Categorical Data Analysis: Contingency Tables and Log-Linear Models
Analyze categorical data using contingency tables and log-linear models for Exam MAS-I.
Contingency Tables
Contingency tables cross-classify observations by two or more categorical variables. The chi-squared test of independence compares observed cell counts O_{ij} to expected counts E_{ij} = (row total * column total) / grand total. The test statistic sum of (O-E)^2/E follows a chi-squared distribution with (r-1)(c-1) degrees of freedom under independence. Fisher's exact test is used for small samples. Measures of association include the odds ratio (for 2x2 tables), Cramer's V, and the contingency coefficient. The odds ratio theta = (n_{11}*n_{22})/(n_{12}*n_{21}) measures the strength of association.
Log-Linear Models
Log-linear models describe the cell counts in multi-way contingency tables. For a two-way table, the saturated model is ln(m_{ij}) = mu + lambda_i^A + lambda_j^B + lambda_{ij}^{AB}. The independence model omits the interaction term. For three-way tables, various models represent conditional independence, joint independence, and homogeneous association. Model selection uses likelihood ratio statistics (G^2) and AIC/BIC. Exam MAS-I tests contingency table analysis, odds ratio calculation and interpretation, and log-linear model fitting for multi-way tables.