Analysis of Differential Expression

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Analysis of Differential Expression

• T-test
• ANOVA
• Non-parametric methods
• Correlation
• Regression

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Research Question

• Do nicotine-exposed rats have different X gene
  expression than control rats in ventral tegmental
  area?
• Design an experiment in which treatment rats
  (Ngt2) are exposed to nicotine and control rats
  (Ngt2) are exposed to saline.
• Collect RNA from VTA, convert to cDNA
• Determine the amount of X transcript in each
  individual.
• Perform a test of means considering the
  variability within each group.

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Observed difference between groups

• May be due to
• Treatment
• Chance

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Hypothesis Testing

• Null hypothesis There is no difference between
  the means of the groups.
• Alternative hypothesis Means of the groups are
  different.

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Hypothesis testing

• You can not accept null hypothesis
• You can reject it
• You can support it

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P-value

• The P stands for probability, and measures how
  likely it is that any observed difference between
  groups is due to chance, alone.

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P-value

• there is a significant difference between groups
  if the P value is small enough (e.g., lt0.05).
• P value equals to the probability of type I
  error.
• Type I error wrongly concluding that there is a
  difference between groups (false positive).
• Type II error wrongly concluding that there is
  no difference between groups (false negative).

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Multiple tests on the same data

• Expression data on multiple genes from the same
  individuals
• Subsets of genes are coregulated thus they are
  not independent.
• Such data requires multiple tests.

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Why not do multiple t-tests? Or if you do, adjust
the p-values

• Because it increases type I error
• a study involving four treatments, there are six
  possible pairwise comparisons.
• If the chance of a type I error in one such
  comparison is 0.05, then the chance of not
  committing a type I error is 1 0.05 0.95.
• then the chance of not committing a type I error
  in any one of them is 0.956 0.74.
• Cumulative type I error 1-0.740.26

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Normal Distribution

• it is entirely defined by two quantities its
  mean and its standard deviation (SD).
• The mean determines where the peak occurs and
• the SD determines the shape of the curve.

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Curves same mean, different stds
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Rules of normal distribution

• 68.3 of the distribution falls within 1 SD of
  the mean (i.e. between mean SD and mean SD)
• 95.4 of the distribution falls between mean 2
  SD and mean 2 SD
• 99.7 of the distribution falls between mean 3
  SD and mean 3 SD.

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Most commonly used rule

• 95 of the distribution falls between mean 1.96
  SD and mean 1.96 SD
• If the data are normally distributed, one can use
  a range (confidence interval) within which 95 of
  the data falls into.

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A sample

• Samples vary
• Samples are collected in limited numbers
• They are representatives of a population.
• A sample
• E.g., nicotine treated rat RNA

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Sample means

• Consider all possible samples of fixed size (n)
  drawn from a population.
• Each of these samples has its own mean and these
  means will vary between samples.
• Each sample will have their own distribution,
  thus their own std.

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Population mean

• The mean of all the sample means is equal to the
  population mean (?).
• SD of the sample means measures the deviation of
  individual sample means from the population mean
  (?)

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Standard error

• It reflects the effect of sample size, larger the
  SE, either the variation is high or sample size
  is small.

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Confidence Intervals

• a confidence interval gives a range of values
  within which it is likely that the true
  population value lies.
• It is defined as follows
• 95 confidence interval (sample mean 1.96 SE)
  to (sample mean 1.96 SE).
• a 99 confidence interval (calculated as mean
  2.56 SE)

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T-distribution

• The t-distribution is similar in shape to the
  Normal distribution, being symmetrical and
  unimodal, but is generally more spread out with
  longer tails.
• The exact shape depends on a quantity known as
  the degrees of freedom, which in this context
  is equal to
• the sample size minus 1.

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T-distribution
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One-sample t-test

• Null hypothesis Sample mean does not differ from
  hypothesized mean, e.g., 0 (Ho ?0)
• A t-statistics (t) is calculated.
• t is the number of SEs that separate the sample
  mean from the hypothesized value.
• The associated P value is obtained by comparison
  with the t distribution.
• Larger the t-statistics, lower the probability of
  obtaining such a large value, thus p is smaller
  and more significant.

