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Statistics 601, Fall 2005
Lab 6 Notes
In this lab we will discuss estimation for the basic beta-binomial mixture model
via maximum likelihood and use of a Newton-Raphson algorithm for numerical
maximization of likelihood or log likelihood functions. The fundamental routine will
be one that is already familiar to you from finding maximum likelihood estimates
for iid samples from gamma distributions. In particular, in this lab we will consider
• estimation under standard α, β parameterization
• estimation under a mean value parameterization µ, φ
• interval estimation of functions of parameters under alternative parameterizations
• comparison of groups
• comparison of groups under restricted model formulation
Lab 6 Assignment
On the course web page under “Laboratory Materials” are links to files that contain data from a paper by Williams (Williams, D.A. 1975, The analysis of binary
responses from toxicological experiments involving reproduction and teratogenicity.
Biometrics 31, 949-952). The study analyzed by Williams involved treating groups
of pregnant rats with a suspected teratogenic compound. There were two groups
– control diet and treated diet. The number of young alive at four days and alive
at 21 days were recorded. The number of young alive at four days is taken as the
binomial sample size for each rat and the number of young alive at 21 days is taken
as the binomial response. The data consist of three columns which denote treatment
2
(1=control, 2=treated), binomial sample size (n) and number of live young at 21
days (y).
Using the data from the Williams paper
1. Fit beta-binomial models to each group of data using a standard α, β parameterization. Compare groups (test for differences), summarize results (a graph
or two would be nice).
2. Fit beta-binomial models to each group of data using a mean-value parameterization with some choice of the additional parameter that is up to you.
Compare groups and summarize results.
3. Compare interval estimates of expectations from the estimation in 1 (plus
the delta method) to direct interval estimation using the inverse information
matrix from 2.
4. Fit a restricted model in which groups (control and treatment) are restricted
to have equal means. Test for a difference between groups using this model.
5. Determine what would happen if we extended the beta-binomial model to
consider both binomial sample sizes and binomial counts as random variables.
That is, observations would now consist of pairs (ni , yi ) for each individual case.
Suppose the random variables {ni : i = 1, . . . , m} follow a Poisson distribution.
That is, would estimates of the beta distribution parameters (either α, β or µ,
φ) change from the beta-binomial model in which the binomial sample sizes
are considered fixed?
6. Summarize your observations about (1) the Williams data and (2) statistical
aspects of the beta-binomial model.