chisquare

Chisquare, commonly written as chi-square and denoted χ², refers to a family of statistical measures based on the chi-square distribution. It quantifies how expectations under a null model diverge from observed categorical data and is used to assess goodness of fit or independence in categorical analyses. The chi-square statistic is an approximation that relies on large samples and nonnegative observed counts.

For a goodness-of-fit test with k categories, observed counts O_i and expected counts E_i under the null hypothesis, the chi-square statistic is χ² = sum over i of (O_i - E_i)² / E_i. The degrees of freedom depend on model specification: df = k - 1 minus the number of estimated parameters from the data. In a test of independence in an r by c contingency table, χ² = sum over i and j of (O_ij - E_ij)² / E_ij, with df = (r - 1)(c - 1). The p-value is obtained from the chi-square distribution with the relevant degrees of freedom.

Assumptions for valid use include that data are counts, observations are independent, and expected frequencies are sufficiently large (commonly at least 5 in each cell) to justify the approximation. The chi-square statistic provides an approximate test; when sample sizes are small or expected counts are low, exact tests or alternative methods may be preferred.

Historically, the chi-square concept was developed by Karl Pearson in the early 20th century and has since become a foundational tool in categorical data analysis, underpinning tests of fit, independence, and homogeneity across diverse disciplines.