Multiple testing in large-scale gene Multiple testing in large-scale gene expression experiments expression experiments

Statistics 246, Spring 2002

Week 8, Lecture 2

OutlineOutline

• Motivation & examples

• Univariate hypothesis testing

• Multiple hypothesis testing

• Results for the two examples

• Discussion

SCIENTIFIC: To determine which genes are differentially expressed between two sources of mRNA (trt, ctl).

STATISTICAL: To assign appropriately adjusted p-values to thousands of genes, and/or make statements about false discovery rates.

MotivationMotivation

• 8 treatment mice and 8 control mice

• 16 hybridizations: liver mRNA from each of the 16 mice (Ti , Ci ) is labelled with Cy5, while pooled liver mRNA from the control mice (C*) is labelled with Cy3.

• Probes: ~ 6,000 cDNAs (genes), including 200 related to lipid metabolism.

Goal. To identify genes with altered expression in the livers of Apo AI knock-out mice (T) compared to inbred C57Bl/6 control mice (C).

Apo AI experiment Apo AI experiment (Matt Callow, LBNL)(Matt Callow, LBNL)

Golub Golub et alet al (1999) experiments (1999) experiments

Goal. To identify genes which are differentially expressed in acute lymphoblastic leukemia (ALL) tumours in comparison with acute myeloid leukemia (AML) tumours.

• 38 tumour samples: 27 ALL, 11 AML.• Data from Affymetrix chips, some pre-processing.• Originally 6,817 genes; 3,051 after reduction.

Data therefore a 3,051 38 array of expression values.

DataData

The gene expression data can be summarized as follows

treatment control

X =

Here xi,j is the (relative) expression value of gene i in sample j. The first n1 columns are from the treatment (T); the remaining n2 = n - n1 columns are from the control (C).

Univariate hypothesis testingUnivariate hypothesis testing

Initially, focus on one gene only.

We wish to test the null hypothesis H that the gene is not differentially expressed.

In order to do so, we use a two sample t-statistic:

€

t=averofn1 trtx − averofn2ctlx

[1n1

(SDofn1trtx)2 +

1n1

(SDofn1ctlx)2]

pp-values-values

The p-value or observed significance level p is the chance of getting a test statistic as or more extreme than the observed one, under the null hypothesis H of no differential expression.

Computing Computing pp-values by permutations-values by permutations

We focus on one gene only. For the bth iteration, b = 1, , B;

1. Permute the n data points for the gene (x). The first n1 are referred to as “treatments”, the second n2 as “controls”.

2. For each gene, calculate the corresponding two sample t-statistic, tb.

After all the B permutations are done;

3. Put p = #{b: |tb| ≥ |tobserved|}/B (plower if we use >).With all permutations in the Apo AI data, B = n!/n1! n2! = 12,870;

for the leukemia data, B = 1.2109 .

Many tests: a simulation studyMany tests: a simulation study

Simulations of this process for 6,000 genes with 8 treatments and 8 controls.

All the gene expression values were simulated i.i.d from a N (0,1) distribution, i.e. NOTHING is differentially expressed.

gene t p-value

index value (unadj.)

2271 4.93 210-4

5709 4.82 310-4

5622 -4.62 410-4

4521 4.34 710-4

3156 -4.31 710-4

5898 -4.29 710-4

2164 -3.98 1.410-3

5930 3.91 1.610-3

2427 -3.90 1.610-3

5694 -3.88 1.710-3

Unadjusted p-values

Clearly we can’t just use standard p-value thresholds (.05, .01).

Multiple hypothesis testing: Multiple hypothesis testing: Counting errorsCounting errors

Assume we are testing H1, H2, , Hm .

m0 = # of true hypotheses R = # of rejected hypotheses

# true # false

null hypo. null hypo.

# non-signif. U T m - R

# significant V S R

m0 m-m0

V = # Type I errors [false positives]

T = # Type II errors [false negatives]

Type I error ratesType I error rates• Per comparison error rate (PCER): the expected value of the number of Type I errors over the number of hypotheses,

PCER = E(V)/m.

