Download Poisson Distribution - GST RNASeq Workshop

Statistics Behind Differential Gene Expression
Arkadipta Bakshi
University of Tennessee-Knoxville
RNA Sequencing Workshop
26th May, 2016
Overview of the Talk
Differential Expression
Fold Change
Distributions
DEseq2
Applications of High-Throughput Sequencing
 Reading Applications: The sequence itself:
 Re-sequencing
 Target-enriched sequencing
 De-novo assembly
 Counting Applications: The ability to count the amounts of reads and
compare these counts:
 ChIP Sequencing
 RNA Sequencing
Question: What do we mean by differential expression of a gene?
Basis of Differential Expression
 Differential expression is the assessment of differences in read counts of
genes between two or more experimental conditions. Genes are
differentially expressed if this difference is statistically significant.
Example:
 There are two samples from the same patient. One sample is from a kidney tumor biopsy. The
other sample is a biopsy from the patient's other kidney, which seems to be perfectly healthy
tissue.
 Theoretically, we would expect that the two samples will have different amounts of certain
messenger RNA transcripts.
 It would be interesting to see which transcripts from the tumor are being synthesized at a
significantly higher or lower number in the tumor tissue compared to that in the healthy
tissue.
Why is Differential Expression in RNAseq different from
Microarray and other High Throughput Data?
 Differences in gene expression in microarray data are based on
numerical intensity values.
 Quantitative Metabolome analysis is based on area of the peak
generated by each metabolite in the sample.
 RNAseq is based on sequence read count distributions.
 RNAseq provides richer information i.e. increased specificity and
sensitivity for enhanced detection of differential gene expression.
Overview of the Talk
Differential Expression
Fold Change
Distributions
DEseq2
Why don’t we just calculate fold change directly?
 Calculating fold change directly can be misleading.
 Low counts can appear to have high fold changes while large
counts are less sensitive.
Question: What are the different methods that have been used to assess
differential expression in RNAseq data?
Overview of the Talk
Differential Expression
Fold Change
Distributions
 Modeling read counts with Poisson distribution
 Overdispersion and the negative binomial distribution
DEseq2
Question: Why it might be appropriate to model read counts as a Poisson
based-process?
Normalization
• Comparing genes to each other brings in many more biases
– If we are comparing the same gene across two data sets (not two genes to each other), we
can make the assumption that length and other biases largely cancel out
– Thus we can ignore these issues:
• Issue 1: Gene length
– At similar expression levels, a longer gene will collect more reads than a shorter gene.
• Issue 2: Uniqueness of mapped reads
– If one gene has a region that is not unique, many reads are lost.
– When compared to another gene of the same length that is entirely unique, no reads
are lost.
• Issue 3: GC content
– If one gene has a much higher GC content or a region of particularly high GC content,
the sequencer will produce fewer or no reads from that region.
– When compared to another gene of normal GC content, where no reads are lost.
Normalization – More units of expression
• Raw counts are sometimes altered in other ways to
reveal the proportion of transcripts in the original pool
of RNA
• FPKM = Fragments Per Kilobase of exon per Million
Mapped reads (paired end reads)
• RPKM = Reads Per Kilobase of exon per million Mapped
reads
• Used by cufflinks (single end reads)
count * 109
transcript length * total reads sequenced
Lior Patcher, Models for transcript quantification from RNA-Seq, ArXiv http://arxiv.org/abs/1104.3889
Justification of Poisson Distribution for RNAseq
http://www.mi.fu-berlin.de/wiki/pub/ABI/GenomicsLecture13Materials/rnaseq2.pdf
 Why do we use the poisson distribution vs. the binomial distribution?
 The binomial distribution is valid when there is a fixed number of events "n"
each with a constant probability of success “p".
Justification of Poisson Distribution (PD) for RNAseq
 PD expresses the probability of a given number of events occurring
in a fixed interval of time and/or space, if these events occur with a
known average rate and independently of the time since the last
event .
 However for poisson distribution,
 we don’t know the number of "n" trials that will happen.
 we don’t know how many times success did not happen.
 we only know the average number of successes per interval.
Poisson Distribution (mean = variance)
 P (x ; µ) =
(e
− µ )(µx )
X!
where x is the number of success and µ is a given region
Poisson Distribution
• The Poisson model assumes that the mean equals the variance.
• Initially confirmed by an RNASeq study with the same initial source of RNA
split into multiple lanes of an Illumina GA sequencer (Marioni et al. 2008).
– Technical replicates only!
• Genuine biological replicates will exhibit higher levels of variation.
• Analyzing biologically replicated data with the Poisson model will likely be
prone to high false-positive rates due to the underestimation of the true
variability (Anders and Huber 2010; Langmead et al. 2010; Robinson and
Smyth 2008).
Technical Variation = Fits Poisson
Mean–variance plot for Marioni et al.
dataset (Marioni et al. 2008). The variability
in technically replicated RNA-seq data can
be adequately captured using a Poisson
model. The grey points in this plot shows
the mean and pooled variance for each
gene, scaled to account for differences in
library size between samples. The black line
displays the theoretical variance under the
Poisson model where the variance is equal
to the mean. The red crosses show binned
variance, where genes are grouped by
mean level.
Biological Variation ≠ Fit Poisson
Mean–variance plot for the Parikh
et al. Dictyostelium dataset (Parikh
et al. 2010). The variability in this
biologically replicated RNAseq
dataset exhibits prominent extraPoisson variability.
Restrictions with Poisson Distribution
 Overdispersion
 Many studies have shown that the
variance grows faster than the mean in
RNAseq data.
Mean count vs variance of RNA seq data
Orange: the fitted observed curve.
Purple: the variance implied by the
Poisson distribution.
Question: How can we address the overdispersion
problem during handling of RNAseq data?
Negative Binomial Distribution
 Can be used as a better substitute for an overdispersed poisson.
 if we define a "1" as failure, all non-"1"s as successes, and we throw a dice
repeatedly until the third time “1” appears (r = three failures), then the probability
distribution of the number of non-“1”s that had appeared will be a negative
binomial.
 Allows mean and variance to be different.
 Requires:
 p – probability of a single success
 r – the total number of successes
Question: What are the different packages that can be used to analyze RNA sequencing
data?
What to do?
• Most people now use the Negative Binomial distribution
– Cuffdiff2 <- The only one that deals with DE isoforms
– Limma
– DESeq2
– EdgeR
– SAMSeq
Why do we need a distribution?
• Why not just use a non-parametric method?
– More difficult to show significance with a non-parametric method
with few replicates
• Rank order statistics will begin working well with ~ 10
biological replicates
– SamSeq (http://www.insider.org/packages/cran/samr/docs/SAMseq)
Overview of the Talk
Differential Expression
Fold Change
Distributions
DEseq2
Modeling dispersion
• Now we have a distribution that allows the dispersion to be
different from the mean.
• But we often still have very low sample numbers (n = 2, 3, 4), which
is not good for modeling variance.
• A variety of ways to handle this – usually share information across
genes to measure variance.
• Both DESeq2 and EdgeR assume that genes of similar average
expression strength have similar dispersion.
– Use this information in slightly different ways to predict reasonable
dispersions
DEseq2
Accepts raw counts of sequencing reads
Requires an associated design formulae
Null hypothesis: the expression change in a gene is 0
Calculates differential expression using negative binomial
distribution
Steps Performed by DEseq Function
 Estimation of size factors
 Estimation of dispersion
 Negative Binomial Generalized Linear Model fitting for βi
and Wald
statistics.
• Generalized linear model is fit for each gene
– Flexible - allows for complex designs
 Plot and fit a curve, adjusting the dispersion parameter toward the curve
(shrinking).
Likelihood Ratio Test vs. Wald Test
 Likelihood Ratio Test
 Compares the likelihood of the data assuming no differential expression
(null model) against the likelihood of the data assuming differential
expression (alternative model).
 Estimates two models and compares the fit of one model to the fit of the
other.
 Wald Test: Default Test
 Uses likelihood ratio but it only estimates one model.
 Tests the null hypothesis that a set of parameters is equal to some value.
Cook’s Distance – Method to Determine the Influential
Points
 Used to remove outliers
 Measures how much a single sample is influencing the fitted coefficients
for a gene.
 p-values and adjusted p-values for genes deemed as outliers are set to NA
in DEseq2 if there are 3 or more replicates.
 POINTS TO BE NOTED:
 Points for which Cook's distance is higher than 1 are to be
considered as influential.
 A threshold of 4/N or 4/(N−k−1), where N is the number of
observations and k is the number of explanatory variables is
also considered.
 The observations 7 and 16 could be considered as influential.
The observation 29 is not substantially different from a couple
of other observations.
Why do we need to adjust p values?
Correct for type 1 error (i.e. false discovery rate) in p values.
Adjusted p values
 Benjamini-Hochberg
 Needs to adjust for multiple testing (of many genes)
 Controls false discovery rate
 Ranks p-values from smallest to largest
 Assumptions and potential problems:
 Individual tests are independent of each other
 May report false negatives
What else can DESeq2 do?
• Vignette and manual available from Bioconductor site
• http://bioconductor.org/packages/release/bioc/html/DESeq2.html
• Other Applications:
– MA plot: Mean (normalized) expression vs log fold change
– Count data transformations: Operates on raw read counts
– Heatmap Analysis
– Sample clustering: Good for Quality Control (QC).
– Principal Components Plot: Use only for QC.
Summary
 RNAseq provides with a richer information, whereas microarray provides only probe
specific information.
 Never calculate fold change directly for RNAseq Data.
 DEseq2 uses the negative binomial distribution but other distributions can be used.
 CuffDiff2 seems to work worse than others. Possibly because it has extra statistics
to deal with isoforms. (If you would like to deal with DE between isoforms this is
probably your best bet).
 Limma, DESeq(2) and EdgeR are pretty similar (even though limma doesn’t use
negative binomial).
 Biological replicates are better than more depth.
 When dealing with large datasets it is very important to adjust the p-values to avoid
type 1 errors.