1. Multiple testing and false discovery rate in feature selection
Workflow of feature selection using high-throughput data
General considerations of statistical testing using high-throughput data
Some important FDR methods
Benjamini-Hochberg FDR
Storey-Tibshirani’s q-value
Efron et al.’s local fdr
Ploner et al.’s Multidimensional local fdr

2. Gene/protein/metabolite expression data
After all the pre-processing, we have a feature by sample matrix of expression indices.
It is like an molecular “fingerprint” of each sample.
The most common use: to find biomarkers of a disease.

3. Workflow of feature selection
Raw data
Preprocessing, normalization,filtering …
Significance of every feature (FWER, FDR, …)
Feature-level expression in each sample
Other information (fold change, biological pathway …)
Statistical testing/model fitting
Test statistic for every feature
Selected features
Compare to null distribution
Biological interpretation, … …
P-value for every feature

4. Workflow of feature selection
Another route.
Raw data
Preprocessing, normalization,filtering …
Feature-level expression in each sample
Feature group information (biological pathway ……)
Statistical testing/model fitting
Group-level significance
Selected features from significant groups

5. Gene/protein/metabolite expression data
The simplest strategy:
Assume each gene is independent from others.
Perform testing between treatment groups for every gene.
Select those that are significant.
When we do 50,000 t-tests, if the alpha level of 0.05 is used, we expect ~50,000x0.05 = 2,500 false-positives !
If we use Bonferonni’s correction? 0.05/50000= 1e-6
Unrealistic!

6. General considerations
Family-wise error rate (FWER)
When we have multiple tests, let V be the number of true nulls called significant (false positives)
FWER = P(V ≥ 1) = 1-P(V=0)
“Family”: a group of hypothesis that are similar in purpose, and need to be jointly accurate.
Bonferroni correction is one version of FWER control.
It is the simplest and most conservative approach.

7. General considerations
Control every test at the level α/m
For each test,
P(Ti significant | H0) ≤ α/m
Then
P(some T are significant | H0) ≤ α
i.e.
FWER = P(V ≥ 1) ≤ α
It has little power to detect differential expression when m is big.

8. References Non-technical Reviews:
Gusnanto A, Calza S, Pawitan Y. Curr Opin Lipidol 2007; 18:
Pounds SB. Brief Bioinf 2005; 7(1):
Saeys Y, Inza I, Larranaga P. Bioinformatics 2007, 23 (19):
Original papers:
Benjamini Y, Hochberg Y. JRSS B 1995; 57(1):289–300.
Storey JD, Tibshirani R. Proc Natl Acad Sci U S A 2003; 100:9440–9445.
Efron B. Ann Stat 2007; 35(4):
Ploner A, Calza S, Gusnanto A, Pawitan Y. Bioinf 2006;22(5):
(A number of figures were taken from these papers.)

9. Differentially expressed
General considerations
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
Simultaneously test M hypotheses.
Q is # true null – genes that didn’t change (unobserved)
R is # rejected – genes called significant (observed)
U, V, T, S are unobservable random variables.
V: number of type-I errors; T: number of type-II errors.

10. Differentially expressed
General considerations
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
In traditional testing, we consider just one test, from a frequentist’s point of view.
we control the false positive rate: E(V/Q)
Sensitivity: E[S/(M-Q)]
Specificity: E[U/Q]

11. Differentially expressed
General
considerations
Significant
Non- significant
No change
False positive
True negative
Total true negative
Differentially expressed
True positive
False negative
Total true positive
Total positive calls
Total negative calls
total
There is always the trade-off between sensitivity and specificity.
Receiver operating characteristic (ROC) curve.
Example from Jiang et al. BMC Bioinformatics 7:417.

12. General considerations

13. Differentially expressed
General considerations
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
False discovery rate (FDR) = E(V/R)
Among all tests called significant, what percentage are false calls?

14. General considerations
Significant
Non- significant
No change 5
49795
49800
Differentially expressed 95
105
200
100
49900
50000
It makes more sense than this, which leans too heavily towards sensitivity:
Significant
Non- significant
No change
320
49480
49800
Differentially expressed
180 20
200
500
49500
50000

15. General considerations
Significant
Non- significant
No change 5
49795
49800
Differentially expressed 95
105
200
100
49900
50000
It makes more sense than this, which leans too heavily towards specificity:
Significant
Non- significant
No change 1
49799
49800
Differentially expressed 14
186
200 15
49985
50000

16. Was the BH definition the first? No. Defined in 1955….
True discovery rate
True positive rate
False positive rate

17. FDR – BH procedure Testing m hypotheses: The p-values are:
Order the p-values such that:
Let q* be the level of FDR we want to control,
Find the largest i such that
Make the corresponding p-value the cutoff value, the FDR is controlled at q*.

