Statistical tests for differential expression in cDNA microarray experiments (2): ANOVA

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Statistical tests for differential expression in
cDNA microarray experiments (2) ANOVA

• Xiangqin Cui and Gary A. ChurchillGenome Biology
  2003, 4210

Presented by M. Carme Ruíz de Villa and Alex
Sánchez Departament dEstadística U.B.
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Introduction
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Remember

• We want to measure how gene expression changes
  under different conditions.
• Only two conditions and an adequate number of
  replicates ? t-tests extensions
• More than two conditions / more than one factor
  several approaches
• Analysis of Variance (ANOVA) (Churchill et al.)
• Linear Models (Smyth, Speed, )

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Sources of variation (1)

• We want to determine when the variation due to
  gene expression is significant, but
• There are multiple sources of variation in
  measurements besides just gene expression.
• We want to know when the variation in
  measurements is caused by
• varying levels of gene expression
• versus other factors.

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Sources of variation (2)

• Some sources of variation in the measurements in
  microarray experiments are
• Array effects
• Dye effects
• Variety effects
• Gene effects
• Combinations

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Relative expression values

• If more than two conditions ? we cannot simply
  compute ratios
• ANOVA modelling yields estimates of the relative
  expression for each gene in each sample
• The ANOVA model is not based on log ratios.
  Rather it is applied directly to intensity data.
  However the difference between two relative
  expression values can be interpreted as the mean
  log ratio for comparing two samples.

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Technical biological replicates

• If inference is being made on the basis of
  biological replicates
• and there is also technical replication ?
• technical replicates should be averaged
• to yield a single value
• for each independent biological unit.

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Derived data sets

• The set of estimated relative expression values,
  one for each gene in each RNA sample, is a
  derived data set that may be subject to a second
  level of analysis.
• The derived data can be analyzed on a gene by
  gene basis using standard ANOVA methods to test
  for differences among conditions. (Oleksiak et
  al. 28)

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Review of ANOVA models
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One way ANOVA

• Suppose you have a model for each measurement in
  your experiment
• yij is jth measurement for ith group.
• µ overall mean effect (constant)
• ai ith group effect (constant)
• eij experimental error term N(0,s2)
• Therefore, observations from group i are
  distributed with mean µ ai and variance s2 .

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Hypothesis Testing
Overall variability
Within group variability
Between group variability
Intuition if between group variability is large
compared to within group variability then the
differences between means is significant.
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Sum of Squares

• Total sum of squares
• Within Sum of Squares
  
• Between Sum of Squares
  

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Mean Sum of Squares

• Between MS Between SS/(k-1)
• Within MS Within SS/(n-k)
• F Between MS / Within SS
• It is summarized in the ANOVA table
• Example 1

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Multiple Factor ANOVA

• The model can be extended by adding more
• Factors (?, ?, )
• Interactions between them (??, )
• Other
• This is used to model the different sources of
  variation appearing in microarray experiments

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Experiment 1 Latin Square
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Random effects models

• If the k factor levels can be considered a random
  sample of a population of factors we have a
  random effect
• ANOVA model Yij ? Ai eij,
• ? overall mean,
• Ai is a random variable instead of a constanty,
• eij experimental error.
• E(Ai)0, E(eij)0, var(Ai)?A2, var(eij) ?2, Ai
  i eij independent? var(Yij) ?A2 ?2.

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Where to find more

• Draghici, S. (2003). ANOVA chapter (7) Data
  analysis tools for microarrays Wiley
• Pavlidis, P. (2003) Using ANOVA for gene
  selection from microarray studies of the nervous
  systemhttp//microarray.cpmc.columbia.edu/pavlidi
  s/ doc/reprints/anova-methods.pdf

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ANOVA Models for Microarray Data
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Kerr Churchills model

• yijkg ? expression measurement from the ith
  array, jth dye, kth variety, and gth gene.
• µ ? average expression over all spots.
• Ai ? effect of the ith array.
• Dj ? effect of the jth dye.
• Vk ? effect of the kth variety (treatment,
  sample, )
• Gg ? effect of the gth gene.
• (AG)ig ? effect of the ith array and gth gene.
• (VG)kg ? effect of the kth variety and gth gene.
• ?ijkg independent and identically distributed
  error terms.

