Bowker Procedure

Development

The responses from a sample of n′ individuals over two periods of time may be tallied into an r × c table (where r, the number of rows, equals c, the number of columns) of cross-classifications, as shown in Table 1.

With respect to the population from which the aforementioned sample was taken, let pij be the probability of responses to the ith category before the treatment condition was imposed and the jth category after. The marginal probabilities before and after treatment sum to unity. That is, p1. + p2. + … + pr. = 1 and p.1 + p.2 + … + p.c = 1.

Testing for Significance of Changes in Related Proportions

In order to investigate changes in repeated measurements, the null hypothesis is that of symmetry:

The alternative is that at least one pair of symmetric probabilities is unequal:

That is, the null hypothesis tested is conditioned on those

individuals whose responses change, where the probability (pij) of a switch from response i to response j is equal to the probability (pji) of a switch from response j to response i, and this probability is 0.5.

The Bowker test statistic B, written as

has a chi-square distribution with u degrees of freedom where u = r(r − 1)/2 = c(c − 1)/2 since r = c. The [Page 108]null hypothesis can be rejected at the α level of significance if

A Posteriori Comparisons

If the null hypothesis is rejected, the researcher Alan Stuart suggested a multiple comparison procedure that permits the development of a post hoc evaluation of changes in the correlated proportions (i.e., marginal probabilities) for each response category versus the c – 1 other categories combined. Thus, regardless of the size of the initial c × c table of cross-classifications, this process allows for the formation of a set of c 2 × 2 tables, one for each of the c categories versus all the other c – 1 categories combined. These 2 × 2 tables take the form for all i = 1,…, c.