Assessing Differential Expression in Mixtures of Cell Types

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Assessing Differential Expression in Mixtures of
Cell Types

• Achim Tresch

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Statistical Testing in a Nutshell
Question / Hypothesis Is the expression of gene g
in cell type 1 higher than in cell type 2?
Data Expression of gene g in several
samples(absolute scale)
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What is a good Statistic?
Probability density
good
Statisctics of genes for which expression of
class1 class 2
Statisctics of genes for which expression of
class1 gt class 2
0
good statistic
Probability density
poor
0
poor statistic
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What is a good Statistic?
Sensitivity
(Proportion of truly higher expressed genes
which were found among the truly higher
expressed genes)
Type I error
(Proportion of non-higher expressed genes that
are erroneously found)
Genes that are foundby the statistics
threshold
5
What is a good Statistic?
Sensitivity
Type I error
Genes that are foundby the statistics
threshold
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What is a good Statistic?
Sensitivity
1
Sensitivity
0
1
Type I error
Type I error
1
Sensitivity
0
1
Type I error
threshold
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What is a good Statistic?
Sensitivity
1
Sensitivity
0
1
Type I error
Type I error
1
Sensitivity
0
1
Type I error
threshold
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Receiver Operating Characteristic (ROC)
ROC curve
1
good
Sensitivity
0
1
Type I error
ROC curve
1
poor
Sensitivity
0
1
Type I error
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Statistics for the Detection of Differential Gene
Expression
Bad Idea Subtract the estimated sample means
Problem d is not scale invariant
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Problems in Mixed Cell Samples
Varying cell proportions lead to varying
observations
Examples

• Tumor biopsies with varying proportions of tumor
  tissue in each sample. An estimate of the tumor
  tissue proportion is provided by the pathologist.
• RNAi experiments in which transfection efficiency
  (resp. knockdown efficiency) is substantially
  below 100. The proportion of cells in which RNAi
  works is estimated via TaqMan analysis of the
  target RNA.

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Problems in Mixed Cell Samples
Let S be a collection of samples. Let xs be the
relative proportion of cells of type 1 in this
sample s.
Then,
Expression
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Connections to Linear Regression
Idea Observe that can be estimated as the
slope of the linear fit of the data
Though formally identical to the t-statistic, the
difference is in the calculation of d.
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Ingredients for the Calculation of d Shrinkage
estimation of d using penalized regression
The standard regression estimate for d is the
minimizer of the quadratic loss
measurement
linear fit
A way to make linear regression more robust is to
add a linear penalty term for d (Lasso,
R.Tibshirani 1998)
Here, ? is the so-called shrinkage parameter. It
is comparable to s0 in the SAM-statistics and
avoids overfitting.
How do we choose ??
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Ingredients for the Calculation of d Selection
of the shrinkage parameter ?
Idea Taking a Bayesian view, d can be derived as
the mode of a posterior distribution
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Ingredients for the Calculation of d Selection
of the shrinkage parameter ?
If we assume that the entries in d follow a
Laplacew,0 distribution, then ths shrinkage
parameter ? can be derived from the shape
parameter of this distribution as ?
1/(2w) Apply a Kolmogorov-Smirnov test to justify
this assumption and fit the shape parameter w.
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Ingredients for the Calculation of d Adjustment
to heteroscedasticity of the measurements
Variance depends upon expression intensity
Use the error model proposed in vsn, use vsn to
estimate the individual variances.
Expression
Perform a weighted penalized linear regression
instead of a simple linear regression for d.
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Ingredients for the Calculation of d The
Two-Component Model
(Taken from W.Huber)
measured intensity offset gain ?
true abundance
A robust fitting method for the estimation of the
parameters ai ,bi ,s1 ,s2 has been developed by
W.Huber and A.v.Heydebreck. It has been
implemented in the R package vsn.
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The final algorithm

• Estimate the empirical distribution of the
  entries of d by a simple linear regression
• Fit a Laplace distribution to obtain the
  shrinkage parameter ?
• Estimate the individual measurement variances by
  the vsn procedure, determine regression weights
  ws
• For each gene, calculate the t-statistics

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Validation by Simulation (1)

• Data generation
• Fix all gene expression values in class 1 to one
  value
• Alter half of the genes in class 2 by some
  constant value
• Add normally distributed noise.

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Validation by Simulation (2)

• Data generation
• Draw from a log-normal distribution
• Draw a fold change vector f by
• Calculate as the product of with the
  fold change vetor
• Generate 8 samples with mixture proportions
  evenly distributed in 0,1
• Add normally distributed noise

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Validation by Simulation (2)
ROC curve
Predictive power
t
SAM
(fraction of truly higher expressed genes among
found genes)
t
Sensitivity
Positive predictive value PPV
Type I error
found genes
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Validation by simulated data (3)

• Generate 16 mixture samples as in example (2)
• Calculate the statistics for 4 configurations

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Outlook

• Prove superiority over method proposed by Ghosh
• Accomplish the paper
• Write R software package
• Apply methodology to RNAi data

Thanks to ...

• Tim Beissbarth for data acquisition and
  preprocessing
• Andreas Buness, Markus Ruschhaupt for helpful
  discussions
• Gordon Smyth, Dileepa Diyagama, Andrew Holloway
  from the WEHI (Melbourne) for the data
• Annemarie Poustka, Holger Sültmann