Confidence Intervals and Statistical Tests

Building on hypothesis testing fundamentals, this lecture addresses CIs and tests for comparing two binary variables (using RR and OR), the chi-squared test, and critical appraisal concepts including internal and external validity.

For relative risk (RR): the sampling distribution of RR is highly positively skewed (RR cannot be negative). To make it approximately normal, the natural log transformation is applied: ln(RR) is approximately normally distributed. CIs are constructed for ln(RR), then back-transformed using e^x to obtain the CI for RR. The null value for RR is 1; the null value for ln(RR) is 0. If the 95% CI for RR contains 1, there is no evidence of a difference.

The same log-transformation approach applies to the odds ratio (OR). The sampling distribution of OR is also positively skewed; CIs are constructed for ln(OR) then back-transformed.

Confidence intervals for RR and OR: if 1 is in the interval, no evidence of a difference. If the interval is entirely > 1: evidence that the exposed group has a higher risk/odds. If entirely < 1: evidence that the exposed group has a lower risk/odds.

The chi-squared (χ²) test is used on contingency tables (2×2 or larger) to test whether there is an association between two categorical variables. It compares observed cell counts to expected cell counts (the values we would expect if H₀ of no association were true). The χ² test statistic cannot be negative (it is squared). Degrees of freedom = (rows − 1) × (columns − 1), excluding the totals rows and columns. The p-value is not doubled for χ² (unlike Z or T tests) because the χ² distribution only has positive values.

Pooled sample proportion: used in hypothesis tests comparing two proportions when they are assumed equal under H₀. Calculated as (total successes in both samples) / (total participants in both samples), to provide a common estimated parameter for SE calculation.

Critical appraisal of a study involves: internal validity (accuracy — how well study findings reflect true associations in the study population, affected by chance, bias, confounding); external validity (generalisability — can results be applied to other populations?); considering alternative explanations (chance, bias, confounding, or a true relationship).

Systematic reviews and meta-analyses (from POPH M5): internal validity considers chance, bias, and confounding; systematic reviews are reproducible and transparent; meta-analysis uses forest plots and weighted averages; challenges include poor-quality primary studies, publication bias, and heterogeneity.