Partitioning Two-Way Tables.

Lancaster (1949) and Irwin (1949) present a method of partitioning tables that was used in Exercise 2.7.4.

We now establish the validity of this method. Consider a two-dimensional I × (J + K − 1) table. The partitioning method tests for independence in two subtables. One table is a reduced I × K table consisting of the last K columns or the full table.

The other table is an I × J table that uses the first J − 1 columns of the full table and also includes a column into which the last K columns of the full table have been collapsed. Write the data with three subscripts as nijk, i = 1,...,I, j = 1, . . . , J, and k = 1,...,Lj , where Lj =

" 1 if j = J K if j = J.

Consider the models log(mijk) = αi + βjk, (1)

log(mijk) = βij + βjk, (2)

and log(mijk) = γijk. (3)

Model (3) is the saturated model so mˆ (3)
ijk = nijk.
Model (1) is the model of independence in the I × J + K − 1 table, so mˆ (1)
ijk = ni··n·jk

n···.

(a) Show that the maximum likelihood estimates for model (2) are mˆ (2)
ijk = " nijk if j = J niJ·n·Jk

n·J· if j = J.

(b) Show that both G2 and X2 are the same for testing model (2) against model (3) as for testing the reduced table for independence.

(c) Show that both G2 and X2 are the same for testing model (1) against model (2) as for testing the collapsed table for independence.

(d) Extend

(b) and

(c) by showing that all power divergence statistics are the same, cf. Exercise 2.7.8.