Crowder (1978) Beta-Binomial ANOVA for Proportions

Summary

Crowder proposes a method for the regression analysis of proportions based on the beta-binomial distribution, to handle data in which the proportion varies more than a simple binomial allows. The motivating problem — seed-germination experiments with several seed types, extracts, dilutions, and replicates — shows clear heterogeneity of proportions between replicates (over-dispersion). Modeling the binomial success probability as a Beta random variable that varies across units captures this extra-binomial variation, and Crowder develops an ANOVA/regression structure and maximum-likelihood estimation for the resulting model.

Key Claims

• Over-dispersion is the problem. When replicate proportions are more variable than a common binomial would predict (confirmed by $\chi^2$ tests on the $2\times m$ tables), fitting an ordinary binomial understates the variance and overstates precision.
• Beta-binomial model. Let the number of successes be binomial given a success probability $p$, and let $p$ itself follow a Beta$(\alpha,\beta)$ distribution across units; marginalizing gives the beta-binomial, whose variance is inflated by an intraclass-correlation factor relative to the binomial.
• ANOVA / regression for proportions. The systematic part (seed type, extract, dilution effects) is modeled through the beta-binomial parameters, giving an analysis-of-variance / regression framework for proportion data with over-dispersion.
• Maximum-likelihood estimation. The parameters (including the over-dispersion parameter) are estimated by ML, providing valid tests that account for the extra-binomial variation.

Concepts Introduced or Extended

• Beta-Binomial Model

Entities Mentioned

• Martin J. Crowder

Quotes

"A method is proposed for the regression analysis of proportions based on the Beta-binomial distribution … it is clear … that there is heterogeneity of proportions between replicates."

My Take

A small, practical paper that gave applied statisticians the standard tool for the very common situation where counted proportions are "too variable" — clustered binary outcomes, litters, replicates, batches. The beta-binomial's appeal is that it adds exactly one over-dispersion parameter with a clean interpretation (an intraclass correlation) while keeping a likelihood you can maximize, which is why it remains a default alongside quasi-likelihood and logistic-normal alternatives. On the wiki it underlies the beta-binomial dependence structure used in the latent class literature (within-class exchangeable over-dispersion) and sits beside the GLMM route to the same problem. Crowder's seed-germination data, like Cressie-Chan's SIDS counts, became a recurring teaching example — another case of a good dataset outliving its original paper.