Analysis of residuals in contingency tables: another nail in the coffin of conditional approaches to significance testing.

Omnibus tests of significance in contingency tables use statistics of the chi-square type. When the null is rejected, residual analyses are conducted to identify cells in which observed frequencies differ significantly from expected frequencies. Residual analyses are thus conditioned on a significant omnibus test. Conditional approaches have been shown to substantially alter type I error rates in cases involving t tests conditional on the results of a test of equality of variances, or tests of regression coefficients conditional on the results of tests of heteroscedasticity. We show that residual analyses conditional on a significant omnibus test are also affected by this problem, yielding type I error rates that can be up to 6 times larger than nominal rates, depending on the size of the table and the form of the marginal distributions. We explored several unconditional approaches in search for a method that maintains the nominal type I error rate and found out that a bootstrap correction for multiple testing achieved this goal. The validity of this approach is documented for two-way contingency tables in the contexts of tests of independence, tests of homogeneity, and fitting psychometric functions. Computer code in MATLAB and R to conduct these analyses is provided as Supplementary Material.
Ministerio de Ciencia e Innovación (MICINN)
Ministerio de Economía y Competitividad (MINECO)
Ministerio de Ciencia e Innovación and FEDER
Universidad del PaísVasco UPV/EHU
Departamento de Educación del Gobierno Vasco
UPV/EHU Econometrics Research Group
Depto. de Psicobiología y Metodología en Ciencias del Comportamiento
Fac. de Psicología
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