Chi-square Test

Chi-square Test

1. Definition, scope, and historical background

• Chi-square (χ²) is a family of non-parametric inferential tests that compare observed (O) frequencies with expected (E) frequencies in categorical data, summarising the discrepancy in a single statistic against a known reference distribution (Swinscow & Campbell 10e Ch.08, pp.65–66).
• The test name refers to the underlying χ² sampling distribution — the distribution of the sum of squares of k independent standard-normal variables, plotted with degrees of freedom (df) as the only shape parameter (Swinscow & Campbell 10e Ch.08, pp.65–66).
• Introduced by Karl Pearson (1900) as the Pearson goodness-of-fit χ² statistic — the first general method for testing whether observed proportions deviate from a theoretical model by more than chance (Swinscow & Campbell 10e Ch.08, p.65).
• Yates (1934) added the continuity correction for 2×2 tables to compensate for the discontinuous nature of count data being approximated by a continuous χ² distribution (Swinscow & Campbell 10e Ch.08, pp.69–70).
• R.A. Fisher subsequently developed the exact probability test as a small-sample alternative when χ² approximation fails (Swinscow & Campbell 10e Ch.09, pp.80–81).
• McNemar (1947) adapted the framework for paired binary data in matched studies (Swinscow & Campbell 10e Ch.08, pp.74–75).
• In MD Pharmacology, χ² is the first-line test for categorical outcomes — drug-vs-placebo response rates, ADR incidence, genotype frequencies in pharmacogenetic studies, and category-level survey responses (Medhi Ch.11, pp.130–131; Shargel 8e Ch.3, Nonparametric data section).
• Where the chi-square test sits in the analytical hierarchy: parametric tests (t-test, ANOVA) require continuous, normally distributed data; χ² is its non-parametric counterpart for nominal/ordinal categorical data that cannot be reduced to a mean ± SD (Medhi Ch.11, p.131; Shargel 8e Ch.3, Statistical inference techniques for nonparametric data).
• Recommended by ICMR's National Ethical Guidelines (2017) and accepted by RGUHS, NMC, and ICH-GCP as a standard test of association for categorical clinical-trial endpoints (Medhi Ch.11, p.131).