Introduction to Statistical Inference

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Introduction to Statistical Inference

• J. Verducci
• MBI Summer Workshop
• August, 2005

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Recommended Text

• Fred L. RAMSEY and Daniel W. SCHAFER.
• The Statistical Sleuth
• A Course in Methods of Data Analysis
• Belmont, CA Duxbury, 2002, xxvi 742 pp.,
• 97.95 (H CD),
• ISBN 0-534-38670-9.

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Outline

• Underlying philosophies about data
• Basics of probability and random variables
• Estimating a population proportion
• Inferring a difference between two distributions
• Assumptions about distributional forms
• Normal Theory
• Nonparametrics
• Hypothesis Testing
• T-test
• Mann-Whitney-Wilcoxon test
• Multiple Comparisons
• Bonferroni
• False Discovery Rate

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Affymetrics Mas 5.0 Expression Set
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Expression Data Matrix30,000 genes x 30 patients
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Philosophy

• Frequentist Observed data X is an imperfect
  representation of an underlying idealized fixed
  truth q.
• Law of Large Numbers When experiments are
  repeated faithfully, the average of observations
  comes closer to their idealization.
• Bayesian Observed data X is fixed, and the
  unknown generating parameter q is random
• Certainty about q depends on both empirical
  information X and prior knowledge about q.

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Examples q as a Population Percentage or Average

• Parameter q
• Percent of population with a particular allele
• Percent of free throws made by Shaq over his
  entire career
• Mean expression level of BRCA1 gene in breast
  cancer cells

• Statistic x
• Percent observed in a sample of 100 people
• Set of Shaqs yearly free-throw percentages up to
  June, 2005
• Sample averages from patients in Stages 1-4
  patients with high and low HER2 expression

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Key Terms

• Population (Sample Space W) set of all possible
  outcomes of an experiment
• Sample subset of the population (Event) that
  is observed
• (Generative / Probability) Model description of
  how samples are obtained from the population
• Parameter a feature of the population used to
  describe the model
• Statistic a summary of the sample that conveys
  information about the parameter of interest.

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Axioms of Probability

• Needed to specify sampling and modeling
• Definition A probability measure P is a
  function from the set of all possible events into
  0,1 such that
• P(f) 0
• P(W) 1
• P( U Ai ) S P(Ai) for countable collections of
  disjoint events Ai

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Random Variables

• A random variable X is a function from the sample
  space W into the real numbers R
• XW ? R
• X(w) x
• The value x is called a realization of the random
  variable X. It can also be thought of as a
  statistic, since it is a function/summary of the
  sample w.

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Example

• Experiment
• role two dice (one red,one green)
• W (i,j) i 1,,6 j 1,,6
• Probability Model (based on symmetry)
• P((i,j)) 1/36 for each ordered pair
  (i,j)
• Random variable X((i,j)) i j
• The probability model induces a probability
  function fX on the possible values x of X.
• fX(x) (6 - x-7) / 36 , x 2,,12

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Probability Function for Sum of Two Dice
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Independence

• Two random variables Y and Z are independent if,
  for all possible y,z
• P(Yy and Zz) P(Xx) P(Yy)
• Dice Example
• Let
• Y( (i,j) ) i
• Z( (i,j) ) j
• Then, for y,z in 1,2,3,4,5,6,
• P(Yy and Zz) 1/36 1/6 1/6 P(Xx)
  P(Yy)

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iid Sample

• iid independent, identically distributed
• Model
• X1, X2, , Xn are mutually independent, that is,
• P(X1 x1, , Xn xn) P(X1 x1) P(Xn
  xn)
• X1, X2, , Xn have the same probability function
  f(. q)
• Xi f(x q), i 1,,n

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Estimating Population Proportion q

• Code Xi
• 1 if the ith observation has the characteristic
  of interest
• 0 otherwise i 1,,n.
• Bernoulli Distribution
• fX(x q)
• q for x 1
• (1-q) for x 0
• 0 otherwise
• Xi are iid Bernoulli(q), i 1,,n.

