Download Differential Expression II

Differential Expression II
Adding power by modeling all the
genes
Oct 06
Linear Model for Several
Conditions
Yij = m + ai + errorij
m is the mean expression for the gene over
the entire experiment.
ai is the deviation of the mean of the ith
condition from the overall mean Si ai=0
Notice that this is a model for 1 gene at a
time.
A basic feature of models
The more items we estimate from the data, the less
precisely we can do the estimation.
The fewer items we estimate from the data, the more we
have to rely on the correctness of the model.
e.g. Yij = m + ai + errorij
Because we are interested in hypotheses about the
means, and are less interested in the variability, we
often assume that the variability does not depend on
the treatment.
When we do an analysis, we check this assumption, but
we do not worry about small violations.
A basic feature of models
Because we are interested in hypotheses about the means, and
are less interested in the variability, we often assume that the
variability does not depend on the treatment.
When we do an analysis, we check this assumption, but we do not
worry about small violations.
For the 2-sample t-test, we have only 2 means, so if we
have sufficient observations, we may allow the
variability to depend on the treatment. (This is the
default.)
If we have only a few observations per treatment, we
might assume that the variances are the same
("pooled variance") which gives more power (if the
assumption is correct.)
Precision and Degrees of
Freedom
The precision of t and F tests is determined in
part by the (error) degrees of freedom.
Assuming constant variance gives more d.f.
t - distributions
F - distributions
Adding Power by Modeling
The main idea:
Since differential expression focuses on differences in
means, use a model for the variance that uses data
from all the genes.
1) all genes have the same variance under all
treatments (frequentist)
2) the variances have a known distribution (Bayesian)
3) the variances have a distribution that can be
estimated (Empirical Bayes)
4) regularization or shrinkage - combine information from
individual genes with information from all the genes
Adding Power by Modeling
1) all genes have the same variance under all
treatments (frequentist)
expression
log 2(expression)
Adding Power by Modeling
as a function of mean
We could assume that
Var(yij) = s2mij
Then the t-test would be
t* 
X Y
X Y
sˆ (  )
n m
2
where we would use a
pooled estimate over all
genes to estimate s2
Adding Power by Modeling
We could assume that Var(yij) = s2 or Var(yij) = s2mij
E.g. with 2 treatments and 2 samples per treatment,
we have only (n+m-2)=2 d.f. for the usual t-test.
If we use the pooled estimate of variance, we obtain
2 x number genes d.f. for the test (practically infinite)
Adding Power by Modeling
as a distribution
This leads to Bayes and Empirical
Bayes models.
Bayes: We assume that we know the
distribution before we even see the
data.
Empirical Bayes: We assume a
shape for the distribution (e.g.
Normal or a multiple of a chi-square)
and estimate parameters such as the
mean and variance or we use a
smoothed version of the sample
histogram
The Effect of (Empirical) Bayes
Let Sp2 be the (estimate of the) mean of the
prior. Then the (empirical) Bayes estimate of
variance of X is a weighted average of Sx2 and
Sp2 and similarly for Y. The associated d.f. for is
a weighted average of the d.f. of Sx2 (sample
size -1) and the pooled variance (the d.f.
depend on the model for the prior).
The t-test then becomes:
X Y
t* 
I am not aware of any software that uses
2
2
the formula here, as the software assumes
ˆ
ˆ
s x , Bayes s y , Bayes
that the variance depends on the gene, but
(

)
not on the treatment
n
m
Adding Power by Regularization
The idea comes from what is called regularization in
matrix inversion - adding a small constant before
taking inverses to avoid dividing by numbers close to
zero.
So, we might replace Sx2 by Sx2+s where s is some small
number.
What number should be chosen? Usually s is based on
the histogram of within gene variances, so this method
is similar to empirical Bayes
But: there are no d.f. associated with the estimate, so
resampling (permutation or bootstrap) is used to
estimate the null distribution.
Some Available Software
MAANOVA: available for Matlab and in
Bioconductor
Fits a very general ANOVA model with fixed and
random effects with the following options:
a) a variance for each gene
b) all genes have the same variance
c) regularized variance estimate
Some Available Software
SAM: available for Excel
samr: in Bioconductor
Fits a few ANOVA models and a survival model
using a regularized test statistic and a null
distribution based on permutations
(simple interface in Excel)
Some Available Software
CyberT: http://visitor.ics.uci.edu/genex/cybert/
Fits paired, 2-sample and one-way ANOVA
models using a full Bayesian model
Can be run on a web interface, or you can
download the R code.
