Why a Three-Way Model Can Have Homogeneous Association
In a three-way contingency table, the association between two variables may differ across the levels of the third. For example, an exposure and an outcome may have one odds ratio among younger patients and another among older patients. A model with no three-factor interaction rules out that change: it allows each pair to be associated, but requires each pair’s conditional association to be the same in every stratum of the remaining variable.
This is called a homogeneous-association model. “Homogeneous” refers to the pairwise associations being constant across strata; it does not mean that the variables are independent, or that the cell means must be equal. The focus here is the relationship between the model’s algebraic form and its conditional odds ratios.
Model Specification
Let \(Y_{ijk}\) denote the count in cell \((i,j,k)\) of an \(I \times J \times K\) table, with factors \(A\), \(B\), and \(C\). Assume the cell counts are independent Poisson variables with positive means \(\mu_{ijk}=E[Y_{ijk}]\). Write \(\eta_{ijk}=\log\mu_{ijk}\). Use reference levels indexed by 0 for each factor; effects involving a reference level are set to zero.
The reference-level convention makes the parameterization identifiable: for instance, \(d_{i0}=d_{0j}=0\). The model does not require any of its two-factor terms to be nonzero. It says those terms are available to represent pairwise association, while the three-factor component is absent.
Worked Example: Reading the Model Terms
Suppose \(A\) is treatment group, \(B\) is outcome status, and \(C\) is age group. In the model above, \(d_{ij}\) allows treatment and outcome to be associated. The term \(e_{ik}\) allows treatment-group proportions to vary by age group, and \(f_{jk}\) allows outcome proportions to vary by age group.
The absence of a three-factor term means that the treatment–outcome association does not change with age group. It does not remove the \(AB\) term, so it does not assert treatment–outcome independence. Nor does it require the age distributions or outcome rates to be the same in all treatment groups.
Conditional Odds Ratios from Expected Counts
For fixed level \(k\) of \(C\), the conditional odds ratio between \(A\) and \(B\), comparing levels \(i\) and \(i'\) of \(A\) and \(j\) and \(j'\) of \(B\), is defined from the positive cell means by
This definition uses the expected counts in the stratum \(C=k\). The same cross-product ratio can be formed from observed counts, but those observed ratios vary because of sampling. The model’s structural claim concerns the means.
Proof. Take logarithms of the conditional odds ratio. Substituting the model expression for each log mean gives
The intercept, all main effects, and both interactions involving \(C\) cancel in this contrast. The remaining expression contains no \(k\), so the conditional odds ratio is the same in every \(C\)-stratum. Repeating the same calculation for the \(AC\) and \(BC\) cross-product ratios shows that those are also constant across levels of \(B\) and \(A\), respectively. \(\square\)
Worked Example: Calculating Homogeneous Odds Ratios
Consider a \(2\times2\times2\) table with levels 0 and 1 for each factor, and suppose its expected counts follow the model
This is the specified model: each factor and each pair can contribute a multiplicative effect, and there is no term involving \(ijk\). For the association between \(A\) and \(B\), the conditional odds ratio at either level of \(C\) is
The value 2 can also be checked directly from the means. At \(k=0\), the four means in the \(AB\) table are \(10,30,20,120\), in row-major order, so the odds ratio is \(10(120)/[30(20)]=2\). At \(k=1\), the means are \(40,60,120,360\), giving \(40(360)/[60(120)]=2\).
The association is not the only feature that can vary by stratum. For example, the expected count for \(A=0,B=0\) changes from 10 to 40 across the two \(C\)-levels. Homogeneous association concerns the cross-product ratio, not equality of cell counts.
The Three-Factor Contrast
The no-three-factor condition can also be expressed directly as a contrast of log means. For any levels \(i,i'\), \(j,j'\), and \(k,k'\), define
This contrast is the difference between the log \(AB\) odds ratio at \(C=k\) and the log \(AB\) odds ratio at \(C=k'\). Thus it measures how much the \(AB\) association changes between those two strata.
