Testing Multiple Hypotheses and False Discovery Rate · Testing Multiple Hypotheses and False...

Testing Multiple Hypotheses and

False Discovery RateModels, Inference, and Algorithms

Primer

Manuel A. Rivas Broad Institute

Let’s assume that we wish to examine the association between a response and

m different covariates

When m tests are performed, the aim is to decide which of the nulls should be rejected.

Not flagged Flagged

H0

H1

Not flagged Flagged

H0

H1

K

This table shows the possibilities when m tests are performed and K are flagged as

requiring further attention.

Not flagged Flagged

H0 m0

H1

K

m0 is the number of true nulls

Not flagged Flagged

H0 B m0

H1

K

B is the number of type I errors

Not flagged Flagged

H0 B m0

H1 C

K

C is the number of type II errors

Not flagged Flagged

H0 A B m0

H1 C D m1

m - K K m

m1 is the number of true alternatives

Not flagged Flagged

H0 A B m0

H1 C D m1

m - K K m

Each of these quantities is unknown. The aim is to select a rule on the basis of some

criterion and this in turn will determine K.

To illustrate the multiple testing problem we focus on GWAS as an example where we

typically test the null hypothesis

H0 : � = 0

i.e. the effect of the genetic variant is 0.

In a single test situation the historical emphasis has been on the control of the

type I error rate (false positives).

In a multiple testing situation there are a variety of criteria that may be considered.

In a multiple testing situation there are a variety of criteria that may be considered:

1. Bonferroni method 2. Sidák correction 3. Benjamini and Hochberg (FDR) 4. Storey (FDR)

Frequentist analysis




Bayesian analysis1. Bayesian Bonferroni-type correction 2. Mixture models 3. Matthew Stephens’ FDR approach


1. Bonferroni method2. Sidák correction3. Benjamini and Hochberg (FDR) 4. Storey (FDR)


Bayesian analysis 1. Bayesian Bonferroni-type correction 2. Mixture models 3. Matthew Stephens’ FDR approach

Family-wise error rate (FWER): the probability of making at least one type I error


Family-wise error rate (FWER): the probability of making at least one type I error


P (B � 1|H1 = 0, . . . , Hm = 0)

Bonferroni method Frequentist analysis

Let be the event that the ith null is incorrectly rejected,so that, B, the random variable representing the number of incorrectly rejected nulls, corresponds to the union of all incorrectly rejected nulls, i.e.

Bi

[mi=1Bi


With a common level for each test ↵⇤

the family-wise error rate (FWER) is

↵F = P (B � 1|H1 = 0, . . . , Hm = 0) = P ([mi=1Bi|H1 = 0, . . . , Hm = 0)

mX

i=1

P (Bi|H1 = 0, . . . , Hm = 0)

= m↵⇤


↵F = P (B � 1|H1 = 0, . . . , Hm = 0) = P ([mi=1Bi|H1 = 0, . . . , Hm = 0)

mX

i=1

P (Bi|H1 = 0, . . . , Hm = 0)

= m↵⇤

The Bonferroni method takes ↵⇤ = ↵F /m

to give FWER ↵F .


Preferred approach for GWAS where to control the FWER at a level of alpha = 0.05 with m = 1,000,000 tests, we would take

↵⇤ = .05/1, 000, 000 = 5⇥ 10�8.

Sidák correction Frequentist analysis

Overcomes conservatism introduced by inequality If test statistics are independent,

P (B � 1) = 1� P (B = 0)

= 1� P⇣\mi=1B

0

i

⌘

= 1�mY

i=1

P⇣B

0

i

⌘

= 1� P (1� ↵⇤)m

Sidák correction Frequentist analysis

Overcomes conservatism introduced by inequality If test statistics are independent,

↵⇤ = 1� (1� ↵F )1/m .

In GWAS, assuming 1,000,000 tests were independent this would change it slightly to 5.13e-8 as a p-value threshold.

False Discovery Rate (FDR) Frequentist analysis

A simple way to overcome the conservative nature of the control of FWER is to increase ↵F .



One measure to calibrate a procedure is via the expected number of false discoveries:

EFD = m0 ⇥ ↵⇤

m⇥ ↵⇤



One measure to calibrate a procedure is via the expected number of false discoveries:

EFD = m0 ⇥ ↵⇤

m⇥ ↵⇤Recall m0 is the number of true nulls.


