Introduction to Statistical Inference

1. Introduction to Statistical Inference J. Verducci MBI Summer Workshop August, 2005

2. 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. 

3. 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

4. Affymetrics Mas 5.0 Expression Set

5. Expression Data Matrix30,000 genes x 30 patients

6. 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.

7. 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 Shaq�s yearly free-throw percentages up to June, 2005 Sample averages from patients in Stages 1-4; patients with high and low HER2 expression

8. 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.

9. 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}

10. Random Variables A random variable X is a function from the sample space W into the real numbers R X:W ? 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}.

11. 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

12. Probability Function for Sum of Two Dice

13. Independence Two random variables Y and Z are independent if, for all possible y,z P(Y=y and Z=z) = P(X=x) * P(Y=y) Dice Example: Let Y( (i,j) ) = i Z( (i,j) ) = j Then, for y,z in {1,2,3,4,5,6}, P(Y=y and Z=z) = 1/36 = 1/6 * 1/6 = P(X=x) * P(Y=y)

14. 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

15. 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.

16. Maximum Likelihood Estimate Y = S Xi has the Binomial(n, q) distribution: