The Bradley-Terry Model.

“Let’s suppose, for a moment, that you have just been married and that you are given a choice of having, during your entire lifetime, either x or y children. Which would you choose?” Imrey, Johnson, and Koch (1976)

report results from asking this question of 44 Caucasian women from North Carolina who were under 30 and married to their first husband. Women were asked to respond for pairs of numbers x and y between 0 and 6 with x

For example, in comparing 0 children with 1 child there are only 17+2 = 19 responses rather than 44.

TABLE 8.6. Family Size Preference Alternative Preferred Number of Children Choice 0123456 0 — 17 22 22 15 26 25 1 2 — 19 13 10 9 11 2 1 0 — 11 11 6 6 3 3 1 7— 6 2 6 4 1 10 12 13 — 4 0 5 1 11 18 15 17 — 11 6 2 13 20 22 14 12 —

This data collection technique is called the method of paired comparisons.

It is often used for such things as taste tests. Subjects find it easier to distinguish a preference between two brands of cola than to rank their preferences among half a dozen. David (1988) provides a good survey of the literature on the analysis of preference data along with notes on the history of the subject. One particular model for preference data assumes that each item has a probability πi of being preferred. Thus, in a paired comparison, the conditional probability that i is preferred to j is πi/(πi + πj ). There are many ways to arrive at this model; the one given above is simple but restrictive. In other developments, the parameters πi need not add up to one, but it is no loss of generality to impose that condition. Bradley and Terry (1952) rediscovered the model and popularized it. The Bradley-Terry model was put into a log-linear model framework by Fienberg and Larntz (1976). For I items being compared, their framework consists of fitting the incomplete I(I − 1)/2 × I table in which the rows consist of all pairs of items and the columns consist of the preferred item. A test of the model log(mij ) = u + u1(i) + u2(j) is a test of whether the Bradley-Terry model holds.

(a) Rewrite Table 8.6 in the Fienberg-Larntz form.

(b) Test whether the Bradley-Terry model fits.

(c) Show that under the log-linear main effects model the odds of preferring item j to item j
is the ratio of a non-negative number depending on j and a non-negative number depending on j

. Show that this is equivalent to the Bradley-Terry model.

(d) Estimate the probabilities πi.