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Maximum Likelihood Estimate

• Y S Xi has the Binomial(n, q) distribution

• The value of q that maximizes this probability
  is called the maximum likelihood estimate it has
  the form

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Generalization to Continuous Means

• Let N be the size of the whole population.
• Estimate the population average

using the sample average
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Central Limit Theorem

• Yi iid Binomial(n,q) , n large, 0 lt q lt 1

Simulating1,000,000 such Zi produces a bell
shaped curve, whose limiting form is called the
Gaussian (also called Normal) Distribution
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Normal Family of Distributions
For any interval (a,b), the area under a density
function between a and b represents
the probability of that a random variable is in
that interval.

• A density function f is a nonnegative function f
  such that
• Standard normal distribution is described by the
  density function
• If Z has a standard normal distribution, then X
  q sZ has a N(q, s2) distribution with mean q
  and standard deviation s.

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Normal Family of Density Functions
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Inferring a Difference in Means Between Two
Normal Populations

• Take independent samples from two populations
• Xi iid N(q1,s2) , i 1,,n1
• Yj iid N(q2,s2) , j 1,,n2
• For simplicity assume that the standard deviation
  (scale parameter) s is the same in both
  populations.

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Example Expression of RNA coding MCL1 protein

• Samples (Golub study)
• Xi sample from n1 27 ALL patients
• Yj sample from n2 11 AML patients
• Is the mean level of MCL1 RNA expression
  different for the two populations of patients?
• Sample means (Max Likelihood Estimates)
• ALL 6.8
• AML 8.1
• Is this enough evidence to conclude that the mean
  MCL1 expression is higher for the AML population
  than for the ALL population?

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Hypothesis Testing

• Null Hypothesis
• Suppose AML mean is less than or equal to ALL
  mean
• Least Favorable Configuration is when both
  means are equal (q1 q2)
• P-value
• Under the least favorable configuration, what
  is the probability that the sample means would be
  so far apart in the direction as what was
  observed (8.1 6.8 1.3)?

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Sampling Distribution

• Under the least favorable configuration, how
  would the difference D between
  the sample means vary over different samples of
  the same sizes?
• D N(0, s2/n1 n2)

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Importance of s

• If s 5,
• sd(D) .81
• P(D gt 1.3)
• .054

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Estimating s2

• In any distribution, s2 is the mean squared
  distance of an observation from its mean.
• Estimate this by the average squared difference
  of a sample observation from the sample mean.
• Adjust to compensate for the fact that
• S (xi c)2 is minimized by c the sample
  average.
• Pooled estimate of common s2 is

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Students t test statistic
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MCL1 Expression in ALL AML

• Sample averages
• ALL 6.8 (n1 27)
• AML 8.1 (n2 11)
• Pooled estimate of s
• Spooled .622
• Test statistic
• Tobserved 5.76
• P-value
• P(T gt 5.76) lt 10-7
• Conclusion
• Mean value of MCL1 is higher in AML population

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Checking Assumptions

• Xi and Yj each have normal distributions
• Check using boxplots. Look for
• Skewness
• Outliers
• If necessary, correct by either
• Transformation
• Use of Non-Parametric Test (Mann-Whitney-Wilcoxon)
  
• Xi and Yj distributions have a common
  standard deviation
• Check using boxplots
• If necessary, correct using Welch modification

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Box Plots of Raw MCL1 Expression
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Box Plots of Log MCL1 Expression
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Mann-Whitney-Wilcoxon Test

• Based on number of pairs (Xi,Yj)
• with Xilt Yj
• Results are invariant to transformations
• Conclusion applies to population medians
• For MCL1 data, P-value is also lt 10-7

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Multiple Comparisons

• Chalk Board Presentation or Discussed Later