Some Available Software
in Bioconductor
Limma: Fits an empirical Bayes model based on
modeling the variances for each gene as
Sp2Chi-squared, where Sp2 and the d.f. are
estimated from the data. Allows 1 random
effect estimated as a fraction of the variance.
Some Available Software
in Bioconductor
LPE: Estimates the variance of each gene as a
function of the mean expression level
multtest: does permutation tests based on t and F
statistics (including regularized versions)
geneTS: for time-course experiments
Some Other DE Software
in Bioconductor
EBayes: Bioconductor
LMGene:
Multiple Comparisons
Regardless of our choice of analysis method, the
end result is a measure of statistical
significance for each gene.
For frequentist methods, the measure is:
P(observed test statistic| H0 is true)
For Bayesian methods, the measure is
P(H0 is true | observed data)
Multiple Comparisons
To understand the multiple comparisons problem,
we start with the p-value for the ith gene:
P(observed ith test statistic| H0i is true)=pi
Typically, we select a cut-off such as a=.05 or
a=.01 and declare that gene i differentially
expresses (reject H0i) if pi≤ a
Types of Error
m=number of genes
Accept
Reject
Total
H0 true
T
V
m0
HA true
U
S
m-m0
Total
m-R
R
If we test at level a,
E(V)=am0
m
m0= number of
genes that do not
differentially express
R=number of genes
for which we reject
H0
e.g. a=0.05
m=10,000
b=0.8
m0=9000
E(V)=0.05*9000=450
If is the average power
is b,
E(U)=b(m-m0)
E(U)=0.2*1000=200
Most MC procedures focus on V
3.5 main ideas (x 2)
1. Family-wise error rate: adjust the value of a to force
P(V>0) ≤ a* some predefined level such as .05 or .01
2. False discovery rate: choose a value of a (estimated
from the data) to keep E(V/R) ≤ f* some predefined
level such as .05 or .01
3. Estimate m0 (or p0 = m0/m) from the data, and then
apply 1 or 2.
3.5. Pick a and report the
estimate of f*.
Accept Reject
Total
H0 true T
V
m0
HA true U
S
m-m0
Total
R
m
m-R
3.5 main ideas (x 2)
1. Family-wise error rate: adjust the value of a to force P(V>0) ≤ a* some
predefined level such as .05 or .01
2. False discovery rate: choose a value of a (estimated from the data) to keep
E(V/R) ≤ f* some predefined level such as .05 or .01
3. Estimate m0 (or p0 = m0/m) from the data, and then apply 1 or 2.
3.5. Pick a and report the estimate of f*.
x2 Apply the same ideas to the false negatives
Accept Reject
Total
H0 true T
V
m0
HA true U
S
m-m0
Total
R
m
m-R
Family-wise Error Rate
adjust the value of a to force P(V>0) ≤ a* some
predefined level such as .05 or .01
The best-known method is the Bonferroni method.
If you are doing m tests, use a a*/m.
Problem: This is much to conservative and explodes the
U, the number of non-detections.
The adjusted p-value is
e.g. a* =0.05
pi/m
m=10,000
aa*/10000=.0000005
Family-wise Error Rate
With 5 d.f., reject for p<0.05
rejects when |t*|>2.57
reject for p<.0000005 rejects when |t*|>20.59
If we can estimate m0 we only need to adjust by dividing
by m0 which provides a slight improvement.
e.g. m0=9000
reject when |t*|>20.16
e.g. a* =0.05
m=10,000
aa*/10000=.0000005
False Discovery Rate
FDR=E(V/R) where V/R==0 if R=0.
There is a related method called the positive FDR
pFDR=E(V/R | R>0)
Controlling False Discovery Rate
Benjamini and Hochberg (1995)
If the p-values are independent, then the following procedure
controls the FDR at level f
Sort the p-values: p(1)≤p(2) ... ≤p(m)
Let k be the largest value of i for which p(i) ≤ if/m.
We then reject all the hypotheses with
p-value ≤ p(k) (BH method)
Benjamini and Hochberg (2000)
Same procedure with an estimate of m0
Benjamini and Yekutieli (2001)
The BH procedure works with some types of dependent tests.
