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ISCAM: Investigating Statistical Concepts, Applications, and Methods

Beth L. Chance, Allan J. Rossman

Section 24.5 Section 5.1 Summary

This section introduced the chi-squared test in three related settings: comparing proportions across populations/treatments, comparing full categorical distributions across populations/treatments, and testing association between two categorical variables in one random sample.
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Across all settings, the mechanics are the same: compare observed cell counts to expected counts under the null model and summarize discrepancies with the chi-squared statistic.
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Summary of Chi-squared Tests.

Test of homogeneity: Data are independent random samples from multiple populations/processes or from a randomized experiment with multiple treatments.
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\(H_0\text{:}\) group distributions are the same (or \(p_1 = p_2 = \cdots = p_I\) in the binary-response case).
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\(H_a\text{:}\) at least one group distribution differs.
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Test of association: Data are one random sample cross-classified by two categorical variables.
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\(H_0\text{:}\) no association between variable 1 and variable 2 in the population.
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\(H_a\text{:}\) there is an association.
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Test statistic: \(\chi^2 = \sum \frac{(O-E)^2}{E}\)
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Degrees of freedom: \((r-1)(c-1)\)
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Technical conditions: at least 80% of expected counts are at least 5, and all expected counts are at least 1.
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Technology Notes.

R: Build a matrix of counts and use chisq.test(matrix). Inspect $expected and $residuals for diagnostics.
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Minitab: Stat > Tables > Chi-Square Test for Association with raw or summarized data.
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JMP: Analyze > Fit Y by X with nominal variables (and frequency column if tallied).
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Applet: Paste raw data or a two-way table into the Analyzing Two-way Tables applet and inspect table, \(\chi^2\text{,}\) and contributions.
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When the test is significant, examining individual cell contributions helps describe the nature of the association. The data collection method still determines what conclusions are justified about causation and generalizability.
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