Differential Expression and Tree-based Modeling

Class web site: http://statwww.epfl.ch/davison/teaching/Microarrays/

Statistics for Microarrays

cDNA gene expression data

Data on G genes for n samples

Genes

mRNA samples

Gene expression level of gene i in mRNA sample j

= (normalized) Log( Red intensity / Green intensity)

sample1 sample2 sample3 sample4 sample5 …

1 0.46 0.30 0.80 1.51 0.90 ...2 -0.10 0.49 0.24 0.06 0.46 ...3 0.15 0.74 0.04 0.10 0.20 ...4 -0.45 -1.03 -0.79 -0.56 -0.32 ...5 -0.06 1.06 1.35 1.09 -1.09 ...

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!

Differentially Expressed Genes

• Simultaneously test m null hypotheses, one for each gene j :

Hj: no association between expression level of gene j and covariate/response

• Combine expression data from different slides and estimate effects of interest

• Compute test statistic Tj for each gene j

• Adjust for multiple hypothesis testing

Test statistics

• Qualitative covariates: e.g. two-sample t-statistic, Mann-Whitney statistic, F-statistic

• Quantitative covariates: e.g. standardized regression coefficient

• Survival response: e.g. score statistic for Cox model

QQ-PlotUsed to assess whether a sample follows a particular (e.g. normal) distribution (or to compare two samples)

A method for looking for outliers when data are mostly normal

Recall that for the normal distribution, approximately:68% within 1 SD of the mean95% within 2 SDs99.7% within 3 SDs

Sam

ple

Theoretical

Sample quantile is 0.125

Value from Normal distribution which yields a quantile of 0.125 (= -1.15)

Typical Deviations from Straight Line Patterns

• Outliers

• Curvature at both ends (long or short tails)

• Convex/concave curvature (asymmetry)

• Horizontal segments, plateaus, gaps

Outliers

Long Tails

Short Tails

Asymmetry

Plateaus/Gaps

Example: Apo AI experiment(Callow et al., Genome Research, 2000)

GOAL: Identify genes with altered expression in the livers of one line of mice with very low HDL cholesterol levels compared to inbred control mice

Experiment: • Apo AI knock-out mouse model• 8 knockout (ko) mice and 8 control (ctl) mice

(C57Bl/6)• 16 hybridisations: mRNA from each of the 16

mice is labelled with Cy5, pooled mRNA from control mice is labelled with Cy3

Probes: ~6,000 cDNAs, including 200 related to lipid metabolism

Which genes have changed?

This method can be used with replicated data:

1. For each gene and each hybridisation (8 ko + 8 ctl) use M=log2(R/G)

2. For each gene form the t-statistic:

average of 8 ko Ms - average of 8 ctl Mssqrt(1/8 (SD of 8 ko Ms)2 + 1/8 (SD of 8 ctl Ms)2)

3. Form a histogram of 6,000 t values4. Make a normal Q-Q plot; look for values “off

the line”5. Adjust for multiple testing

Histogram & Q-Q plot

ApoA1

Plots of t-statistics

Assigning p-values to measures of change

• Estimate p-values for each comparison (gene) by using the permutation distribution of the t-statistics.

• For each of the possible permutation of the trt / ctl labels, compute the two-sample t-statistics t* for each gene.

• The unadjusted p-value for a particular gene is estimated by the proportion of t*’s greater than the observed t in absolute value.

816 12,870

Multiple Testing

# not rej # rejected totals

# true H U V (False +)

m0

# false H T (False -) S m1

totals m - R R m

* Per-comparison = E(V)/m * Family-wise = p(V ≥ 1)

* Per-family = E(V) * False discovery rate = E(V/R)

Apo AI: Adjusted and unadjusted p-values for the 50 genes with the larges absolute t-statistics

Genes with adjusted p-value 0.01

Gene Adjustedp-values

t Num Den

Apo A1 0 -22.9 -3.2 0.14

Sterol C5desaturase

0 -13.1 -1.1 0.08

Apo A1 0 -12.2 -1.9 0.16

Apo CIII 0 -11.9 -1.0 0.09

ApoA1 0 -11.4 -3.1 0.2

EST 0 -9.1 -1.0 0.11

Apo CIII 0 -8.4 -1.0 0.12

Sterol C5desaturase

0 -7.7 -1.0 0.13

Single-slide methods

• Model-dependent rules for deciding whether (R,G) corresponds to a differentially expressed gene

• Amounts to drawing two curves in the (R,G)-plane; call a gene differentially expressed if it falls outside the region between the two curves

• At this time, not enough known about the systematic and random variation within a microarray experiment to justify these strong modeling assumptions

• n = 1 slide may not be enough (!)

