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?