•Per-family error rate (PFER): the expected number of Type I errors,

PFE = E(V).

•Family-wise error rate: the probability of at least one type I error

FEWR = pr(V ≥ 1)

•False discovery rate (FDR) is the expected proportion of Type I errors among the rejected hypotheses

FDR = E(V/R; R>0) = E(V/R | R>0)pr(R>0).

• Positive false discovery rate (pFDR): the rate that discoveries are false

pFDR = E(V/R | R>0).

Two types of control of Type I errorTwo types of control of Type I error

• strong control: control of the Type I error whatever the true and false null hypotheses. For FWER, strong control means controlling

max pr(V ≥ 1 | M0)

M0H0C

where M0 = the set of true hypotheses (note |M0| = m0);

• weak control: control of the Type I error only under the complete null hypothesis H0

C = iHi . For FWER, this is control of pr( V ≥ 1 | H0

C ).

Adjustments to Adjustments to pp-values-values

For strong control of the FWER at some level , there are procedures which will take m unadjusted p-values and modify them separately, so-called single step procedures, the Bonferroni adjustment or correction being the simplest and most well known. Another is due to Sidák.

Other, more powerful procedures, adjust sequentially, from the smallest to the largest, or vice versa. These are the step-up and step-down methods, and we’ll meet a number of these, usually variations on single-step procedures.

In all cases, we’ll denote adjusted p-values by , usually

with subscripts, and let the context define what type of adjustment has been made. Unadjusted p-values are denoted by p.

What should one look for in aWhat should one look for in a multiple testing procedure? multiple testing procedure?

As we will see, there is a bewildering variety of multiple testing procedures. How can we choose which to use? There is no simple answer here, but each can be judged according to a number of criteria:

Interpretation: does the procedure answer a relevant question for you?

Type of control: strong or weak?

Validity: are the assumptions under which the procedure applies clear and definitely or plausibly true, or are they unclear and most probably not true?

Computability: are the procedure’s calculations straightforward to calculate accurately, or is there possibly numerical or simulation uncertainty, or discreteness?

pp-value adjustments: single-step-value adjustments: single-step

Define adjusted p-values π such that the FWER is controlled at level where Hi is rejected when πi ≤ .

• Bonferroni: πi = min (mpi, 1)

• Sidák: πi = 1 - (1 - pi)m

Bonferroni always gives strong control, proof next page.

Sidák is less conservative than Bonferroni. When the genes are independent, it gives strong control exactly (FWER= ), proof later. It controls FWER in many other cases, but is still conservative.

Proof for BonferroniProof for Bonferroni((single-step adjustment)single-step adjustment)

pr (reject at least one Hi at level | H0C)

= pr (at least one i ≤ | H0C)

≤ 1m pr (i ≤ | H0

C) , by Boole’s inequality

= 1m pr (Pi ≤ /m | H0

C), by definiton of i

= m /m , assuming Pi ~ U[0,1])= .

Notes:

1. We are testing m genes, H0C is the complete null hypothesis, Pi is the

unadjusted p-value for gene i , while i here is the Bonferroni adjusted p-value.

2. We use lower case letters for observed p-values, and upper case for the corresponding random variables.

Proof for Sidák’s methodProof for Sidák’s method(single-step adjustment)(single-step adjustment)

pr(reject at least one Hi | H0C)

= pr(at least one i ≤ | H0C)

= 1 - pr(all i > | H0C)

= 1-∏i=1m pr(i > | H0

C) assuming independence

Here i is the Sidák adjusted p-value, and so i > if and only if Pi > 1-(1- )1/m (check), , giving

1-∏i=1m pr(i > | H0

C)

= 1-∏i=1m pr(Pi > 1-(1- )1/m | H0

C)

= 1- { (1- )1/m }m since all Pi ~ U[0,1],=

Single-step adjustments (ctd)Single-step adjustments (ctd)

The minP method of Westfall and Young:

i = pr( min Pl ≤ pi | H)

1≤l≤m

Based on the joint distribution of the p-values {Pl }. This is the most powerful of the three single-step adjustments.

