Factorial Interaction Analysis Worked Example

Why an Interaction Is a Difference of Differences

In a factorial microarray experiment, the response for a gene may depend on two factors, such as treatment and genotype. An interaction asks whether the treatment effect differs between genotypes. It is not enough to observe that expression changes in one group or that a gene has a large expression value: the interaction is a comparison of changes.

For a two-by-two design, let \(m_{00},m_{01},m_{10},m_{11}\) denote the four expected log-expression levels, where the first index indicates factor A and the second factor B. The interaction of interest can be written as the change in B's effect across A levels:

If \(\theta=0\), the expected difference between the B levels is the same at both A levels. A nonzero value represents a departure from additivity on the modeled log-expression scale. The sign depends on the ordering of the factor levels, so it must be stated.

Model and Moderated Test

For each gene \(g\), model the normalized log-expression measurements as

Here \(\mathbf{y}_g\) contains the sample measurements for gene \(g\), \(X\) is the design matrix shared across genes, and \(\boldsymbol{\beta}_g\) contains its coefficients. The model assumes that the mean is correctly represented by \(X\boldsymbol{\beta}_g\), that residuals are independent with mean zero, and that the residual variance is constant across samples for this gene. Normal residuals support the small-sample reference distribution. The design must also have enough independent information to estimate the requested coefficients; confounded factor effects cannot be separated by fitting an interaction term.

A cell-means parameterization assigns one coefficient to each factor combination. Then the contrast vector for the two-by-two interaction is \(\mathbf{c}=(1,-1,-1,1)^\mathsf{T}\), when cell means are ordered as \((m_{00},m_{01},m_{10},m_{11})\). Thus \(\hat{\theta}_g=\mathbf{c}^\mathsf{T}\hat{\boldsymbol{\beta}}_g\). Reference-level coding gives different coefficient names, but the fitted cell means and the specified contrast do not change.

Let \(s_g^2\) be the gene-wise residual variance estimate with \(df_g\) residual degrees of freedom. Empirical-Bayes moderation combines this estimate with the across-gene prior variance \(s_0^2\) and prior degrees of freedom \(d_0\):

For a prespecified contrast matrix \(C\) with rank \(q\), the moderated F-statistic is

Under the linear-model and variance-prior assumptions, the reference distribution is \(F_{q,\,df_g+d_0}\). For a single contrast, \(q=1\), the statistic equals the square of the moderated t-statistic. The test concerns \(H_0:C\boldsymbol{\beta}_g=\mathbf{0}\); it does not test whether the gene has any expression or any effect outside the selected contrast.

Worked Analyses

Justification and Design Checks

The standard error calculation in the balanced two-by-two example follows directly from variance propagation. If the four sample cell means are independent and each is based on \(n\) samples with common residual variance \(\sigma_g^2\), each cell-mean variance is \(\sigma_g^2/n\). For \(\hat{\theta}=\bar{Y}_{11}-\bar{Y}_{10}-\bar{Y}_{01}+\bar{Y}_{00}\), independence makes the variance of the sum equal the sum of the four variances; the signs disappear on squaring:

Replacing \(\sigma_g^2\) by the moderated estimate gives \(SE=\sqrt{4s_{g,\mathrm{post}}^2/n}\). In an unbalanced design, or with a more complex design matrix, do not use this shortcut: calculate the contrast variance as \(s_{g,\mathrm{post}}^2 C(X^\mathsf{T}X)^{-1}C^\mathsf{T}\). This accounts for the actual allocation and coefficient covariance.

Before interpreting a result, verify that all four cells are represented, the design matrix is full rank for the requested effects, and sample labels and factor reference levels are correct. The linear model's independence assumption is especially important: repeated measurements or matched samples require an appropriate design representation, not a claim that all observations are independent. Inspect residual patterns for serious outliers or changing variance. Empirical-Bayes moderation stabilizes gene-wise variance estimates using information across genes; it does not repair a misspecified mean model, confounding, dependence, or poor normalization.

Choosing and Interpreting the Interaction Test

A prespecified contrast answers a focused question, such as whether the treatment effect differs between two selected genotypes. An omnibus interaction test instead tests a collection of interaction coefficients jointly. In a two-by-two design, there is only one interaction degree of freedom, so the planned contrast and omnibus interaction test coincide when they express the same comparison. With more factor levels, an omnibus test can detect some interaction pattern without identifying which particular contrast is responsible; a targeted contrast is more directly interpretable but only tests the specified pattern.

The interaction is scale-dependent. In this tutorial the model is fitted to log2 expression, so additivity and interaction refer to that scale. Exponentiating a log2 difference-in-differences yields a ratio of fold changes. It does not establish that the treatment causes an expression change unless the experimental design and study assumptions support a causal interpretation. Nor does a small p-value imply a large biological effect; report \(\hat{\theta}_g\), its units, and uncertainty alongside the test.

• Wrong: “Treatment is significant in A1, but not significant in A0, so the treatment-by-genotype interaction is significant.”
• Correction: A difference between separate significance decisions is not itself a test of the difference. Test the contrast \((m_{11}-m_{10})-(m_{01}-m_{00})\) directly.
• Wrong: Treating a positive interaction coefficient as inherently stronger treatment response without stating the factor coding.
• Correction: State the factor order and contrast direction, then interpret the sign in those terms.