Why an Interaction Can Be an Odds Ratio
In a two-way table, a row effect changes the mean count across a row and a column effect changes it across a column. An interaction describes what remains when those separate effects do not account for the joint pattern. For a two-by-two table, that remainder has a direct interpretation: it is the logarithm of the cross-product odds ratio.
This connection is useful because the odds ratio is unchanged by multiplying an entire row or column by a positive constant. It isolates association between the two table variables from their separate overall levels. In a loglinear model, the corresponding interaction contrast removes the row and column contributions by subtraction.
When the table records two binary outcomes, this cross-product ratio is the ratio of the odds in one row to the odds in the other. More generally, it is called the odds ratio for the two-by-two table even when the row and column categories are not themselves outcomes. The cell order matters: exchanging the two rows or the two columns changes the odds ratio to its reciprocal.
The Two-by-Two Loglinear Interaction
Let \(\mu_{ij}>0\) be the mean count in cell \((i,j)\), and write \(\eta_{ij}=\log\mu_{ij}\). A saturated two-by-two loglinear model can be written using reference-cell coding:
Here \(\lambda\) is the log mean in the reference cell \((1,1)\); \(\alpha_i\) and \(\beta_j\) are row and column effects relative to that cell. The interaction parameter \(\gamma_{22}\) is the part specific to the combination of row 2 and column 2. It is a contrast, not simply the log mean of cell \((2,2)\).
Proof. Under the stated coding, \(\eta_{11}=\lambda\), \(\eta_{21}=\lambda+\alpha_2\), \(\eta_{12}=\lambda+\beta_2\), and \(\eta_{22}=\lambda+\alpha_2+\beta_2+\gamma_{22}\). Therefore,
Since every mean is positive, its logarithm is defined. The logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of logarithms. Thus \(\eta_{11}+\eta_{22}-\eta_{12}-\eta_{21} =\log(\mu_{11}\mu_{22}/(\mu_{12}\mu_{21}))\), proving the claim. \(\square\)
This result also explains the phrase “no interaction” in a two-by-two loglinear model: \(\gamma_{22}=0\) exactly when \(\operatorname{OR}=1\). It does not mean that all four cell means are equal. Row and column effects can still make them quite different; the condition says that their cross-product ratio is one.
Worked Example: A Two-by-Two Table
Suppose a two-by-two table has positive expected counts \(\mu_{11}=30,\ \mu_{12}=10,\ \mu_{21}=12,\ \mu_{22}=24\). Calculate the odds ratio and the loglinear interaction.
First calculate the cross-product odds ratio:
Now take its logarithm. The reference-cell interaction is \[ \gamma_{22}=\log(6)\approx 1.7918. \] The same answer follows from the log-mean contrast: \[ \log(30)+\log(24)-\log(10)-\log(12) =\log\left(\frac{30(24)}{10(12)}\right) =\log(6). \] Thus the interaction is positive, corresponding to an odds ratio greater than one. Reversing either the row comparison or the column comparison would give \(1/6\), with log odds ratio \(-\log(6)\).
Interaction Contrasts in Larger Tables
A table need not have only two rows and two columns for this relationship to apply. Choose any two distinct rows \(i,i'\) and any two distinct columns \(j,j'\). Those levels form a two-by-two subtable. Its odds ratio, using row order \(i,i'\) and column order \(j,j'\), is \(\mu_{ij}\mu_{i'j'}/(\mu_{ij'}\mu_{i'j})\). The corresponding log-mean contrast is the log odds ratio.
Proof. Because \(\eta_{ab}=\log\mu_{ab}\), the contrast on the right is
For the second equality, substitute \(\eta_{ab}=\lambda+\alpha_a+\beta_b+\gamma_{ab}\) into the four terms. The intercept occurs twice with a plus sign and twice with a minus sign. Each row effect and each column effect also occurs once with each sign. They all cancel, leaving \(\gamma_{ij}+\gamma_{i'j'}-\gamma_{ij'}-\gamma_{i'j}\). This proves both equalities. \(\square\)
With reference row 1 and reference column 1, the contrast comparing row \(i\) with row 1 and column \(j\) with column 1 reduces to \(\gamma_{ij}\). But if neither selected level is the reference, the log odds ratio generally involves a difference of two interaction parameters, or the full four-term contrast above. A single interaction coefficient is not automatically the log odds ratio for every pair of levels.