If Pi U [0,1], it gives a FWER exactly = (see next page).

It always confers weak control, and gives strong control under subset pivotality (definition next but one slide).

Proof for (single-step) minP adjustmentProof for (single-step) minP adjustment

Given level let c be such that

pr(min1 ≤ i ≤ m Pi ≤ c| H0C) = .

Note that {i ≤ } {Pi ≤ c} for any i.

pr(reject at least one Hi at level | H0C)

= pr (at least one i ≤ | H0C)

= pr (min1 ≤ i ≤ m i ≤ | H0C)

= pr (min1 ≤ i ≤ m Pi ≤ c | H0C)

=

Strong control and subset pivotalityStrong control and subset pivotality Note the above proofs are under H0

C, which is what we term weak control.

In order to get strong control, we need the condition of subset pivotality.

The distribution of the unadjusted p-values (P1, P2, …Pm) is said to have the subset pivotality property if for all subsets L {1,…,m} the distribution of the subvector {P i: i L} is identical

under the restrictions {Hi: i L} and H0C .

Using the property, we can prove that for each adjustment under their conditions, we have

pr (reject at least one Hi at level i M0 = pr (reject at least one Hi at level i M00

C ≤ pr (reject at least one Hi at level for all i 0

C ≤

herefore, we have proved strong control for the previous three adjustments, assuming subset pivotality.

Permutation-based single-step minP Permutation-based single-step minP adjustment of p-valuesadjustment of p-values

For the bth iteration, b = 1, , B;

1. Permute the n columns of the data matrix X, obtaining a matrix Xb. The first n1 columns are referred to as “treatments”, the second n2 columns as “controls”.

2. For each gene, calculate the corresponding unadjusted p-values, pi,b , i= 1,2, m, (e.g. by further permutations) based on the permuted matrix Xb.

After all the B permutations are done.

3. Compute the adjusted p-values πi = #{b: minl pl,b ≤ pi}/B.

The computing challenge: The computing challenge: iterated permutationsiterated permutations

The procedure is quite computationally intensive if B is very large (typically at least 10,000) and we estimate all unadjusted p-values by further permutations.

Typical numbers:

• To compute one unadjusted p-value B = 10,000

• # unadjusted p-values needed B = 10,000

• # of genes m = 6,000. In general run time is O(mB2).

Avoiding the computational difficulty of Avoiding the computational difficulty of single-step minP adjustmentsingle-step minP adjustment

• maxT method: (Chapter 4 of Westfall and Young)

πi = Pr( max |Tl | ≥ | ti | | H0C )

1≤l≤m

needs B = 10,000 permutations only.

However, if the distributions of the test statistics are not identical, it will give more weight to genes with heavy tailed distributions (which tend to have larger t-values)

•There is a fast algorithm which does the minP adjustment in O(mBlogB+mlogm) time.

Proof for the single-step maxT adjustmentProof for the single-step maxT adjustment

Given level , let c such that pr(max1 ≤ i ≤ m |Ti | ≥ c | H0C) = .

Note the { Pi ≤ } { |Ti | ≥ c} for any i. Then we have (cf. min P)

pr(reject at least one Hi at level | H0C)

=pr (at least one Pi ≤ | H0C)

=pr ( min1 ≤ i ≤ m Pi ≤ | H0C)

=pr (max1 ≤ i ≤ m |Ti | ≥ c | H0C)

= . To simplify the notation we assumed a two sided test by using the

statistic Ti .We also assume Pi ~ U[0,1].

More powerful methods: More powerful methods: step-down adjustmentsstep-down adjustments

The idea: S Holm’s modification of Bonferroni.

Also applies to Sidák, maxT, and minP.

