Statistical Methods for Analyzing Statistical Methods for Analyzing Ordered Gene Expression Ordered Gene Expression

Microarray DataMicroarray Data

Shyamal D. PeddadaBiostatistics Branch

National Inst. Environmental Health Sciences (NIH)

Research Triangle Park, NC

An outlineAn outline Ordered gene expression data

Common experimental designs

A review of some statistical methods

An example

Demonstration of ORIOGEN – a software for ordered gene expression data

Some examples of ordered Some examples of ordered gene expression datagene expression data

Comparison of gene expression by:

– various stages of cancer Normal - Hyperplasia – Adenoma – Carcinoma

– tumor size New tumor – Middle Size – Large tumor (with necrosis)

– dose of a chemical (dose-response study)

– duration of exposure to a chemical (time-course experiments)

– dose & duration

Some commonly used experimental Some commonly used experimental designsdesigns

Experimental unit: Tissues/cells/animals Single chemical/treatment

– Dose response study– Time course study

single dose but responses obtained at multiple time points after treatment

experimental units are treated at multiple time points using the same dose.

– Dose response x Time course study Multiple doses at multiple time points

Multi chemicals/treatments

Possible objectivesPossible objectives

– Investigate changes in gene expression at certain biologically relevant category.

E.g. Hyperplasia to Adenoma to Carcinoma E.g. “early time point” to “late time point” since the

exposure to a chemical

– Identify/cluster genes with similar expression profiles over time/dose.

Correlation coefficient based Correlation coefficient based methodsmethods

Correlation coefficient based methods match genes with similar observed patterns of expression across dose/time points.

Gene 1

Gene 2

Correlation coefficient based Correlation coefficient based methodsmethods

A number of variations to this general principle exist in the literature. Here we outline some prominent ones.

A. Chu et al. (Science, 1998): Pre-select a set of biologically relevant patterns of gene

expressions over time. Identify a sample of about 3 to 8 genes for each pattern. Compute the correlation coefficient of each candidate

gene in the microarray data with the above pre-selected genes.

Cluster each candidate gene into the cluster with highest correlation coefficient

Correlation coefficient based Correlation coefficient based methods …methods …

B. Kerr and Churchill (PNAS, 2001):

They correctly recognized the uncertainty associated with Chu et al. ‘s clustering algorithm. Hence they proposed a bootstrap methodology to evaluate Chu et al.’s clusters.

C. Heyer et al. (Genome Research, 1999):

Rather than using the standard correlation coefficient between genes, they employ jackknife version which robustifies against outliers.

Unlike Chu et al.’s strategy, they classify genes on the basis of pairwise correlation coefficients.

Correlation coefficient based Correlation coefficient based methods …methods …

Strengths Familiarity among biologists Easy to compute and interpret (although it is often

misinterpreted too!)

Weakness Non-linearity in the data can lead to misinterpretation Outliers and influential observations can affect the

numerical value of the correlation coefficient. Heterogeneity between genes can also affect the numerical

value of the correlation coefficient. It is also important to note that correlation coefficient is

typically estimated on the basis of a very small number of points.

Regression based proceduresRegression based procedures

Basic assumption among these methods:Basic assumption among these methods:

The “conditions” are numerical, The “conditions” are numerical, e.g. dose or timee.g. dose or time

Polynomial regressionPolynomial regression Liu et al. (BMC Bioinformatics, 2005)Liu et al. (BMC Bioinformatics, 2005)

For each gene Liu et al. fitted a quadratic regression model:

They cluster each gene into a particular cluster depending upon the sign and statistical significance of the regression parameters.

If for a gene none of the regression coefficients are significant then such a gene is declared un-important.

tggggtg ttY ,2

2,1,0,, εβββ +++=

Polynomial regressionPolynomial regression Liu et al. (BMC Bioinformatics, Liu et al. (BMC Bioinformatics,

2005)2005) Strengths:

Biologists are reasonably familiar with quadratic regression analysis.

Regression coefficients are easy to interpret.

For small number of doses or time points and for evenly spaced doses, a quadratic model may be a reasonable approximation.

An easy to use EXCEL based software is available.

Polynomial regressionPolynomial regression Liu et al. (BMC Bioinformatics, Liu et al. (BMC Bioinformatics,

2005)2005) Two major limitations because it is fully parametric:

1. Departure from quadratic model is common:

In such cases thequadratic modelmay not be correct.

2. Normality assumption need not be valid.Time

““Semi-parametric” regression Semi-parametric” regression methodsmethods

Several authors have tried semi-parametric regression approach to gene expression data.

E.g. deHoon et al. (Bioinformatics, 2002) Bar-Joseph et al. (PNAS, 2003, Bioinformatics, 2004) Luan and Li et al. (Bioinformatics, 2003) Storey et al. (PNAS, 2005)

Storey et al. (2005)Storey et al. (2005)

Basic idea:

For each gene, they fit mixed effects model with a B-spline basis. This methodology is largely based on Brumback and Rice (JASA, 1998).

Statistical significance of each gene is evaluated using an F like test statistic with P-value (q-value) determined by bootstrap.

