1. Differential Gene Expression
Xiaole Shirley Liu
STAT115 / STAT215

2. Identification of Diagnostic Genes
Classical study of cancer subtypes
Golub et al. (1999)

3. Differential Expression
Naïve method: Fold change
Avg(X) / Avg(Y)
Note on scale:
Natural scale: MAS4, MAS5, dChip
Log scale: RMA, need to take exp()

4. Fold Change Problems
Does not give confidence of differential expression
Better statistical test?

5. Test Normality Normal distribution QQ Plot Normal: T*-test
Non-normal: non-parametric test

6. Wilcoxon Rank Sum Test Break
Rank all data in row, count sum of ranks TT or TC
Significance calculated from permutation as well
E.g. 10 normal and 10 cancer
Min(T) = 55
Max(T) = 155
Significance(T=150)
Check U table
(transformation of T)
for stat significance
Non-parametric, less
power with fewer samples
Break

7. Linear Model for Differential Expression
Yijk = mj + aij + errorijk
Separate model for each gene j.
mj is the mean expression for gene j over the entire experiment (RMA ExprIndex).
aij is the deviation of the mean of the ith condition from the overall mean Si aij=0
k is a specific sample.
For 3 rep (mutant) over 3 rep (wildtype ctrl), we care whether amu-awt=0 (null hypothesis H0)

8. Ordinary t-tests c based on sample size in the two conditions
How to determine sg?

9. Variance Estimates Same variance across treatment: standard t-test
Variance in different treatment is different: Welch-t test
Big |t|, small p, reject H0

10. Variance Stabilization
Problem with estimating variance when the sample size is small (e.g. 2-3 replicates in each condition)
Statistical Analysis of Microarrays (SAM)
Modified t*, increase sg based on sg of other genes on the array (i.e. lowest 5 percentile of sg)
LIMMA: Smyth 2004
Empirical Bayes: borrow info from all genes

11. LIMMA: Design Matrix Specifies RNA samples used on arrays >Mat
Treat1 Treat2 Control
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample

12. LIMMA: Contrast Matrix
Specifies which comparisons are of interest
> contrast
Treat1-Control Treat2-Control
Treat
Treat
Control
Flexibility of the generalized linear model can consider many different conditions
Yijk = mj + aij + errorijk

13. LIMMA: Contrast Matrix
Smooth gene-wise variance towards a common (typical) value in a graduated way by borrowing information from all the genes, but allow flexibility for individual genes
Assume gene’s variance follows inverse gamma distribution, large variance shrunk down, small variance shrunk up
Break

14. Multiple Hypotheses Testing
How many differential genes to report?

15. Multiple Hypotheses Testing
We test differential expression for every gene with p-value, e.g. 0.01
For ~20 K genes on the array, potentially 0.01 x 20K = 200 genes wrongly called
H0: no diff expr; H1: diff expr
Reject H0: call something to be differential expressed
Should control family-wise error rate or false discovery rate

16. Family-Wise Error Rate
P(false rejection at most one hypothesis) < α
P(no false rejection ) > 1- α
Bonferroni correction: to control the family-wise error rate for testing m hypotheses at level α, we need to control the false rejection rate for each individual test at α/m
If α is 0.05, for 20K gene prediction, p-value cutoff is 0.05/20K = 2.5E-6
Too conservative for differential expressed gene selection

17. False Discovery Rate Break U V m0 T S m1 m - R R m
# not rejected
Not called
# rejected
Called
Total
# H0
Two groups similar U V m0
# H1
Two groups different T S m1
m - R R m
V: type I errors, false positives
T: type II errors, false negatives
FDR = V / R, FP / all called
Break

18. False Discovery Rate Less conservative than family-wise error rate
Benjamini and Hochberg (1995) method for FDR control, e.g. FDR ≤ *
Assume all the p-val from different tests are independent
Draw all m genes (x), ranked by p-val (y)
Draw line y = x * / m, x = 1…m
Call all the genes below the line

19. FDR Threshold
Genes ranked by p-val
p-value
x * / m line
index / m

20. Q-value
Teaser: what’s the pvalue distribution if there are no differential genes
Storey & Tibshirani, PNAS, 2003
Empirically derived q-value
Every p-value has its corresponding q-value (FDR)

21. Practical Use of FDR
Very useful concepts in most of genomics or high throughput studies
Pvalue and FDR are monotonic
Common FDR: 1%, 5%, 10%, also filter by fold change
Give rough estimate of signal / noise and experimental quality
For expression, most people are comfortable with ~ differentially expressed genes

22. Summary Differential Expression
Fold change
T* test on normally distributed data
LIMMA uses hierarchical model to stabilize gene-wise variance
FDR: adjust for multiple hypotheses testing
FWER: conservative
Benjamini-Hochberg
qvalue