Checking Global Compatibility in Three Variables

Why Check All Three Conditionals Together?

In fully conditional specification, each variable is updated from a model conditional on the other variables. When there are three variables, it is not enough for one pair of conditionals to be compatible: the restrictions must agree around all three pairs. A mismatch in any pair rules out a single joint distribution that has all three specified conditionals.

We develop an exact check for three binary variables \(X\), \(Y\), and \(Z\). It compares the pairwise conditional odds ratios in both directions and across the strata defined by the third variable. The result is a finite set of coefficient equalities that is both necessary and sufficient under the assumptions below.

Conditional Logistic Models and Pairwise Restrictions

Assume each variable takes values 0 and 1, all conditional probabilities are specified for every configuration of the other two variables, and all probabilities are strictly between 0 and 1. Write the three conditional models as saturated logistic regressions:

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). Each conditional includes both main effects and the interaction between its two predictors, so it can represent any strictly positive binary conditional table. The coefficient \(c_X\), for example, allows the association between \(X\) and \(Y\) to differ between \(Z=0\) and \(Z=1\).

For a fixed value \(z\), the conditional odds ratio relating \(X\) and \(Y\) can be calculated from the \(X\)-conditional. Changing \(Y\) from 0 to 1 changes the log odds of \(X=1\) by \(b_{XY}+c_Xz\), so the odds ratio is \(\exp(b_{XY}+c_Xz)\). The \(Y\)-conditional gives the same pairwise odds ratio as \(\exp(b_{YX}+c_Yz)\). If both models come from the same positive joint distribution, these odds ratios must agree at each value of \(z\). Equality at \(z=0\) gives \(b_{XY}=b_{YX}\); equality at \(z=1\) then gives \(c_X=c_Y\).

Applying this argument to the other two pairs yields the complete set of restrictions:

Each pairwise coefficient must agree in the two conditionals for that pair. In addition, the three conditional interaction coefficients must agree. Checking only one stratum of a pair tests the relevant \(b\) equality but cannot establish equality of the interaction coefficients.

Why These Conditions Are Sufficient

Necessity follows from the odds-ratio argument. To show sufficiency, suppose the displayed equalities hold. Form a joint distribution with unnormalized probability

Because every \(w(x,y,z)\) is positive, the normalizing constant \(K=\sum_{x,y,z\in\{0,1\}}w(x,y,z)\) is finite and positive. Thus \(P(X=x,Y=y,Z=z)=w(x,y,z)/K\) defines a positive joint distribution. To obtain the conditional odds of \(X\), take the ratio of its joint probabilities at \(x=1\) and \(x=0\), holding \(y,z\) fixed. The normalizing constant cancels:

The same calculation for \(Y\) and \(Z\) recovers their specified conditional logits, using the coefficient equalities. Therefore the constructed joint distribution has all three stated full conditionals. This proves that, under positivity and the stated saturated logistic forms, the restrictions are necessary and sufficient.

Checking the Restrictions in Practice

A systematic check has three pairwise comparisons and an interaction comparison. For each pair, compare the corresponding coefficients in the two conditionals; then verify that all three interaction coefficients agree. Equivalently, compare conditional odds ratios within each stratum of the third variable. The coefficient method is convenient when the logistic models are fitted directly; the stratum method makes the meaning of the restrictions explicit.

Diagnostics, Alternatives, and Limitations

For specified conditional models, the restrictions are exact algebraic checks. For fitted models, coefficients are estimates and may differ because of sampling variability, model fitting, or rounding. Inspect the matched coefficients and their uncertainty, and compare implied odds ratios in both strata. A small difference in rounded output is not, by itself, evidence of incompatibility; conversely, a substantial discrepancy should not be dismissed because the chain appears numerically stable.

A direct alternative is to specify a joint model first and derive all three conditionals from it. The joint log-linear form used in the sufficiency argument makes the shared pairwise and three-way terms explicit. This enforces compatibility by construction, whereas separately specified full conditional models require checking. The conditional approach can be more convenient in FCS, but the compatibility check does not decide whether either modeling strategy is substantively appropriate.

The result relies on positivity. If some configurations have probability zero by design or because of structural constraints, certain conditional probabilities and odds ratios may be undefined, and these equalities do not apply without modification. It also relies on the stated saturated logistic representation for binary full conditionals. For variables with other distributions, or models with different predictors or constraints, the corresponding compatibility conditions must be derived for those models rather than borrowed from this binary case.