For example, we could specify ↵⇤ such that EFD <= 1

We choose: ↵⇤ = 1/m.

↵⇤ = 1⇥ 10�6 for GWAS with 1,000,000 markers.


We introduce the false discovery proportion (FDP) as the proportion of incorrect rejections:

FDP =B

K.

B is the number of type I errors


We introduce the false discovery proportion (FDP) as the proportion of incorrect rejections:

FDP =B

K.

K is the number flagged for additional attention


False Discovery Rate (FDR), the expected proportion of rejected nulls that are actually true:

FDR = E [FDP] = E [B/K|K > 0]P (K > 0)

False Discovery Rate (FDR) Benjamini and Hochberg (1995) procedure


For independent p-values, each of which is uniform under the null.

P(1) < · · · < P(m)1. Let denote the ordered p-values.



For independent p-values, each of which is uniform under the null.

P(1) < · · · < P(m)1. Let denote the ordered p-values.

2. Assume we would like FDR control at ↵ = 0.05


Frequentist analysisP(1) < · · · < P(m)1. Let denote the ordered p-values.

2. Assume we would like FDR control at ↵ = 0.05

li = i↵/m R = max

�i : P(i) < li

Let and

3. We use p-value threshold at P(R).



GWAS example

1. Assume m = 1,000,000 independent tests.2. Assume P(10) = 4.5e-7 and P(11) = 5.7e-7

10*.05/1,000,000 = 5e-7P(10) < 5e-7



GWAS example


10*.05/1,000,000 = 5e-7P(10) < 5e-7

11*.05/1,000,000 = 5.5e-7P(11) 5.5e-7⌅



GWAS example


10*.05/1,000,000 = 5e-7P(10) < 5e-7

11*.05/1,000,000 = 5.5e-7P(11) 5.5e-7⌅

3. Use p-value threshold at P(10).



FDR is controlled at

If procedure is applied, then regardless of how many nulls are true (m0) and regardless of the distribution of the p-values when the null is false

FDR m0

m↵ < ↵.

↵.



FDR is controlled at ↵.

Bonferroni = 5%




FDR = 5%




EFD = 1

False Discovery Rate (FDR) Storey (2002)


Introduced the q-value

For each observed statistic we can obtain an associated q-value, which tells us the proportion of false positives incurred at a thresholded statistic.

q (t) = P (H = 0|T > t)




Bayesian analysis1. Bayesian Bonferroni-type correction2. Mixture models3. Matthew Stephens’ FDR approach

Bayes Factors

Bayesian analysis

Defined as ratios of marginal likelihoods of the data under two models.

Model 0 can be the null model.

Bayes Factori = P (Data|M0) /P (Data|M1)

Bayes Factors

Bayesian analysis

We apply the same procedure m times (can be genetic variants for instance).

Bayes Factors

Bayesian analysis

Combine with prior probabilities

Posterior Oddsi = Bayes Factori ⇥ Prior Oddsi

where

Prior Oddsi = ⇡0i/ (1� ⇡0i) .

Bayes Factors

Bayesian analysis

Combine with prior probabilities

Posterior Oddsi = Bayes Factori ⇥ Prior Oddsi

where

Prior Oddsi = ⇡0i/ (1� ⇡0i) .

Bayesian Bonferroni-type correction

Bayesian analysis

If the prior probabilities of each of the nulls are independent with ⇡0i = ⇡0 for i = 1, …, m.

Then prior probability that all nulls are true is

⇧0 = P (H1 = 0, . . . , Hm = 0) = ⇡m0

Bayesian Bonferroni-type correction

Bayesian analysis

If the prior probabilities of each of the nulls are independent with ⇡0i = ⇡0 for i = 1, …, m.

Suppose that we wish to fix the prior probability that all of the nulls are true at . We can fix ⇧0 ⇡0i = ⇧1/m

0

Mixture model

Bayesian analysis

Estimate common parameters like the proportion of null tests

Gibbs sampler.

