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Statistics and Epidemiology Robert 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. Reasons to use Biostatistics.
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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
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
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
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
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”
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.
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.
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)
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
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?
Measures of Dispersion • Old story of two surgical students! • Variance • Standard Deviation
Properties of Standard Deviation • + & - 1s from mean • 68.27% • + & -2s from mean • 95.45% • + & -3s from mean • 99.73% • Problem: Heights in Class
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)
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
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
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
How can we tell if something is wrong? • Chi-square • Compares observed with expected
Statistical Decision Theory • P value • Statistical significance • One-tailed vs. Two-tailed test • Confidence intervals • Standard Error • Standard deviation
The Correlation • Two independent variables • Ice cream in Georgia story
The Regression • Dependent with Independent Variable • Least Squares analysis
Multiple Linear Regression Analysis (MLRA) • One dependent and multiple independent variables • Predictive? • Problems • Variables normally distributed • Equal variances • True independence between independent variables
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?
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
Trend Analysis • Autocorrelation • Predictive with assumptions
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?