Statistics for Microarrays
Linear and Nonlinear Modeling
Class web site: http://statwww.epfl.ch/davison/teaching/Microarrays/
cDNA gene expression data
Data on G genes for n samples
mRNA samples
sample1 sample2 sample3 sample4 sample5 …
Genes
1
2
3
4
5
0.46
-0.10
0.15
-0.45
-0.06
0.30
0.49
0.74
-1.03
1.06
0.80
0.24
0.04
-0.79
1.35
1.51
0.06
0.10
-0.56
1.09
0.90
0.46
0.20
-0.32
-1.09
...
...
...
...
...
Gene expression level of gene i in mRNA sample j
= (normalized) Log( Red intensity / Green intensity)
Identifying Differentially
Expressed Genes
• Goal: Identify genes associated with
covariate or response of interest
• Examples:
– Qualitative covariates or factors:
treatment, cell type, tumor class
– Quantitative covariate: dose, time
– Responses: survival, cholesterol level
– Any combination of these!
Modeling Introduction
• Want to capture important features
of the relationship between a (set of)
variable(s) and one or more responses
• Many models are of the form
g(Y) = f(x) + error
• Differences in the form of g, f and
distributional assumptions about the
error term
Examples of Models
• Linear: Y = 0 + 1x + 
• Linear: Y = 0 + 1x + 2x2 + 
• (Intrinsically) Nonlinear:
Y = x1x2 x3 + 
• Generalized Linear Model (e.g. Binomial):
ln(p/[1-p]) = 0 + 1x1 + 2x2
• Proportional Hazards (in Survival Analysis):
h(t) = h0(t) exp(x)
Linear Modeling
• A simple linear model:
E(Y) = 0 + 1x
• Gaussian measurement model:
Y = 0 + 1x + ,
where  ~ N(0, 2)
• More generally:
Y = X + ,
where Y is n x 1, X is n x G,  is G x 1,
 is n x 1, often assumed N(0, 2Inxn)
Analysis of Designed
Experiments
• An important use of linear models
• Here, the response (Y) is the gene
expression level
• Define a (design) matrix X so that
E(Y) = X,
where  is a vector of contrasts
• Many ways to define design matrix/contrasts
Contrasts (I)
• Example: One-way layout, k classes
Yij =  + j + ij, i = 1,…,nj; j = 1,…,k
X = 1 1 0 … 0 = [1 Xa]
101…0
…
100…1
• Problem: this specification is overparametrized
Contrasts (II)
• Could resolve by removing column of 1’s:
Yij = j + ij, i = 1,…,nj; j = 1,…,k
X = 1 0 … 0 = [Xa]
01…0
…
00…1
• Here, parameters are class means
Contrasts (III)
• Define design matrix
X* = [1 XaCa],
with Ca (k x (k-1)) chosen so that X*
has rank k (the number of columns)
• Parameters may become difficult to
interpret
• In the balanced case (equal number of
observations in each class), can choose
orthogonal contrasts (Helmert)
Model Fitting
• For the standard (fixed effects)
linear model, estimation is usually by
least squares
• Can be more complicated with random
effects or when x-variables subject
to measurement error as well (so that
estimates are not biased)
Model Checking
• Examination of residuals
–
–
–
–
Normality
Time effects
Nonconstant variance
Curvature
• Detection of influential observations
Do we need robust methods?
• Tukey (1962):
“A tacit hope in ignoring deviations from
ideal models was that they would not
matter; that statistical procedures which
were optimal under the strict model
would still be approximately optimal under
the approximate model. Unfortunately it
turned out that this hope was often
drastically wrong; even mild deviations
often have much larger effects than were
anticipated by most statisticians.”