Matthew Stephens’ FDR approach

Bayesian analysis


Bayesian analysis

Open source R package

http://github.com/stephens999/ashr

ashr

http://github.com/stephens999/ashr


Bayesian analysis

Three key ideas

1. Assumes distribution of effects is unimodal, with a mode at 0.


Bayesian analysis

Three key ideas


2. Takes as input two numbers: i) effect size estimate and ii) corresponding standard error.


Bayesian analysis

Model outlineData corresponds to effect size estimates and corresponding (estimated) standard errors, i.e.

� =⇣�1, . . . , �m

⌘

s = (s1, . . . , sm) .


Bayesian analysis

Model outline

Goal is to compute a posterior distribution

p⇣�|�, s

⌘/ p (�|s) p

⇣�|�, s

⌘.

LikelihoodPrior


Bayesian analysis

Model outline


p⇣�|�, s

⌘/ p (�|s) p

⇣�|�, s

⌘.

Key: For assumption is that the betas are independent from a unimodal distribution.

p (�|s)

“Unimodal assumption”


Bayesian analysis

Model outlineAssume that it is a mixture of point mass at 0 and a mixture of zero-mean normal distributions:


p (�|s,⇡) = ⇡0�0 (·) +KX

k=1

⇡kN�·; 0,�2

k

�

Estimate mixture proportions mixture component standard deviations is a grid


Bayesian analysis

Model outline

For the likelihood p⇣�|�, s

⌘

p⇣�|�, s

⌘=

mY

j=1

N⇣�j ;�j , s

2j

⌘


Bayesian analysis

Model outline


p⇣�|�, s

⌘/ p (�|s) p

⇣�|�, s

⌘.



Bayesian analysis

Model outline


p⇣�|�, s

⌘/ p (�|s) p

⇣�|�, s

⌘.

Measurement precision in likelihood


Bayesian analysis

Three key ideas


2. Takes as input two numbers: i) effect size estimate and ii) corresponding standard error.

3. local false sign rate - probability of getting sign of effect wrong.


Bayesian analysis

Three key ideas


lfdrj := P⇣�j = 0|�, s, ⇡

⌘.

local false discovery rate (“local FDR”)

probability, given the observed data, that effect j would be a false discovery, if we were to declare it a discovery.


Bayesian analysis

Three key ideas

lfdrj := P⇣�j = 0|�, s, ⇡

⌘.

some statisticians argue that it is inappropriate because that null hypothesis is often implausible.


Bayesian analysis

Three key ideas

lfdrj := P⇣�j = 0|�, s, ⇡

⌘.

Can obtain an estimate of the average error rate over subsets of observations, for example if you declared all tests in as significant.

[FDR (�) := (1/ |�|)X

j2�

lfdrj


Bayesian analysis

Three key ideas

Tukey stated:

All we know about the world teaches us that the effects of A and B are always different - in some

decimal place - for any A and B.


Bayesian analysis

Three key ideas


Tukey suggested:

Is the evidence strong enough to support a belief that the observed difference has the correct sign?


Bayesian analysis

Three key ideas3. local false sign rate - probability of getting sign of effect wrong.

lfsrj := minhp⇣�j � 0|�, s

⌘, p

⇣�j 0|⇡, �, s

⌘i.


Bayesian analysis

Three key ideas3. local false sign rate - probability of getting sign of effect wrong.

lfsrj := minhp⇣�j � 0|�, s

⌘, p

⇣�j 0|⇡, �, s

⌘i.

Gelman proposed focusing on “type S errors”, errors in sign, rather than traditional type I errors.


Bayesian analysis

Other results/observations covered, but not in this primer

1. Computation/Implementation details 2. Comparisons to other approaches

Multiple tests of the same null hypothesis

Bayesian analysis

In genetics we may be interested in applying: 1) additive model, 2) dominant, and 3) recessive model.

How to correct?


Bayesian analysis


Bonferroni?


Bayesian analysis


Bonferroni? Too conservative.


Bayesian analysis


Minimum p-value?


Bayesian analysis


Minimum p-value? not valid p-value since it is not uniform under the null. (can modify null though)


Bayesian analysis


Minimum p-value? not valid p-value since it is not uniform under the null.

Permutation.


Bayesian analysis


Bayesian model averaging.

Hoeting, Madigan, Raftery and Volinksy Statisticial Science 1999

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