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Paired t-test

• Used with paired data.
• Paired data arise in a number of different
  situations,
• a matched casecontrol study in which individual
  cases and controls are matched to each other, or
• A repeat measures study in which some measurement
  is made on the same set of individuals on more
  than one occasion

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Paired t-test
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Two-sample t-test

• Comparison of two groups with unpaired data.
• E.g., comparison of individuals of treatment and
  those of control for a particular variable.
• Now there are two independent populations thus
  two STDs

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Calculation of pooled STD

• The pooled SD for the difference in means is
  calculated as follows

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Calculation of pooled SE

• the combined SE gives more weight to the larger
  sample size (if sample sizes are unequal) because
  this is likely to be more reliable. The pooled SD
  for the difference in means is calculated as
  follows

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Two sample T-test

• Comparison of means of two groups based on a
  t-statistics and its students t-distribution.
• dividing the difference between the sample means
  by the standard error of the difference.

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T-statistic

• A P value may be obtained by comparison with the
  t distribution on n1 n2 2 degrees of freedom.
• Again, the larger the t statistic, the smaller
  the P value will be.

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Example
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Calculation of SD
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Calculation of SE
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T-statistic

• t (95-81)/2.41 14/2.41 5.81,
• with a corresponding P value less than 0.0001.
• Reject null hypothesis that states that sample
  means do not differ.

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Analysis of Variance

• ANOVA
• A technique for analyzing the way in which the
  mean of a variable is affected by different types
  and combinations of factors.
• E.g., the effect of three different diets on
  total serum cholesterol

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Sample Experiment
Variance
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Sum of squares calculations
total
within
between
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Degrees of freedom
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Sources of variation
P value of 0.0039 means that at least two of the
treatment groups are different.
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Multiple Tests

• Post hoc comparisons between pairs of treatments.
• Overall type I error rate increases by increasing
  number of pairwise comparisons.
• One has to maintain the 0.05 type I error rate
  after all of the comparisons.

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Bonferroni Adjustment

• 0.05/of tests
• Too conservative

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NonParametric methods

• Many statistical methods require assumptions.
• T-test requires samples are normally distributed.
• They require transformations
• Nonparametric methods require very little or no
  assumptions.

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Wilcoxon signed rank test for paired data
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Wilcoxon signed rank test
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Central venous oxygen saturation on admission and
after 6 h into ICU.

• Take the difference between the paired data
  points.
• Patients have SvO2 values on admission and after
  6 hours.

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Central venous oxygen saturation on admission and
after 6 h into ICU.

• Rank differences regardless of their sign.
• Give a sign to the ranked differences

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Calculate

• Sum of positive ranks (R)
• Sum of negative ranks (R-)

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Sum of positive and negative ranks
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Critical values for WSR test when n 10
5
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Wilcoxon sum or Mann-Whitney test

• Wilcoxon signed rank is good for paired data.
• For unpaired data, wilcoxon sum test is used.

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Steps of Wilcoxon rank-sum test
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Total drug doses in patients with a 3 to 5 day
stay in intensive care unit.

• Rank all observations in the increasing order
  regardless of groupings
• Use average rank if the values tie
• Add up the ranks
• Select the smaller value, calculate a p-value for
  it.

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Critical values
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Correlation and Regression

• Correlation quantifies the strength of the
  relationship between two paired samples.
• Regression expresses the relationship in the form
  of an equation.
• Example whether two genes, X and Y are
  coregulated, or the expression level of gene X
  can be predicted based on the expression level of
  gene Y.

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Product moment correlation
r lies between -1 and 1
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Age and urea for 20 patients in emergency unit
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Scattergram
r 0.62
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Confidence intervals around r
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Confidence of r
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Misuse of correlation

• There may be a third variable both of the
  variables are related to
• It does not imply causation.
• A nonlinear relationship may exist.

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Regression
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Method of least squares

• The regression line is obtained using the method
  of least squares. Any line y a bx that we
  draw through the points gives a predicted or
  fitted value of y for each value of x in the
  dataset.
• For a particular value of x the vertical
  difference between the observed and the fitted
  value of y is known as the deviation or residual.
• The method least squares finds the values a and b
  that minimizes the sum of squares of all
  deviations.

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Age and urea level
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Residuals
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Method of least squares