24.4 Inference for Comparisons, Chi-Square & Regression

Comparing Two Groups

Many PA-CAT Statistics items compare two populations. The comparison tool depends on the data type:

• Two means, independent groups: two-sample t-test (Welch preferred).
• Two means, paired data: paired t-test on differences.
• Two proportions: two-proportion z-test; CI for (p̂₁ − p̂₂).
• Two variances: F-test (rare on PA-CAT).

The two-proportion z statistic is z = (p̂₁ − p̂₂)/√(p̂_pool(1 − p̂_pool)(1/n₁ + 1/n₂)), where p̂_pool = (x₁ + x₂)/(n₁ + n₂). Example: comparing surgical site infection rates between laparoscopic (12/200) and open (28/200) appendectomy gives p̂₁ = 0.06, p̂₂ = 0.14, p̂_pool = 0.10; z ≈ −2.67, p ≈ 0.0076. Reject H₀ at α = 0.05: the open approach has a significantly higher infection rate.

Analysis of Variance (ANOVA)

ANOVA compares the means of three or more groups simultaneously. Doing repeated two-sample t-tests would inflate the family-wise Type I error; ANOVA controls it in one test. The null is H₀: μ₁ = μ₂ = … = μₖ; the alternative is that at least one mean differs.

The test partitions total variability into between-group (treatment) and within-group (error) components:

• F = MS_between / MS_within
• MS_between = SSB/(k − 1), MS_within = SSW/(n − k)

Large F values indicate the group means differ more than expected by chance. The F-distribution has (k − 1, n − k) degrees of freedom. A significant ANOVA tells us that at least one mean differs, but not which; post-hoc procedures (Tukey HSD, Bonferroni, Scheffé) perform pairwise comparisons while controlling the family-wise error rate.

Example: comparing mean pain scores across three analgesics (acetaminophen, ibuprofen, ketorolac) in 90 postsurgical patients uses a one-way ANOVA with (2, 87) df. A significant F prompts pairwise Tukey tests to identify which analgesics differ.

Assumptions: independent SRS from each group, approximately normal populations (or large n), equal variances (homoscedasticity). Violations of equal variance are addressed by Welch's ANOVA; severe non-normality by the Kruskal-Wallis nonparametric alternative.

Chi-Square Tests

The chi-square (χ²) test handles categorical data. Two forms appear on the PA-CAT:

Goodness-of-fit tests whether observed counts match a hypothesized distribution for one categorical variable. Statistic: χ² = Σ (O − E)²/E, df = k − 1. Example: testing whether blood type frequencies in a clinic match the expected US distribution (O: 42%, A: 31%, B: 15%, AB: 12%).

Test of independence examines whether two categorical variables are associated, using a two-way contingency table. df = (r − 1)(c − 1). Example: testing whether smoking status (never/former/current) is associated with COPD stage (0/1/2/3/4) in 500 patients. A large χ² rejects independence; variables are associated. Expected cell counts should be ≥5 for the approximation to be valid; otherwise use Fisher's exact test.

Chi-square measures association, not strength. For strength, use Cramér's V or the phi coefficient for 2×2 tables.

Correlation

The Pearson correlation coefficient r measures the linear association between two quantitative variables, ranging from −1 to +1. Values near ±1 indicate strong linear relationships; near 0, weak or nonlinear. Correlation does not imply causation: a strong r may reflect a confounder (e.g., ice cream sales and drowning incidents are both driven by temperature).

Assumptions: linear relationship, quantitative variables, no extreme outliers, approximate bivariate normality. For ordinal data or outliers, use Spearman's rank correlation.

Linear Regression

Simple linear regression fits a line y = b₀ + b₁x that minimizes the sum of squared residuals (ordinary least squares). b₁ estimates how much y changes per unit increase in x. The coefficient of determination r² is the proportion of variation in y explained by the model; r² = 1 − SS_residual/SS_total.

Example: regressing systolic blood pressure (mm Hg) on BMI (kg/m²) in 120 adults yields b₁ = 1.8, r² = 0.42. Interpretation: each 1-unit increase in BMI is associated with a 1.8 mm Hg rise in systolic BP; BMI explains 42% of the variability in systolic BP.

Inference for the slope: test H₀: β₁ = 0 using t = b₁/SE(b₁), df = n − 2. A significant slope indicates a linear relationship. The CI for β₁ quantifies the plausible range of the effect. Extrapolation beyond the observed range of x is hazardous—the linear model may not hold outside the data.

Nonparametric Overview

When assumptions of normality or equal variance are untenable, nonparametric tests rank the data instead of using raw values:

Nonparametric tests are less powerful when parametric assumptions hold but are more robust when they do not. The PA-CAT may ask you to select the appropriate nonparametric analog for a given scenario.