Worked Example: Selecting Two Columns in a Two-by-Three Table
Consider expected counts in a table with two rows and three columns:
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | 20 | 10 | 5 |
| Row 2 | 12 | 12 | 12 |
Compare row 1 with row 2 using columns 1 and 3, in that order. The selected subtable has cells 20 and 5 in row 1, and 12 and 12 in row 2. Its odds ratio is
The log-mean contrast gives the same result: \[ \eta_{11}+\eta_{23}-\eta_{13}-\eta_{21} =\log(20)+\log(12)-\log(5)-\log(12) =\log(4). \] With column 1 as the reference, this contrast is the interaction parameter \(\gamma_{23}\). The comparison concerns columns 1 and 3, not columns 1 and 2; using the wrong pair of columns would calculate a different odds ratio.
Worked Example: Comparing Two Nonreference Rows
Suppose a three-by-three table has positive expected counts:
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | 12 | 6 | 3 |
| Row 2 | 8 | 8 | 8 |
| Row 3 | 4 | 8 | 16 |
Compare rows 2 and 3 across columns 1 and 3, in that order. The odds ratio is
To express this using reference-cell interactions, take row 1 and column 1 as the references. The row-and-column terms cancel in each baseline interaction: \[ \gamma_{23}=\log\left(\frac{\mu_{23}\mu_{11}}{\mu_{21}\mu_{13}}\right) =\log\left(\frac{8(12)}{8(3)}\right)=\log(4), \] and \[ \gamma_{33}=\log\left(\frac{\mu_{33}\mu_{11}}{\mu_{31}\mu_{13}}\right) =\log\left(\frac{16(12)}{4(3)}\right)=\log(16). \] The selected comparison uses two nonreference rows, so its log odds ratio is their interaction-parameter difference: \[ \gamma_{33}-\gamma_{23}=\log(16)-\log(4)=\log(4). \] This agrees with the direct calculation. The check matters: \(\gamma_{33}\) alone compares row 3 with the reference row, not row 3 with row 2.
Means, Observed Counts, and Interpretation
The defining loglinear relationship is a statement about positive cell means. In an observed table, the sample cross-product ratio is calculated from the four observed counts. If the table has independent Poisson cell counts and a saturated model is fitted, the fitted means equal the observed counts when all cells have positive counts; in that case the observed log odds ratio is also the fitted interaction contrast. If an observed cell is zero, the sample cross-product ratio may be zero, infinite, or undefined, and its logarithm need not be finite. Do not treat that as a finite interaction estimate without additional modeling choices.
The odds ratio describes a multiplicative comparison. An odds ratio of 4 says that the cross-product in the numerator is four times the cross-product in the denominator; its log, \(\log(4)\), is the additive contrast on the log-mean scale. Neither value by itself establishes a causal effect. Interpretation depends on what the rows and columns represent and how the table was obtained.
Common Mistake: Reading One Cell as the Interaction
A common error is to call \(\log(\mu_{22})\) the interaction in a two-by-two table, or to identify an interaction parameter with the odds ratio itself. But \(\log(\mu_{22})\) includes the intercept and both main effects, while the loglinear interaction is a log contrast. In the reference-cell coding above, \(\gamma_{22}\) is the log odds ratio; exponentiating it gives the odds ratio.
Another source of error is reversing only part of a comparison. The numerator must pair one selected row with one selected column and the other selected row with the other selected column. For the stated order, use \(\mu_{ij}\mu_{i'j'}/(\mu_{ij'}\mu_{i'j})\). Swapping exactly one pair of levels reverses the ratio and changes the sign of its logarithm; swapping both pairs leaves it unchanged.
Check Your Understanding
Work through these questions using the stated row and column order.
- A two-by-two table has means \(\mu_{11}=15,\ \mu_{12}=5,\ \mu_{21}=9,\ \mu_{22}=9\). Calculate the odds ratio and loglinear interaction.
- For rows \(i,i'\) and columns \(j,j'\), write the log-mean contrast that equals the log odds ratio.
- In a reference-cell model, why is \(\gamma_{ij}\) the log odds ratio for row \(i\) versus row 1 and column \(j\) versus column 1?
- If an odds ratio is 5, what are the odds ratio and log odds ratio when exactly one of the row or column comparisons is reversed?
- Why can a zero observed count make the sample log odds ratio nonfinite even though the loglinear identity is defined for positive means?