The BH procedure works with arbitrary dependency if we
replace f by f/S(i/m)≈ f/ln(m) (Compare with Bonferroni)
Estimating pFDR
Storey and Tibshirani (2001)
Let n0 be an estimate of m0.
An estimate of pFDR = a n0/R if we reject for p<a.
They also introduced the q-value - for each observed value of p,
the q-value is the largest estimated pFDR if we reject for
smaller p-values.
The q-value (or the FDR equivalent) is sometimes called an
adjusted p-value, but it is not a p-value=prob(observed|H0)
The pFDR cannot be "controlled" because Pr(R=0)≠0, but it is
readily estimated.
Estimating m0
When m0=m, all the genes are independent, and the test
statistics follow the theoretical null distributions, the pvalues are uniformly distributed.
When m0<m the differentially expressed genes tend to
have smaller p-values
Estimating m0
When m0<m the differentially expressed genes tend to
have smaller p-values.
The flat part of the histogram is used to estimate m0.
There are several methods, and they work quite well as
long as the peak near zero is sharp.
Problems in estimating m0
and FDR
When the power is low (small sample size or noisy data)
we may estimate m0=m and FDR>50% even for the
smallest p-value.
Even if we estimate m0<m (so m-m0 genes DE) we will
not "detect m-m0 genes.
If m0 is very small, there may not be a large enough flat
region of the histogram for estimation.
Occasionally, the distribution of p-values is bi-modal probably due to the distribution of expression too far
from normal.
Bayesian Methods and
Local FDR
So far, we have discussed Bayesian ANOVA
methods which use a prior for the expression
variance.
To adjust for multiple comparisons, Bayesians
model the log2(expression ratio), t-value or pvalue.
As an example, I will show how the method works
for modeling p-values.
Distribution of p-values
The p-values of the nonDE genes is uniform. The pvalues of the DE genes are skewed so that small pvalues are more prevalent. The percentage of nonDE
genes is p0=m0/m.
Distribution of p-values
The distribution of p-values is pofo(p) + (1-po)f1(p).
f0(p)
f1(p)
p0f0(p)+(1-p0)f1(p)
Bayesian Interpretation
p 0 f 0 ( pi )
Pr( H 0i true | pi ) 
p 0 f 0 ( pi )  (1  p 0 ) f1 ( pi )
If you reject when pi<a, the FDR will be
p 0 f0 ( p )
0 p 0 f0 ( p)  (1  p 0 ) f1 ( p) dp
a
So
p 0 f 0 ( pi )
p 0 f 0 ( pi )  (1  p 0 ) f1 ( pi )
can be thought of as a
local FDR
It seems like we should reject H0i when local FDR is small
more importantly than when FDR is small
Frequentists, Bayesians
and FDR
In the samex Bayesian context, and
with F ( x)   f ( x)dx we find that
p 0 F0 (a )
pFDR (a ) 
p 0 F0 (a )  (1  p 0 ) F1 (a )
0
p 0a

p 0a  (1  p 0 ) power
So, pFDR, the rejection region, power and the
Bayesian posterior probability that the gene does not
differentially expression are all connected in a way that
makes intuitive sense.
Frequentists, Bayesians
and q-values
For each observed value of p, the q-value is the largest
estimated pFDR if we reject for smaller p-values.
In the same Bayesian context, the q-value is the largest
probability that H0 is true, if we reject for smaller pvalues. Storey 2003
Efron 2006
Obtains e-Bayes estimates of the distributions of the test
statistic under the null and alternative hypotheses,
defines local FDR using this Bayesian interpretation
and notes the connection between local FDR and qvalues.
Testing for DE and Multiple
Comparisons Adjustments
Most of the software previously mentioned
includes the capability to incorporation FWER,
FDR or some variant.
Each uses a slightly different definition of these
quantities.
The qvalue library in Bioconductor computes
variants of FDR from a list of p-values.
Number of genes = 68665
estimated p0= 0.5669248
estimated m-m0=29737
number of rejections if pFDR≈0.05
0.1
pFDR if we reject at a=0.05
2619
9202
pFDR≈ 0.14
Number of genes = 68665
estimated p0= 0.934315
estimated m-m0=4510
number of rejections if pFDR≈0.05
0.1
pFDR if we reject at a=0.05
0
0
pFDR≈ 0.78