S Holm’s modification of BonferroniS Holm’s modification of Bonferroni

Order the unadjusted p-values such that pr1 ≤ pr2 ≤ ≤ prm. . The indices r1, r2, r3,.. are fixed for given data.

For control of the FWER at level , the step-down Holm adjusted p-values are

πrj = maxk {1,…,j} {min((m-k+1)prk, 1).

The point here is that we don’t multiply every prk by the same factor m, but only the smallest. The others are multiplied by successively smaller factors: m-1, m-2, ..,. down to multiplying prm by 1.

By taking successive maxima of the first terms in the brackets, we can get monotonicity of these adjusted p-values.

Exercise: Prove that Holm’s adjusted p-values deliver strong control. Exercise: Define step-down adjusted Sidák p-values analogously.

Step-down adjustment of minPStep-down adjustment of minP

Order the unadjusted p-values such that pr1 ≤ pr2 ≤ ≤ prm.

Step-down adjustment: it has a complicated formula, see below, but in effect is

1. Compare min{Pr1, , Prm} with pr1 ;

2. Compare min{Pr2, , Prm} with pr2 ;

3 Compare min{Pr3 , Prm} with pri3 …….

m. Compare Prm with prm . Enforce monotonicity on the adjusted pri . The formula is

πrj = maxk{1,,…,j} {pr(minl{rk,…rm} Pl ≤ prk | H0C )}.

Proof for step-down minP adjustmentProof for step-down minP adjustment

Let M0 is the set of null hypotheses.

Given level . Define gK to be the quantile of mint K Pt | H0C , for any subset K, and define

gi= gSi, where Si={ri, …, rm} {There is at least one Type I error}

={Reject at least one H0t: t M0}

{ mint M0 pt gj} (j m- |M0| is defined by mint M0pt =prj)

={mint M0 pt gM0} (note gj gM0 )

Therefore,

pr( at least one type I error | Ht, t M0)

pr(mint M0 pt gM0 | Ht, t M0)

= pr(mint M0 pt gM0 | H0C) (using subset pivotality)

= (from the definition of gK).

False discovery rateFalse discovery rate(Benjamini and Hochberg 1995)(Benjamini and Hochberg 1995)

Definition: FDR = E(V/R |R>0) P(R >0).

Rank the p-values pr1 ≤ pr2 ≤ …≤ prm.

The adjusted p-values are to control FDR when Pi are independently distributed are given by the step-up formula:

ri= mink {i…m} { min (mprk/k ,1) }.

We use this as follows: reject pr1 ,pr2 ,…, ,prk* where k* is the largest k such that prk ≤ k/m. . This keeps the FDR ≤ under independence, proof not given.

Compare the above with Holm’s adjustment to control FWE, the step-down version of Bonferroni, which is ri = maxk {1,…i} { min (kprk ,1) }.

False discovery rateFalse discovery rate(Benjamini and Yekutieli 2001)(Benjamini and Yekutieli 2001)

This variant controls FDR (1995 definition) for certain regular dependence structures, but it is mostly very conservative.

i= mink=im { min (m/k prk {h=1

m 1/h} ,1) }.

Positive false discovery rate Positive false discovery rate (Storey, 2001, independent case)(Storey, 2001, independent case)

A new definition of FDR, called positive false discovery rate (pFDR) pFDR= E(V/R | R >0)

The logic behind this is that in practice, at lease one gene should be expected to be differentially expressed.

The adjusted p-value (called q-value in Storey’s paper) are to control pFDR.

ri = mink {1..,i} {m/k prk 0}

Note 0 = m0 /m can be estimated by the following formula for suitable 0 = #{i: pi>}/ {(1-) m}.

One choice for is 1/2; another is the median of the observed (unadjusted) p-values.

Positive false discovery rate Positive false discovery rate ( Storey, 2001, dependent case)( Storey, 2001, dependent case)

In order to incorporate dependence, we need to assume identical distributions.

