Pearson's chi-squared test ($X^2=\sum (O-E{)}^2/E$)

The Pearson chi-squared test compares observed categorical counts with null expected counts using the displayed statistic. Its applications include Pearson chi-squared goodness-of-fit test for specified probabilities and Pearson chi-squared test of independence for contingency tables; the fitted restrictions determine the limiting degrees of freedom.

For $K$ independent strata, each containing an $r\times c$ contingency table, test independence of the two table variables within every stratum. Under adequate expected counts and the usual multinomial distribution sampling assumptions, each stratum's Pearson chi-squared statistic has asymptotic chi-squared distribution with $(r-1)(c-1)$ statistical degrees of freedom. Their sum has asymptotic chi-squared distribution with $K(r-1)(c-1)$ statistical degrees of freedom, by the additivity of independent chi-squared distributions. This is a test of conditional independence, not of equality of a common effect across strata; the latter needs an interaction contrast or another specified homogeneity test.

For observed cell counts $O_j$ and null expected counts $E_j$, the statistic isWith $k$ specified null cell probabilities and sufficiently large expected counts, its null limit is ${\chi }_{k-1}^2$.

For $N$ independent groups each containing $m$ independent Bernoulli trials with common probability $\theta$, the group count has a binomial distribution. If $O_j$ groups have count $j$, the maximum-likelihood estimate is $\hat{\theta }={\sum }_{j}j{O}_j/(mN)$. Form the Pearson chi-squared statistic using $E_j=N\left(\genfrac{}{}{0ex}{}{m}{j}\right){\hat{\theta }}^j(1-\hat{\theta }{)}^{m-j}$. With all $m+1$ categories retained, an interior true parameter, and sufficiently large expected counts, its null limit is a chi-squared distribution with $m-1$ statistical degrees of freedom: one degree is removed by the fixed total and another by fitting $\theta$. Small expected counts require care with this approximation. If categories are pooled, the fitted parameter and calibration should correspond to the pooled statistical model; simply pooling a statistic after fitting the unpooled model does not automatically give the usual chi-squared limit.

For an $r$ by $c$ contingency table, the statistichas asymptotically a chi-squared distribution with $(r-1)(c-1)$ degrees of freedom under independence.