Robust Regression
• Idea: downweight observations that
produce large residuals
• More computationally intensive than
least squares regression (which gives
equal weight to each observation)
• Use maximum likelihood if can assume
specific error distribution
• When not, use M-estimators
M-estimators
• ‘Maximum likelihood type’ estimators
• Assume independent errors with
distribution f()
• Robust estimator minimizes
i(ei/s) = i{(Yi – xi’)/s},
where (.) is some function and s is an
estimate of scale
• (u) = u2 corresponds to minimizing the
sum of squares
M-estimation Procedure
• To minimize i{(Yi – xi’)/s} wrt the ’s,
take derivatives and equate to 0
• Resulting equations do not have an
explicit solution in general
• Solve by iteratively reweighted least
squares
Examples of Weight Functions
Generalized Linear Models
(GLM/GLIM)
• Response Y assumed to have exponential
family distribution:
f(y) = exp[a(y)b() + c() + d(y)]
• Parameters  and explanatory variables X;
linear predictor  = 1x1 + 2x2 + … pxp
• Mean response , link function l () = 
• Allows unified treatment of statistical
methods for several important classes of
models
Some Examples
Link
logit
probit
cloglog
identity
inverse
Log
Sqrt
Binomial
Default
X
X
Gamma
Normal Poisson
X
X
Default
X
X
Default
X
(BREAK)
Survival Modeling
• Response T is a (nonnegative) lifetime
• Cumulative distribution function (cdf)
F(t), density f(t)
• More usual to work with the survivor
function
S(t) = 1 – F(t) = P(T > t)
and the instantaneous failure rate, or
hazard function
h(t) = limt->0P(t  T< t+t | T  t)/ t
Relations Between Functions
• Cumulative hazard function
H(t) = 0t h(s) ds
• h(t) = f(t)/S(t)
• H(t) = -log S(t)
Censoring
• Incomplete information on the lifetime
• A censored observation is one whose
value is incomplete due to random
factors for each individual
• Most commonly, observation begins at
time t = 0 and ends before the outcome
of interest is observed (right-censoring)
Estimation of Survivor Function
• Most commonly used estimate is
Kaplan-Meier (also called product
limit) estimator
• Risk set r(t) = number of cases alive
just before time t
^
• S(t)
= tit [r(ti) – di]/r(ti)
Cox Proportional Hazards
Model
• Baseline hazard function h0(t)
• Modified multiplicatively by covariates
• Hazard function for individual case is
h(t) = h0(t) exp(1x1 + 2x2 + … + pxp)
• If nonproportionality:
– 1. Does it matter
– 2. Is it real
Strategies for Gene
Expression-based Modeling
• The biggest problem is the large number
of variables (genes)
• One possibility is to first reduce the
number of genes under consideration
(e.g. consider variability across samples,
or coefficient of variation)
• Screening/Prioritizing: One gene at a
time approach
• Two at a time, perhaps plus interaction
Example: Survival analysis
with expression data
• Bittner et al. dataset:
– 15 of the 31 melanomas had
associated survival times
– 3613 ‘strongly detected’ genes
Average Linkage Hierarchical
Clustering
unclustered
‘cluster’
Association of Variables
• Variables tested for association with
cluster:
– Sex (p = .68, n = 16 + 11 = 27)
Age (p = .14, n = 15 + 10 = 25)
Mutation status (p = .17, n = 12 + 7 = 19)
– Biopsy site (p = .88, n = 14 + 10 = 24)
– Pigment (p = .26, n = 13 + 9 = 22)
– Breslow thickness (p = .26, n = 6 + 3 = 9)
– Clark level (p = .44, n = 6 + 5 = 11)
Specimen type (p = .11, n = 11 + 12 = 23)
Survival analysis: Bittner et al.
• Bittner et al. also looked at differences
in survival between the two groups (the
‘cluster’ and the ‘unclustered’ samples)
• ‘Cluster’ seemed associated with longer
survival
Kaplan-Meier Survival Curves
Average Linkage Hierarchical
Clustering, survival samples only
unclustered
cluster
Kaplan-Meier Survival
Curves, new grouping
Identification of Genes
Associated with Survival
For each gene j, j = 1, …, 3613, model the
instantaneous failure rate, or hazard
function, h(t) with the Cox proportional
hazards model:
h(t) = h0(t) exp(jxij)
and look for genes with both:
^
• large effect size j
^
^
• large standardized effect size j/SE(j)
Sites Potentially Influencing
Survival
Image
UniGene
UniGene Cluster Title
Clone ID Cluster
137209 Hs.126076 Glutamate receptor
interacting protein
240367 Hs.57419 Transcriptional repressor
838568
825470
841501
Hs.74649
Cytochrome c oxidase
subunit Vlc
Hs.247165 ESTs, Highly similar to
topoisomerase
Hs.77665 KIAA0102 gene product
Findings
• Top 5 genes by this method not in
Bittner et al. ‘weighted gene list’ - Why?
• weighted gene list based on entire
sample; our method only used half
• weighting relies on Bittner et al. cluster
assignment
• other possibilities?
Statistical Significance of
Cox Model Coefficients
Advantages of Modeling
• Can address questions of interest directly
– Contrast with what has become the
‘usual’ (and indirect) approach with
microarrays: clustering, followed by
tests of association between cluster
group and variables of interest
• Great deal of existing machinery
• Quantitatively assess strength of evidence
Limitations of Single Gene Tests
• May be too noisy in general to show much
• Do not reveal coordinated effects of
positively correlated genes
• Hard to relate to pathways
Not Covered…
• Careful followup
– Assessment of proportionality
– Inclusion of combinations of genes,
interactions
– Consideration of alternative models
• Power assessment
– Not worth it here, there can’t be much!
Some ideas for further work
• Expand models to include more genes,
possibly two-way interactions
– Issue of automation
– Still very small scale compared to
probable pathway size, number of
genes involved, etc.
• Nonparametric tree-based modeling
– Will require much larger sample sizes
Acknowledgements
• Debashis Ghosh
• Erin Conlon
• Sandrine Dudoit
• José Correa