Specify 0 to be a small number, say 0.2, where most t-statistics will fall between (-0, 0) for a null hypothesis, and to be a large number, say 3, where we reject the hypotheses whose t-statistics exceed .

For the original data, find the W = #{i: |ti| 0} and R= #{i: |ti| }.We can do B permutations, for each one, we can compute Wb and Rb similarly

by: Wb = #{i: |ti| 0} and Rb= #{i: |ti| }, b=1,…, B.The we can compute the proportion of genes expected to be null

0=W/{(W1+W2+…+Wb)/B)

An estimate of the pFDR at the point will be 0{(R1+R2+…+RB)/B}/R.Further details can be found in the references.

ResultsResults

gene t unadj. p minP plower maxT

index statistic (104) adjust. adjust.

2139 -22 1.5 .53 8 10-5 2 10-4

4117 -13 1.5 .53 8 10-5 5 10-4

5330 -12 1.5 .53 8 10-5 5 10-4

1731 -11 1.5 .53 8 10-5 5 10-4

538 -11 1.5 .53 8 10-5 5 10-4

1489 -9.1 1.5 .53 8 10-5 1 10-3

2526 -8.3 1.5 .53 8 10-5 3 10-3

4916 -7.7 1.5 .53 8 10-5 8 10-3

941 -4.7 1.5 .53 8 10-5 0.65

2000 +3.1 1.5 .53 8 10-5 1.00

5867 -4.2 3.1 .76 0.54 0.90

4608 +4.8 6.2 .93 0.87 0.61

948 -4.7 7.8 .96 0.93 0.66

5577 -4.5 12 .99 0.93 0.74

The gene namesThe gene names

Index Name

2139 Apo AI

4117 EST, weakly sim. to STEROL DESATURASE

5330 CATECHOL O-METHYLTRANSFERASE

1731 Apo CIII

538 EST, highly sim. to Apo AI

1489 EST

2526 Highly sim. to Apo CIII precursor

4916 similar to yeast sterol desaturase

Callow data with some FDR values included

Some pFDR figuresSome pFDR figures

With the Apo AI data we find that 0 = 0.8 when 0 = 0.2.

For values of equal to 2, 3, 4, 5 and 10, we find estimate

the pFDR with 100 permutations to be 0.572, 0.425, 0.281,

0.082 and 0.0085 respectively. These values are fairly

robust to the number of permutations.

Discussion

The minP adjustment seems more conservative than the maxT adjustment, but is essentially model-free.

With the Callow data, we see that the adjusted minP values are very discrete; it seems that 12,870 permutations are not enough for 6,000 tests.

With the Golub data, we see that the number of permutations matters. Discreteness is a real issue here to, but we do have enough permutations.

The same ideas extend to other statistics: Wilcoxon, paired t, F, blocked

F.Same speed-up works with the bootstrap.

Discussion, ctd.

Major question in practice: Control of FWER or some form of FDR? In the first case, use minP, maxT or something else? In the second case, FDR, pFDR or something else. If minP is to be used, we need guidelines for its use in terms of sample sizes

and number of genes.

Another approach: Empirical Bayes. There are links with pFDR.

Selected referencesSelected references

Westfall, PH and SS Young (1993) Resampling-based multiple testing: Examples and methods for p-value adjustment, John Wiley & Sons, Inc

Benjamini, Y & Y Hochberg (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing JRSS B 57: 289-300

J Storey (2001): 3 papers (some with other authors), www-stat.stanford.edu/~jstorey/

The positive false discovery rate: a Bayesian interpretation and the q-value.

A direct approach to false discovery rates

Estimating false discovery rates under dependence, with applications to microarrays

Y Ge et al (2001) Fast algorithm for resampling based p-value adjustment for multiple testing

AcknowledgementsAcknowledgements

Yongchao Ge

Yee Hwa (Jean) Yang

Sandrine Dudoit

Ben Bolstad

SoftwareSoftware

C and R code available for different tests:

http://www.stat.berkeley.edu/users/terry