of
the test statistic TS
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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.

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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)
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Sample 1 Normal (100, 1) Sample 2 Normal (103,
1) CFC 1.0164
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CFC2.89
CFC3.53
CFC1.45
CFC1.07
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P-values and FC contains different information
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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)

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The False Positive Rate (FPR)

• If we select genes with p-value lt 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.

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Dealing with Multiple Comparison

• Bonferroni inequality To control the family-wise
  error rate for testing m hypotheses at level a,
  we need to control the FPR for each individual
  test at a/m
• Then P(false rejection at least one hypothesis)
  lt a
• or P(no false rejection) gt 1- a
• 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.