Independence in Two-Way Tables

Independence as a Loglinear Model

A two-way table records counts classified by a row variable and a column variable. The independence model says that the expected count pattern has no special row-column pairing: row and column effects combine additively on the log-mean scale. This does not require the row or column effects to be absent. It requires the cell mean to be built from those effects without an additional term specific to a row-column combination.

Let \(Y_{ij}\) be the count in row \(i\) and column \(j\), for \(i=1,\ldots,I\) and \(j=1,\ldots,J\). In the Poisson loglinear formulation, the cell counts are independent Poisson random variables with means \(\mu_{ij}>0\). A convenient reference-cell parameterization of the independence model is:

Here \(\lambda\) is the log mean in the reference cell, while \(\alpha_i\) and \(\beta_j\) describe row and column effects relative to the first row and column. The constraints set the reference effects to zero; they do not change the model’s fitted means.

To see the multiplicative form, exponentiate the additive specification:

where \(a_i=\exp(\lambda+\alpha_i)\) and \(b_j=\exp(\beta_j)\), for example. A row effect multiplies the means throughout its row; a column effect multiplies the means throughout its column. Independence is the absence of any further cell-specific adjustment.

What the Model Says About the Table

A positive mean array has this multiplicative form precisely when every pair of rows and columns satisfies the same cross-product relation:

This relation says that comparing two columns gives the same mean ratio in every row. It is one way to check the absence of interaction without relying on a particular choice of reference categories.

Proof. If \(\mu_{ij}=a_i b_j\), then \(\mu_{ij}\mu_{k\ell}=a_i b_j a_k b_\ell=a_i b_\ell a_k b_j=\mu_{i\ell}\mu_{kj}\). Conversely, suppose all the cross-product equalities hold. Fix reference indices \(1,1\). Taking \(k=1,\ell=1\) gives \(\mu_{ij}\mu_{11}=\mu_{i1}\mu_{1j}\), so \(\mu_{ij}=(\mu_{i1}/\mu_{11})\mu_{1j}\). This is a row factor times a column factor. Taking logarithms of positive factors gives an additive row-plus-column form. Thus all three conditions are equivalent. \(\square\)

The mean structure also describes independence of the classifications. Let \(p_{ij}=\mu_{ij}/\mu_{++}\), where \(\mu_{++}=\sum_i\sum_j\mu_{ij}\). Under the multiplicative form, the cell probabilities factor into a row probability and a column probability:

Thus, conditional on the grand total in the independent-Poisson sampling formulation, the row and column classifications of a count are independent. This is the substantive row-column independence represented by the model. It is not the same claim as independence of the Poisson cell counts: under the usual Poisson sampling formulation, the cell counts are specified to be independent across cells whether or not their means satisfy the independence model.

Fitted Means and Observed Margins

Write \(y_{i+}=\sum_j y_{ij}\) and \(y_{+j}=\sum_i y_{ij}\) for the observed row and column totals. The independence model has a useful maximum-likelihood property: when a finite interior maximum-likelihood estimate exists, its fitted means \(\hat\mu_{ij}\) reproduce these margins. This is a statement about fitted totals, not about matching each observed cell.

The reason follows from the score equations. Under independent Poisson sampling, omitting terms that do not depend on the parameters, the log-likelihood is \(\ell=\sum_{i,j}(y_{ij}\log\mu_{ij}-\mu_{ij})\). In the reference-cell parameterization, differentiating with respect to the intercept and each non-reference row and column effect gives:

Proof. At an interior maximum, the displayed derivatives are zero. The intercept equation gives equality of the fitted and observed grand totals. Each non-reference row score equation gives \(\sum_j\hat\mu_{ij}=y_{i+}\) for \(i>1\). Subtracting these row totals from the grand-total equality yields the same equality for row 1. Each non-reference column score equation gives \(\sum_i\hat\mu_{ij}=y_{+j}\) for \(j>1\); subtracting those column totals from the grand-total equality gives the equality for column 1. Hence all fitted row and column margins equal their observed counterparts. The log-likelihood is concave in the loglinear parameters: its Hessian is the negative of a sum of terms \(\mu_{ij}x_{ij}x_{ij}^{T}\), where \(x_{ij}\) is the parameter-design vector for cell \((i,j)\). Therefore a stationary point is a maximum. The stated margin identities hold at any such interior maximum. \(\square\)

The qualification matters. If the data have zero margins or other features that prevent a finite interior estimate, a maximum may lie on the boundary rather than at finite parameter values. The score equations above apply at a finite interior maximum; they should not be used to claim finite estimates in every possible table.

Common Mistakes and Practical Meaning

• Confusing no interaction with no effects. The independence model permits row and column effects. It excludes only an extra row-column term. Equal means throughout the table would be a much stronger restriction.
• Equating independent Poisson counts with row-column independence. Independence of the random cell counts is part of the usual Poisson sampling setup. Row-column independence is a separate restriction on how their means combine.
• Expecting every fitted cell to equal its observed count. The maximum-likelihood fit under independence matches row and column margins, not necessarily individual cells.
• Treating a reference-cell constraint as a scientific assumption. Setting \(\alpha_1=\beta_1=0\) chooses a parameterization. It does not assert that the first row and column have no meaningful effect.

The no-interaction specification is useful because it translates a substantive claim—no association between two classifications—into a compact model for expected counts. Its multiplicative form gives a clear interpretation of the fitted mean pattern, while margin matching explains which summaries the maximum-likelihood fit preserves. Whether the restriction describes the data adequately is a separate question from writing and interpreting the model.