Storey et al. (2005)Storey et al. (2005)

Strengths:

It is semi-parametric A user friendly software called EDGE is available

Limitations: It does not perform well for “threshold” patterns of gene

expression The “conditions” should be numerical Unequal dose or time spacing can have an impact on

the performance of the procedure

OOrder rder RRestricted estricted IInference for nference for OOrdered rdered GGene ene EExpressioxpressioNN

(ORIOGEN)(ORIOGEN)

Peddada et al. (Bioinformatics, 2003, 2005)Simmons and Peddada (Bioinformation, 2007)

Temporal Profile /Dose ResponseTemporal Profile /Dose Response

Pattern of the (unknown) mean expression of a gene

over time (dose) is known as the temporal profile (dose response) of a gene.

– ORIOGEN: uses mathematical (in)equalities to describe a profile.

)(μ

Some ExamplesSome Examples Null profile:

654321 μμμμμμ =====

Examples Continued …Examples Continued …

Up-down profile with maximum at 3 hours

654321 μμμμμμ ≥≥≥≤≤

Examples Continued …Examples Continued …

Non-increasing profile

Cyclical profile

654321 μμμμμμ ≥≥≥≥≥

654321 μμμμμμ ≤≥≥≤≤

ORIOGENORIOGEN

Step 1 (Profile specification):

Pre-specify the shapes of profiles of interest.

Some Examples Of Pre-specified Some Examples Of Pre-specified ProfilesProfiles

ORIOGEN …ORIOGEN …

Step 2 (profile fitting): Fit each pre-specified profile to each gene using the estimation procedure described in:

Hwang and Peddada (1994, Ann. of Stat.)

A Brief Description Of The Estimation A Brief Description Of The Estimation Procedure …Procedure …

DefinitionsDefinitions

Linked parameters: Two parameters are said to be linked if the inequality between them is known a priori.

Nodal parameter: A parameter is said to be nodal if it is linked to all parameters in the graph.

For any given profile, the estimation always starts at the nodal parameter.

Pool the Adjacent Violator Algorithm Pool the Adjacent Violator Algorithm (PAVA)(PAVA)

Hypothesis:

Observed data

Isotonized data (PAVA)

Estimation: The General IdeaEstimation: The General Idea

1

2 4

5

3

1

25

4

3

3 is the only nodal parameter

Estimation Continued …Estimation Continued …

From this sub-graph we estimate 1 and 2.

1

2

3

Step 3: Determine the norm of a gene corresponding

to each temporal profile.

This is defined as the maximum (studentized) difference between estimates corresponding to linked parameters.

Peddada et al. (2001, Biometrics).

A Measure of “Goodness-of-fit” A Measure of “Goodness-of-fit” NormNorm∞l

∞l

An ExampleAn Example Observed data:

1, 1.5, 2, 2.5, 1.5, 2.25

Two pre-specified temporal profiles:

(a) (b)

Example Continued …Example Continued …

Fit under profile (a)

1, 1.5, 2.25, 2.25, 1.875, 1.875

Fit under profile (b)

1, 1.5, 2, 2.5, 1.875, 1.875

Example Continued …Example Continued …

norm for profile (a) is:

2.25 - 1 = 1.25

norm for profile (b) is:

2.5 - 1 = 1.5

∞l

∞l

““Best Fitting” ProfileBest Fitting” Profile

Step 4: Identify the profile with the largest norm.

In the example, profile (b) has larger norm than profile (a) .

Hence profile (b) is a better fit than (a).

Statistical SignificanceStatistical Significance

Step 5: Statistical significance:

P-value for statistical significance is obtained using the bootstrap methodology:

Illustration …Illustration …

MCF-7 breast cancer cell treated MCF-7 breast cancer cell treated with 17 -estradiol (Lobenhofer et with 17 -estradiol (Lobenhofer et al., 2002, al., 2002, Mol. EndocrinMol. Endocrin.)..).

Gene expressions were measured after: 1hr, 4hrs, 12hrs, 24hrs, 36hrs and 48hrs of treatment.

# of genes on each chip = 1900.

# of samples at each time point = 8

β

Available softwaresAvailable softwares

Linear Regression Method (Liu et al., 2005) EDGE (Storey et al., 2005) EPIG (Chao et al., 2008) ORIOGEN (Peddada et al., 2006)

Concluding remarksConcluding remarks

Methodology Freely available software

Applicable to ordinal “conditions”

Repeated measures and correlated data

Model assumptions

Linear Regression Yes No No Linear regression

EPIG Yes No ? No

EDGE Yes No Yes No

ORIOGEN Yes Yes Yes No

Some open problemsSome open problems

ORIOGEN is potentially subject to Type III error. How do we control FDR & Type III error.

How to deal with

– Dependent samples?– Covariates?

Order restricted inference in the context of mixed effects linear models.

AcknowledgmentsAcknowledgments

– Leping Li – David Umbach– Clare Weinberg– Ed Lobenhofer – Cynthia Afshari

Software developers at Constella Group

– (late) John Zajd– Shawn Harris