The above Bayesian test requires the specification of the hyperparameter P. The R C interaction terms { 
} are assumed to be independent distributed N(0, 
) under the dependence hypothesis. This prior can be better understood in terms of the odds-ratios in the two-way table. The association in the 2 x 2 subtable formed by the i and i+1 rows and j and j+1 columns of the table can be measured by the local log odds ratio
Under our prior specification, each local log odds ratio is distributed N(0, 
). The user can assign a value of P which reflects a priori the size of the local log odds ratios if the two classification variables are dependent. For example, if P = 2, the prior standard deviation of each local odds ratio is 1.4. If the two variables are dependent, then each local odds ratio will lie in the interval (-2.8, 2.8) with probability .95.
Albert (1994) investigated empirically the effect of the choice of the hyperparameter P on the values of the Bayes factors against independence. Several conclusions can be drawn from this investigation. The value of P should be not chosen too small or too large. For a 4 x 4 table, the choice P = 100 gives a dependence hypothesis that is too close to independence and the Bayes factors do not distinguish tables with different dependence structures. On the other hand, the value P = .01 gives a dependence hypothesis that is too diffuse, and the Bayes factors will support independence even for tables that have significant dependence.
In the case where one has little prior information about the size of interaction terms under dependence, Raftery (1993) recommends the range of values RC /25 ;SPMlt; P ;SPMlt; RC, where R C is the number of cells in the table. In the examples in Albert (1994), the values of the Bayes factor are relatively insensitive to values of P within this range.
One concern is the possible sensitivity of the computed Bayes factors with respect to the choice of a normal prior on the dependence hypothesis. Albert (1994) considered the use of a Cauchy prior with median 0 and scale parameter b on the interaction terms. The hyperparameter b can be chosen to reflect prior knowledge about the sizes of the
{ } when the table is not independent. In the case where little prior information exists about dependence, the Bayes factors based on normal and Cauchy prior give similar results. However, in the case where one has significant prior information about
the association structure in the table, the two statistics can disagree. In this situation, the Cauchy statistic is preferable, since the corresponding Bayes factor is relatively insensitive to the small changes in the assessed value of the scale parameter b.