18. FDR – BH procedure
The method assumes weak dependence between test statistics.
In computation, it can be simplified by taking mP(i)/i and compare to q*.
Intuitively,
mP(i) is the number of false-positives expected if the cutoff is P(i)
If the cutoff were P(i), then we select the first i features.
So, mP(i)/i is the expected fraction of false-positives – the FDR.

19. FDR – BH procedure
Higher power compared to FWER controlling methods:

20. Differentially expressed
ST q-value
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
FDR = E[V/(V+S)] = E[V/R]
Let t be the threshold on p-value, then with all p-values observed, V and R become functions of t.
V(t) = # {null pi ≤ t}
R(t) = # {pi ≤ t}
FDR(t) = E[V(t)/R(t)] ≈ E[V(t)]/E[R(t)]
For R(t), we can simply plug in # {pi ≤ t};
For V(t), true null p-values should be uniformly distributed.

21. Differentially expressed
ST q-value
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
V(t) = Qt
However, Q is unknown.
Let π0=Q/M
Now, try to find π0.
Without specifying the distribution of the alternative p-values, but assuming most of them are small, we can use areas of the histogram that’s relatively flat to estimate π0
Density of p-values λ

22. Differentially expressed
ST q-value
Significant
Non- significant
No change V U Q
Differentially expressed S T
M-Q R
M-R M
This procedure involves tuning the parameter λ.
With most alternative p-values at the smaller end, the estimated
Should stabilize when λ is above a certain value.

23. ST q-value
“The more mathematical definition of the q value is the minimum
FDR that can be attained when calling that feature significant”
Given a list of ordered p-values, this guarantees the corresponding q-values are increasing in the same order as the p-values.
The q-value procedure is robust against weak dependence between features, which “can loosely be described as any form of dependence whose effect becomes negligible as the number of features increases to infinity.”

24. ST q-value

25. ST q-value

26. Efron’s Local fdr
The previous versions of FDR make statements about features falling on the tails of the distribution of the test statistic. However they don’t make statements about and individual feature, i.e. how likely is this feature false-positive given its specific p-value ?
Efron’s local FDR uses a mixture model and the empirical Bayes approach. An empirical null distribution is put in the place of the theoretical null.
With z being the test statistic, local FDR:

27. Efron’s Local fdr
The test statistic come from a mixture of two distributions:
The exact form of f1() is not specified. It is required to be longer-tailed than f0().
We need the empirical null. But we only have a histogram from the mixture. So the null comes in a strong parametric form.
And we need the proportion p0, the Bayes a priori probability.

28. Efron’s Local fdr
One way to estimate in the R package locfdr - “central matching”:
Use quadratic form to approximate

29. Efron’s Local fdr
6033 test statistics

30. Efron’s Local fdr
Now we have the null distribution and the proportion. Define the null subdensity around z:
The Bayes posterior probability that a case is null given z,
Compare to other forms of Fdr that focus on tail area,
(the c.d.f.s of f0 and f1)
Fdr(z) is the average of fdr(Z) for Z<z

31. Efron’s Local fdr
A real data example. Notice most non-null cases (bars plotted negatively) are not reported. A big loss of sensitivity to control FDR, which is very common.

32. Multidimensional Local fdr
A natural extension to the local FDR.
Use more than one test statistics to capture different characteristics of the features. Now we have a multidimensional mixture model.
Comment:
Remember the “curse of dimensionality” ? Since we don’t have too many realizations of the non-null distribution, we can’t go beyond just a few, say 2, dimensions.

33. Multidimensional Local fdr
Using t-statistic in one dimension and the log standard error in the other.
Simulated:
Genes with small s.e. tend to have higher FDR.
This approach discounts genes with too small s.e. – similar to the fold change idea but in a theoretically sound way.

34. Multidimensional Local fdr
The null distribution is generated by permutation:
Permute the treatment labels of each sample, and re-compute the test statistics.
Repeat 100 times to obtain the null distribution f0(z).
The f(z) is obtained by the observed Z.
Like local FDR, smoothing is involved. Here two densities in 2D need to be obtained by smoothing. In 2D, the points are not as dense as in 1D. So the choice of smoothing parameters becomes more consequential.

35. Multidimensional Local fdr
To address the problem, the authors did smoothing on the ratio (details skipped):
p is the number of permutations.
Afterwards, the local fdr is estimated by:

36. Multidimensional Local fdr
Real data:

37. Multidimensional Local fdr
Using other statistics: