Contingency Table Analysis

Contingency Table Analysis Mary Whiteside, Ph.D.

Overview • Hypotheses of equal proportions • Hypotheses of independence • Exact distributions and Fisher’s test • The Chi squared approximation • Median test • Measures of dependence • The Chi squared goodness-of-fit test • Cochran’s test

Contingency Table Examples • Countries - religion by government • States – dominant political party by geographic region • Mutual funds - style by family • Companies - industry by location of headquarters

More examples - • Countries - government by GDP categories • States - divorce laws by divorce rate categories • Mutual funds - family by Morning Star rankings • Companies - industry by price earnings ratio category

Statistical Inference hypothesis of equal proportions H0: all probabilities (estimated by proportions, relative frequencies) in the same column are equal, H1:at least two of the probabilities in the same column are not equal Here, for an r x c contingency table, r populations are sampled with fixed row totals, n1, n2, … nr.

Hypothesis of independence H0: no association i.e. row and column variable are independent, H1: an association, i.e. row and column variable are not independent Here, one populations is sampled with sample size N. Row totals are random variables.

Exact distribution for 2 x 2 tables: hypothesis of equal proportions; n1 = n2 = 2

Fisher’s Exact Test • For 2 x 2 tables assuming fixed row and column totals r, N-r, c, N-c: • Test statistic = x, the frequency of cell11 • Probability = hyper-geometric probability of x successes in a sample of size r from a population of size N with c successes

Large sample approximation for either test • Chi squared = S [Observed - Expected]2 /Expected • Observed frequency for cell ij comes from cross-tabulation of data • Expected frequency for cell ij =Probability Cell ij * N • Degrees of freedom (r-1)*(c-1)

Computing Cell Probabilities Assumes independence or equal probabilities (the null hypothesis) • Probability Cell ij = Probability Row i * Probability Column j = (R i/N) * (C j/N) • Expected frequency ij = (R/N)*(C/N)*N = R*C/N.

Distribution of the Sum • Chi Square with (r-1)*(c-1) degrees of freedom • Assumes [Observed - Expected]2 /Expected is standard normal squared

Implies [Observed - Expected] /Square root[Expected] is standard normal • Implies • = s2 and Observed is a Poisson RV • Poisson is approximately normal if m> 5, traditional guideline • Conover’s relaxed guideline page 201

Measures of Strength: Categorical Variables • Phi 2x2 • Cramer's V for rxc • Pearson's Contingency Coefficient • Tschuprow's T

Measures of Strength: Ordinal Variables • Lambda A .. Rows dependent • Lambda B .. Columns dependent • Symmetric Lambda • Kendall's tau-B • Kendall's tau-C • Gamma

Steps of Statistical Analysis Significance - Strength 1- Test for significance of the observed association 2 -If significant, measure the strength of the association

Consider the correlation coefficient a measure of association (linear relationship between two quantitative variables) • significant but not strong • significant and strong • not significant but “strong” • not significant and not strong

r and Prob (p-value) • r = .20 p-value < .05 • r = .90 p-value < .05 • r = .90 p-value > .05 • r = .20 p-value > .05

Concepts • Predictive associations must be both significant and strong • In a particular application, an association may be important even if it is not predictive (I.e. strong)

More concepts • Highly significant , weak associations result from large samples • Insignificant “strong” associations result from small samples - they may prove to be either predictive or weak with larger samples

Examples • Heart attack Outcomes by Anticoagulant Treatment • Admission Decisions by Gender

Summary • Is there an association? • Investigate with Chi square p-value • If so, how strong is it? • Select the appropriate measure of strength of association • Where does it occur? • Examine cell contributions

Contingency Table Analysis