$Z_{ijk}=1$ if the $k$th site ($k$th body part) of the $j$th subject in the $i$th treatment group is free of disease at the end of the study and 0 otherwise for $i=0,1$, $j=1,\dots, m_{+i}$, and $k=1,2$. Dallal’s (reduced) model is characterized by the following assumptions:

Assumption 1:

:   $P\big(Z_{ijk}=1\big) = \lambda_i$ for $i=0,1$ with $0<\lambda_i<1$.

Assumption 2:

:   $P\big(Z_{ijk}=1 \mid Z_{ij(3-k)}=1\big) = 1-\gamma
    $ with $0<\gamma<1$.

Assumption 2 states that the conditional probability of an occurrence of a particular characteristic at one site given an occurrence of that characteristic at the other site to be the same in the two treatment groups. This statement is relaxed and replaced by $P\big(Z_{ijk}=1 \mid Z_{ij(3-k)}=1\big) = 1-\gamma_i$ with $0<\gamma
_i<1,\,i=0,1$ in the full or saturated model. That is, two conditional probability statements are made, one for the treatment group and the other for the control group. We also refer to this saturated model as Dallal’s saturated model. However, the full model has one more parameter than the reduced model.

Let $m_{hi}$ be the number of subjects in the $i$th group with exactly $h$ site(s) cured and $p_{hi}$ be the success probability associated with $m_{hi}$ for $h=0,1,2$ and $i=0,1$. The two group total sample 