Derivation. First, the identity
follows by expanding the two log odds ratios and subtracting. Therefore all such contrasts are zero exactly when the \(AB\) odds ratios are constant across \(C\).
To see why this condition is exactly the no-three-factor model, take the reference levels to be 0 and consider the baseline contrast
If the model has the stated main and two-factor terms, substitution shows that every term cancels in \(\tau_{ijk}\); hence \(\tau_{ijk}=0\). Conversely, if \(\tau_{ijk}=0\) for every \(i,j,k\), rearranging gives
The right-hand side is a sum of a constant, main-effect functions, and pairwise functions, each of which can be expressed with the reference-level constraints in the model. There is no remaining three-factor component. Finally, constancy of all \(AB\) odds ratios across \(C\) makes the contrast \(\tau_{ijk}\) zero by taking \(i'=j'=k'=0\) in the odds-ratio identity. Thus the conditions are equivalent. With the three-factor contrasts zero, the same cancellation applies to the other two pairs, establishing their homogeneity as well. \(\square\)
Worked Example: Detecting a Change in Association
Suppose the observed counts in a \(2\times2\times2\) table are as follows, with rows corresponding to \(A=0,1\), columns to \(B=0,1\), and separate tables for \(C=0\) and \(C=1\):
The observed \(AB\) odds ratio in the first stratum is
In the second stratum it is
The ratio of these odds ratios is \(3/(2/3)=4.5\), so the empirical log-odds-ratio contrast is \(\log(4.5)\), not zero. The observed table therefore does not exhibit homogeneous \(AB\) association across \(C\). This calculation describes the table; deciding whether sampling variation is compatible with a homogeneous-association model requires a separate model assessment.
Homogeneous Association Is Not Conditional Independence
A frequent misinterpretation is to read “no three-factor interaction” as “no interaction” or “independence.” The model retains all three pairwise interactions. If the \(AB\) term is nonzero, \(A\) and \(B\) can be associated within every \(C\)-stratum; the restriction is that the association does not change across those strata. Conditional independence of \(A\) and \(B\) given \(C\) would require every conditional \(AB\) odds ratio to equal 1, a stronger condition.
For example, the odds ratio of 2 in the previous worked example is constant across \(C\), but is not 1. It represents homogeneous association, not conditional independence. Likewise, the model does not imply that any pair is marginally independent: marginal association after summing over a third factor is a different quantity.
Common Mistake and Practical Interpretation
- Wrong reasoning: “There is no three-factor interaction, so all variables are mutually independent.” This confuses the absent three-factor term with the pairwise terms that remain in the model.
- Correction: The model allows \(AB\), \(AC\), and \(BC\) associations. It constrains each conditional pairwise odds ratio to be constant across levels of the third factor.
- Wrong reasoning: “Homogeneous association means the cell counts are the same in each stratum.” The model places no such equality restriction on individual means.
- Correction: Compare cross-product ratios, not the raw counts. Main effects and interactions with the stratifying factor can change the cell means while the pairwise odds ratio remains fixed.
This interpretation is useful when a scientific question concerns whether an association is modified by a third variable. A no-three-factor-interaction model encodes the assumption that it is not modified on the conditional odds-ratio scale. That scale matters: constancy of conditional odds ratios is the model’s claim, not a universal claim that every possible measure of association is constant.
Check Your Understanding
Use the model and contrast definitions above to answer these questions.
- In the no-three-factor-interaction model, which terms cancel when forming the conditional \(AB\) odds ratio?
- Does a constant \(AB\) odds ratio across levels of \(C\) have to equal 1? Explain.
- What does a nonzero three-factor contrast say about the \(AB\) association between two \(C\)-strata?
- Can expected cell counts change across \(C\)-strata while the \(AB\) odds ratio stays fixed? Give a reason.
- Why does observing different sample odds ratios across strata not, by itself, establish that the population model has a three-factor interaction?