Statistics and Epidemiology Robert F. Waters, Ph.D.

1. Statistics and EpidemiologyRobert F. Waters, Ph.D. • Statistics • “status” (manner of standing) • In medicine • Biostatistics • Biometrics • Epidemiology • Epi (upon) demos (people) • Study of health and illness in human populations • Pattern recognition

2. Reasons to use Biostatistics • Evaluation of medical research • Applying study results to patient care • Understanding epidemiological problems • Interpreting information about drugs • Evaluating study protocols • Participating and directing research projects

3. Elementary Probability Theory • Probability of success (p) • p = Pr{E} = h/n • h = # of ways • n = total number of ways • Probability of failure (q) • q = Pr{not E} = n - h/n = 1 - h/n= 1-p • p + q = 1.00

4. Probability • Example: • 1000 tosses of fair coin get 529 heads • Another 1000 tosses gives 493 heads • Keep repeating tosses should approach p = .5 • Cards • Mutually exclusive events..add Pr • What is probability of drawing an Ace? • 4/52 • What is probability of drawing a king? • 4/52 • How about an ace and king? (With Replacement) • 4/52 + 4/52

5. Probability • How about dice? • Throwing two fair die • Probability of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 • Let’s play “craps”

6. Combinatorial Analysis • What is a factorial? • N! = n(n – 1)(n – 2) ….. 1 • What is 5! ? • Permutation • nPr = n(n – 1)(n – 2) …. (n – r + 1) or: • n!/(n – r)! • Problem: How many permutations of the letters a b c be taken 2 at a time? • 3!/(3 – 2)! = 3 x 2 = 6 • List the permutations.

7. Combinatorial Analysis • Combinations (Order does matter) • nCr = n!/r!(n – r)! Or nPr/r! • Number of combinations of a b c taken two at a time. • 3C2 = 3!/2!(3 – 2)! = ? • List the combinations.

8. Data • Continuous variable • Temperature • Discrete variable • Number of children in a family • Can’t have 2.3 children • Nominal data • Pretty, tall, etc. • Ordinal data • 0, 3, 5 (order from worst to best)

9. What is a Population? • Infinite? • Finite • We have a sample! • Sometimes we need to sample from a large population. • Therefore statistics is called.. • Statistical Inference • Inductive Statistics • Trying to characterize infinite population

10. Measures of Central Tendency • Mean • Arithmetic mean • Harmonic mean (RMS) • Geometric mean • Median • Mode • When would the mean median and mode be the same? • What is a variate?

11. Measures of Dispersion • Old story of two surgical students! • Variance • Standard Deviation

12. Properties of Standard Deviation • + & - 1s from mean • 68.27% • + & -2s from mean • 95.45% • + & -3s from mean • 99.73% • Problem: Heights in Class

13. Moments, Skewness, Kurtosis • Major problem in biometrics • Moments about the mean • First moment • Arithmetic mean • Second moment • Variance • Skewness • Degree of asymmetry • Kurtosis • Leptokurtic (narrow) • Platykurtic (flattened) • Mesokurtic (normal)

14. Elementary Sampling Theory • Many problems in biometrics • Random samples • Without bias • Error evenly distributed • Level of significance (usually 0.05 in science) • Degrees of freedom • Orthogonality (Comparisons) • Example: • Ways to account for sources of variation • Patients with different doctors in different clinics • Patients with same doctors different clinics • Patients with same doctors same clinic

15. Application to Epidemiology • Binomial Distribution • p(X) = nCxpxqn-x = [N!/X!(N-X)!] pxqn-x • Problem: • What is the probability in a family of four children there will be at least 1 boy? • 1 boy 4C1 (1/2)1 * (1/2)3 • = 4!/1!(4 – 1)! * ½ * 1/8 = ¼ • 2 boys = 3/8 • 3 boys = ¼ • 4 boys = 1/16 • 4C1(1/2)4 * (1/2)0 • What is probability of at least one boy? • Pr(1boy) + Pr(2boys) + Pr(3boys) + Pr(4boys) • ¼ + 3/8 + ¼ + 1/16 = 15/16

16. Application of Binomial • Out of 2000 families with four children, how many have at least one boy? • 1875 • Out of 2000 families with four children, how many are expected to have two boys? • 750

17. How can we tell if something is wrong? • Chi-square • Compares observed with expected

18. Statistical Decision Theory • P value • Statistical significance • One-tailed vs. Two-tailed test • Confidence intervals • Standard Error • Standard deviation

19. The Correlation • Two independent variables • Ice cream in Georgia story

20. The Regression • Dependent with Independent Variable • Least Squares analysis

21. Multiple Linear Regression Analysis (MLRA) • One dependent and multiple independent variables • Predictive? • Problems • Variables normally distributed • Equal variances • True independence between independent variables

22. Hardy Weinberg Equilibrium • Alleles in populations tend towards H-W equilibrium • Answers the questions: • How can O be the most common of the blood types if it is a recessive trait? • If Huntington's disease is a dominant trait, shouldn't three-fourths of the population have Huntington's while one-fourth have the normal phenotype? • Shouldn't recessive traits be gradually be swamped out so they disappear from the population?

23. Hardy Weinberg Cont: • Hardy Weinberg equilibrium is achieved if: • There is a large population • There is random mating • No selection for a particular allele • No mutations • No migration or isolation

24. Trend Analysis • Autocorrelation • Predictive with assumptions

25. Discussion Questions • What should you expect in a paper (epidemiology) that uses statistics? • Why not just compare means of samples? • Can we always assume statistical assumptions to be correct? • When should a correlation be used? • How about a linear regression? • MLRA? • Binomial? • Hardy Weinberg? • Chi-Square?