Analyzing Expression Data: Clustering and Stats

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Analyzing Expression DataClustering and Stats

• Chapter 16

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Goals

• Weve measured the expression of genes or
  proteins using the technologies discussed
  previously.
• What can we do with that information?
• Identify significant differences in expression
• Identify similar patterns of expression
  (clustering)

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Analysis steps

• Data normalization
• Statistical Analysis
• Cluster Analysis

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I. Data Normalization

• Why normalize?
• Removes systematic errors
• Makes the data easier to analyze statistically

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Sources of Error

• Measurements always contain errors.
• Systematic (oops)
• Random (noise!)
• Subtracting the background level can remove some
  systematic error
• Using the ratio in two-channel experiments does
  this
• Subtracting the overall average intensity can be
  used with one-channel data.
• Taking averages over replicates of the experiment
  reduces the random error.
• Advanced error models are mentioned on p. 628 and
  covered in Further Reading.

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Expression data usually not Gaussian (normal)

• Many statistical tests assume that the data is
  normally distributed.
• Expression microarray spot intensity data (for
  example) is not.
• Intensity ratio data (two-channel) is not normal
  either.
• Both go from 0 to infinity whereas normal data is
  symmetrical.

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Taking the logarithm helps normalize expression
ratio data

• The expression ratio plotted versus the
  expression level (geometric mean) in both
  channels.
• Plotting the log ratio vs. the log expression
  level gives data that is centered around y0 and
  fairly normal looking.

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Taking the log of the expression ratio fixes
the left tail
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LOWESS Normalization

• Sometimes there is still a bias that depends on
  the expression level.
• This can be removed by a type of regression
  called Locally Weighted Scatterplot Smoothing.
• This computes and subtracts the mean locally for
  various values of expression level (RG).

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II. Statistical Analysis

• Determining what differences in expression are
  statistically significant
• Controlling false positives

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When are two measurements significantly different?

• We want to say that an expression ratio is
  significant if it is big enough (gt1) or small
  enough (lt1).
• A two-fold ratio (for example) is only
  significant if the variances of the underlying
  measurements are sufficiently small.
• The significance is related to the area of the
  overlap of the underlying distributions.

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The Z-test

• If the data is approximately normal, convert it
  to a Z-score.
• X can be the log expression ratio ? is then 0
• ? is the sample standard deviation n is the
  number of repeats
• The Z-score is distributed N(0,1) (standard
  normal).
• The significance level is the area in the tail(s)
  of the standard normal distribution.

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The t-test

• The t-test makes fewer assumptions about the data
  than the Z-test
• It can be applied to compare two average
  measurements which can have
• Different variances
• Different numbers of observations
• You compute the t-statistic (see pages 654-655)
  and then look up the significance level of the
  Students T distribution in a table.

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III. Cluster Analysis

• Similar expression patterns
• Groups of genes/proteins with similar expression
  profiles
• Similar expression sub-patterns
• Groups of genes/proteins with similar expression
  profiles in a subset of conditions
• Different clustering methods
• Assessing the value of clusters

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Example Gene Expression Profiles

• Expression level of a gene is measured at
  different time points after treating cells.
• Many different expression profiles are possible.
• No effect
• Immediate increase or decrease
• Delayed increase or decrease
• Transient increase or decrease

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Clustering by Eye

• n genes or proteins
• m different samples (or conditions)
• Represent a gene as a point
• X ltx1, x2, , xmgt
• If m is 1 or 2 (or even 3) you can plot the
  points and look for clusters of genes with
  similar expression.
• But what if m is bigger than 3?
• Need to reduce the dimensionality PCA

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Reducing the Dimensionality of Data Principal
Components Analysis

• PCA linearly map each point to a small set of
  dimensions (components).
• The principal components are dimensions that
  capture the maximum variation in the data.
• The principal components capture most of the
  important information in the data (usually).
• Plotting each points values in two of the
  principal component dimensions allows us to see
  clusters.

2-D Gel Data
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PCA An IllustrationYeast Cell Cycle Gene
Expression

• Singular value decomposition of a matrix X (SVD)
  is
• X U ? VT
• The mapped value of X is
• Y X VT
• The rows of Y give the mapping of each gene.
• Mapped gene i Yi lty1, y2, ., ymgt

_at_PNAS (2000)
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Clustering Using Statistics

• Algorithm identifies groups.
• Example similar expression profiles
• Distance measure between pairs of points is
  needed.

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Distance Measures Between Pairs of Points

• In order to cluster the points (genes or
  conditions), we need some concept of which points
  are close to each other.
• So we need a measure of distance (or,
  conversely,) similarity between two rows (or
  columns) in our n by m matrix.
• We can then compute all the pair-wise distances
  between rows (or columns).

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Standard Distance Measures

• Euclidean Distance
• Pearson Correlation Coefficient
• Mahalanobis Distance

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Euclidean Distance

• Standard, everyday distance
• Treats all dimensions equally
• If some genes vary more than others (have higher
  variance), they influence the distance more.

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Mahalanobis Distance

• The normalized Euclidean distance
• Scales each dimension by the variance in that
  dimension.
• This is useful if the genes tend to vary much
  more in one sample than in others since it
  reduces the affect of that sample on the
  distances.

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Pearson Correlation Coefficient

• Distances are small when two genes have similar
  patterns of change even if the size of the
  changes are different.
• This is accomplished by scaling by the sample
  variance of the genes expression levels under
  different conditions.

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Choice of Distance Matters

• Heirarchical clustering (dentrogram) of tissues.
• Corresponds to clustering the columns of the
  matrix.
• Branches are different (cancer B/C vs A/B).

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Clustering Algorithms

• Hierarchical Clustering
• K-means clustering
• Self-organizing maps and trees

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Hierachical Clustering

• Algorithms progressively merge clusters or split
  clusters.
• Merging criterion can by single-linkage or
  complete-linkage.
• Produce dendrograms
• Can be interpreted at different thresholds.

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Types of Linkage

• A. Single Linkage
• B. Complete Linkage
• C. Centroid Method

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K-means Clutering

• Related to Expectation Maximization
• You specify the number of clusters
• Iteratively moves the means of the clusters to
  maximize the likelihood (minimize total error).