Analysis of Categorical Data and Contingency Tables

Vocabulary flashcards covering key terms from the lecture notes on categorical data analysis, contingency tables, chi-square tests, Fisher’s exact test, and odds ratios.

Categorical Variable

A variable that takes on categories (e.g., eye color).

Numerical Variable

A variable that takes numeric values (e.g., height, age).

Ordinal

Categorical with a meaningful order (e.g., first, second, third).

Nominal

Categorical without an inherent order.

Contingency Table

A table showing frequencies for two categorical variables to assess association.

Observed Frequencies

Frequencies that are actually observed in the data for each cell.

Expected Frequencies

Frequencies expected under the null hypothesis, calculated from margins.

Null Hypothesis (H0) in Contingency Table

There is no association between the variables; they are independent.

Alternative Hypothesis (Ha)

There is an association between the variables.

Pearson Chi-Square Test

Tests for an association between two categorical variables by comparing observed and expected frequencies.

Chi-Square Statistic

Sum over all cells of (O − E)² / E, where O is observed and E is expected.

Degrees of Freedom (df)

For an r × c table, df = (r − 1) × (c − 1).

Yates' Continuity Correction

Adjustment to the chi-square for 2×2 tables in small samples, subtracting 0.5 from |O − E|.

Fisher’s Exact Test

An exact test for count data; preferred when expected counts are < 5; more conservative than chi-square.

Odds Ratio (OR)

A measure of effect size for binary variables; OR = (ad)/(bc) for a 2×2 table.

Interpreting OR > 1 or OR < 1

OR > 1 indicates higher odds of the outcome with the first condition; OR < 1 indicates lower odds.

Independence in Contingency Table

Null hypothesis that row and column variables are independent (no association).

Standardized Residuals

Residuals divided by their standard deviation; help identify cells contributing to the chi-square.

Observed vs Expected in R Output

O = observed frequencies; E = expected frequencies; used to compute chi-square and residuals.

CrossTable (gmodels)

R function to display a contingency table with options like fisher, chisq, expected, and standardized residuals.

Chi-Square Test in R (chisq.test)

R function to perform Pearson's chi-square test; can disable Yates' correction with correct=FALSE.

Fisher's Exact Test in R (fisher.test)

R function to perform Fisher's Exact Test for count data.

Expected Counts in a 2×2 Table

Calculated as (row total × column total) / grand total.

Contingency Test Summary

Use Chi-square when expected counts are sufficient; use Fisher's exact when not; if they disagree, prefer Fisher.

Example Contingency Table (Training vs Dancing in Cats)

A case study with variables like Reward type and whether cats danced, used to illustrate testing and odds ratios.

Sample Size Assumption Met

All expected cell counts ≥ 5 allows use of Pearson's chi-square; otherwise use Fisher's exact.