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Interpreting main effects

• A differences in fluorescent signal from array
  to array (e.g. if arrays are probed under
  inconsistent conditions that increase or reduce
  hybridization of labeled cDNA)
• D differences between two dye fluorescent labels
  (one dye may consistently be brighter than the
  other)
• G differences in fluorescence for equally
  expressed genes.
• V differences of expression level between
  different varieties (samples, tumour types,..).

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Interpreting interactions

• DV If for a particular variety labelling is
  produced in separate runs of the process ?
  Differences in the runs can produce pools of cDNA
  of varying concentrations or quality.
• AG (Spot effect) Spots for a given gene on the
  different arrays vary in the amount of cDNA
  available for hybridization.
• DG if there are differences in the dyes that are
  gene-specific
• VG reflects differences in expression for
  particular variety and gene combinations that are
  not explained by the average effects of these
  varieties and genes.THIS IS THE QUANTITY OF
  INTEREST !!!

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Normalization

• A,D,V terms effectively normalize the data, thus
  the normalization process is integrated with the
  data analysis.
• This approach has several benefits (?)
• The normalization is based on a clearly stated
  set of assumptions
• It systematically estimates normalization
  parameters based on all the data
• The model can be generalized to the situation
  where genes are spotted multiple times on each
  array rather

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Statistically Significant Effects

• Array, Dye , Variety Gene effect
• Goal To estimate their value.
• Need not assess their significance
• Sometimes dont appear (gene-level model)
• Array x Gene, Variety x Gene effects
• May or not be present
• Goal To assess their significance
• Mean effect 0 if fixed
• Effect variance 0 if random

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Test statistics The 3 Fs

• Hypothesis testing involves the comparison of two
  models.
• In this setting we consider a
• null model of no differential expression (all VG
  0) and
• an alternative model with differential expression
  among the conditions (some VG are not equal to
  zero).
• F statistics are computed on a gene-by-gene
  basis based on the residual sums of squares from
  fitting each of these models.

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Example 1

• A gene, which is believed to be related to
  ovarian cancer is investigated
• The cancer is sub-classified in 3 cathegories
  (stages) I, II, III-IV
• 15 samples, 3 per stage are available
• They are labelled with 3 colors and hybridized on
  a 4 channel cDNA array (1 channel empty)(A
  seemingly more reasonable procedure double
  dye-swap reference design)
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

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Example 1. Normalized Data
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Example 1 ANOVA table (1)
If arrays are homogeneous The appropriate model
is 1 factor ANOVA
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Example (1) Blocking
If arrays are not homogeneous? the appropriate
model is 2 factor ANOVA (1 new block factor for
arrays)
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Example 2 CAMDA kidney dataftp//ftp.camda.duke.
edu/CAMDA02_DATASETS/papers/README_normal.html

• 6 mouse kidney samples
• (suppose 6 different treatments)
• Compared to a common reference in a double
  reference design
• Dye swap
• Replicate arrays

2
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2.1. The ANOVA model

• Work only at the gene level no main effects (A,
  D, V, G) as defined
• YijkDGiAGjVGk?ijk
• i1,2 (dyes)
• j1,2 (array)
• K1,,6 (sample)

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Example 3 A 2 factor design Diet X Strain
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3.2. Design
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3.3. The ANOVA model

• YijkDGiAGjStrainlDietm StrainDietlm
  VGk?ijklm
• i1,,2 (dyes)
• j1,,2 (array)
• k1,,12 (sample)
• l 1,,3 (strain)
• m 1,...,2 (diet)

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3.4 Sample R code (1)

• data(paigen)
• paigen lt- createData(rawdata, 2)
• model.full.fix lt- makeModel (data
  paigen,formulaDGAGSG StrainDietStrainDie
  t)
• anova.full.fix lt-fitmaanova (paigen,
  model.full.fix)
• model.noint.fix lt- makeModel (data
  paigen,ormulaDGAGSGStrainDiet)
• anova.noint.fix lt- fitmaanova(paigen,
  model.noint.fix)

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3.4. Sample R code (2)

• permutation tests
• test for interaction effect
• test.int.fix lt- ftest(paigen, model.full.fix,
  model.noint.fix, n.perm500) idx.int.fix lt-
  volcano(anova.full.fix, test.int.fix,
  title"Int. test")