Single-slide methods

• Chen et al: Each (R,G) is assumed to be normally and independently distributed with constant CV; decision based on R/G only (purple)

• Newton et al: Gamma-Gamma-Bernoulli hierarchical model for each (R,G) (yellow)

• Roberts et al: Each (R,G) is assumed to be normally and independently distributed with variance depending linearly on the mean

• Sapir & Churchill: Each log R/G assumed to be distributed according to a mixture of normal and uniform distributions; decision based on R/G only (turquoise)

Matt Callow’s Srb1 dataset (#8). Newton’s, Sapir & Churchill’s and Chen’s single slide method

Difficulty in assigning valid p-values based on a single slide

Another example: Survival analysis with expression data

• Bittner et al. looked at differences in survival between the two groups (the ‘cluster’ and the ‘unclustered’ samples)

• ‘Cluster’ seemed to have longer survival

Kaplan-Meier Survival Curves, Bittner et al.

unclustered

cluster

Average Linkage Hierarchical Clustering, survival only

Kaplan-Meier Survival Curves, reduced 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)

^

^ ^

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

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

Some ideas for further work

• Expand models to include more genes and possibly two-way interactions

• Nonparametric tree-based subset selection – would require much larger sample sizes

(BREAK)

Trees• Provide means to express knowledge• Can aid in decision making• Can be portrayed graphically or by means of

a chart or ‘key’, e.g. (MASS space shuttle):

stability error sign wind magnitude visibility

DECISION

any any any any any no auto

xstab any any any any yes noauto

stab LX any any any yes noauto

stab XL any any any yes noauto

stab MM nn tail any yes noauto

any any any any Out of range

yes noauto

Etc…

Tree-based Methods – References

• Hastie, Tibshirani, Friedman 2001– The Elements of Statistical Learning

• Venables and Ripley, 1999– Modern Applied Statistics with S-Plus (MASS)

• Ripley, 1996– Pattern Recognition and Neural Networks

• Breiman, Olshen, Friedman, Stone 1984– Classification and Regression Trees

Tree-based Methods

• Automatic construction of decision trees dates from social science work in the early 1960’s (AID)

• Breiman et al. (1984) proposed new algorithms for tree construction (CART)

• Tree construction can be seen as a type of variable selection

Response types

• Categorical outcome – Classification tree

• Continuous outcome – Regression tree

• Survival outcome – Survival tree

• Software – Available R packages include tree, rpart (tssa available in S)

Trees Partition the Feature Space

• End point of tree is a (labeled) partition of the (feature) space of possible observations X

• Tree-based methods partition X into rectangular regions; try to make the (average) responses in each box as different as possible

• In logical problems it is assumed that there does exist a partition of X that will correctly classify all observations; task is to find a tree to succinctly describe this partition

Partitions and CART

X1 X1

t1 t3

t2

t4

R2

R1

R3

R5

R4

Yes No

X2 X2 XX

XX

XX

Partitions and CART

X1

t1 t3

t2

t4

R2

R1

R3

R5

R4

X2

X1 t1

X2 t2 X1 t3

X2 t4

R1 R2 R3

R4 R5

Tree Comparison

• Measure how well the partition created by a tree corresponds to the correct decision rule (classification)

• For a logical problem, count number errors

• For statistical problem, usually overlapping class distributions, so that no partition unambiguously describes classes – estimate misclassification prob.

Three Aspects of Tree Construction

• Split Selection Rule

• Split-stopping Rule

• Assignment of predicted values

Split Selection

• Binary splits

• Look only one step ahead – avoids massive computational time by not attempting to optimize whole tree performance

• Choose an impurity measure to optimize each split – Gini index or entropy, rather than misclassification rate for classification tree, deviance (squared error) for regression tree

Split-stopping

• Issue: A very large tree will tend to overfit the data (and therefore lack generalizability), while too small a tree might not capture important structure

• Usual solution: grow large/maximal tree (stop splitting only when some minimum node size, 5 or 10 say, is reached), followed by (cost-complexity) pruning

Pruning

• Sequence of rooted subtrees

• Measure Ri (e.g. deviance) at leaves, R = Ri

• Minimize the cost-complexity measure

R = R + * size

governs tradeoff between tree size and goodness of fit

• Choose to minimize cross-validated error (misclassification or deviance)

Assignment of Predicted Values

• Assign value to each leaf (terminal node)

• In Classification: (weighted) voting among observations in the node

• In Regression: mean of observations in the node

Other Issues (I)

• Loss matrix

– Procedures can be modified for asymmetric losses

• Missing predictor values

– Can create ‘missing’ category

– Surrogate splits exploit correlation between predictors

• Linear combination splits

Other Issues (II)• Tree Instability

– Small changes in the data can result in very different series of splits – difficulties in interpretation

– Aggregate trees to reduce (e.g. bagging)

• Lack of smoothness

– More of an issue in regression trees

– Multivariate Adaptive Regression Splines (MARS)

• Difficulty in capturing additive structure with binary trees

Acknowledgements

• Sandrine Dudoit

• Jane Fridlyand

• Yee Hwa (Jean) Yang

• Debashis Ghosh

• Erin Conlon

• Ingrid Lonnstedt

• Terry Speed