The model for (*A, B, C*) is ln y =* λ + λ ^{A} + λ^{B} + λ^{C}* and has

*abc*– [(

*a*– 1) + (

*b*– 1) + (

*c*– 1) + 1] =

*abc*– (

*a+b+c*) + 2 degrees of freedom. For Example 1 of Three-way Contingency Tables the mutual independence model has 2 ∙ 2 ∙ 3 – (2 + 2 + 3) + 2 = 7 degrees of freedom.

In this model *A* and *B* are independent, *A* and *C* are independent, and *B* and *C* are independent. The probability that an observation will occur in cell *ijk* is

For Example 1 of Three-way Contingency Tables, the expected frequencies for the mutual independence model are given in Figure 1.

**Figure 1 – Expected frequencies for ( C, G, T) model**

Once again, all the marginal totals for the expectations are set equal to the corresponding totals for the observations. Each of the cells in the range J40:L43 are calculated as described above. For example the expected frequency of a positive result for males taking therapy 1 (cell J40) is calculated via the formula =M46*M49*J44/M44^2.

χ^{2} (maximum likelihood) is 64.48 with 7 degrees of freedom, which is significant, and so once again this model is not a good fit for the data.

Note too that

for any values of *i, j, k, i′, j′, k′*. E.g. the formula =(J40*K41)/(J41*K40) has value 1.