Interaction in multidimensional contingency tables: an information theoretic approach

Abstract

The problem of interaction in multidime nsional co ntin ge ncy tables is inves ti gated from the viewpoint of information th eory as developed by Kullbac k. The hypothesis of no rth-order interaction is defin ed in the sense of an hypothes is of "generalized " ind ependence of classifications with fix ed rth order marginal restraints. For a three· way table, with given cell probabilities 7Tijk , the minimum discrimination information for a contin ge ncy table with marginals P jj ., P. jk, and P i ... is given by the se t of cell probabilities P 0 k = a jjbj k e ,,·7Ti j k wh ere aij , b j ,., and eik are fun ctions of th e giv e n marginal probabilities, that is, In (p!jJ7T ij") = In a ij + In bj k + In e ,,·, representing no second-order interaction.The minimum di scrimination inl ormation stati sti c, asymptotically distributed as X" with appropriate degrees of freedom is where Xi jk are the observed cell frequencies and Xijk are the " no inte raction" cell frequencies uniquely de termined by a simple co nve rge nt ite rati on process of th e marginals on 7T ijk . For low e r ord e'r marginal restraints the us ual independ ence hypoth eses are ge ne rated when 7Tijk are taken to be the cell probabilities under uniform di stribution. It is shown th at th e set P !j,. satis fi es definitions of no second order interaction in a 2 x 2 x 2 table give n by Bartl ett and no interaction in a r X s X t table by Roy and Kastenbaum, and is also related to that give n by Good. Res ults of application to the analys is of so me "classical " three-dimensional contingency tables are given , together with full details for two four· dimensional examples.

Nonsynchronous Spatial Overlap of Lizards in Patchy Habitats

“…By inspection of the data and Table 4, it was possible to eliminate interactions roughly in the order in which their deletion would least affect goodness of fit. After each elimination, the difference between models was evaluated for significance according to the method of partitioning expounded in Kullback ( 1959) and Ku and Kullback ( 1968). In this approach, the difference between the log-likelihood ratio chi-squares for the two models can be used to test the hypothesis that the difference between expected values for the two models is due to random variation.…”

“…Both these statistics have the same y! distribution for sample sizes large enough, where the degrees of freedom are computed in the way described by Ku and Kullback (1968) or by Fienberg in the article immediately following this one. In four of the five sets of data, the models containing all two-factor interactions gave satisfactory fits at the 5% level or better as judged by both these criteria (Table 5) .…”

Detecting higher-order interactions among the spiking events in a group of neurons

“…This is a special case of an approximation algorithm proposed by Kullback et al (Gokhale andKullback 1978: Ireland andKullback 1968;Ku andKullback 1969: Kullback 1948) for the purpose of hypothesis lesting in multidimensiomd conlingency tables. A detailed treatment of this kind of algorithm can be found in Bishop el al.…”

“…10.2 and 10.2-1). We presenl the ilerative solution given m Ku andKullback (1968, 1969) and Csiszfir (1975),concentrating on the case k 2, which is of interest for us. With all second-order marginals lixed, p* is obtained as the limit of the iterates described by the equations:…”

Minimum Information Updating With Specified Marginals in Probabilistic Expert Systems

“…Therefore, we ought to deal with the incomplete information as the latest true marginal probabilities, and we update the prior probabilities subject to the specified marginal probability restriction such that discrimination information between the prior and the posterior probabilities is minimized. Ku and Kullback (1968) and Ireland and Kullback (1968) discussed an updating method for saturated models of contingency tables based on minimum discrimination information . In this paper, we shall extend their method to log-linear models , discuss localization of global updating, and we shall show that Deming and Stephan's iterative procedure can be used to find the posterior probabilities.…”