Products And Interaction Contrasts From Regression Models

Why Products Need Their Own Pooling Calculation

Some scientific questions concern a product of regression coefficients: for example, whether the product of two estimated slopes differs between groups, or the size of a product-defined pathway. With multiple imputation, the product must first be calculated in each completed data set. Its uncertainty within that imputation depends on the uncertainties of both coefficients and on their covariance. The resulting product estimates and variances can then be pooled as a scalar estimand.

This is not the same as multiplying two coefficients that have already been pooled. In general, the average of imputation-specific products is not the product of the two average coefficients. Nor is a product automatically the same thing as a regression model’s usual interaction coefficient. In a linear model containing \(A\), \(Z\), and \(AZ\), for instance, the coefficient of \(AZ\) describes the model’s specified interaction term; the product \(\beta_A\beta_Z\) is a different, nonlinear estimand.

The coefficients must refer to a clearly defined pair of parameters from the same model or from models whose joint covariance is available. Their units determine the product’s units. For example, if the coefficients have units of outcome per unit \(A\) and outcome per unit \(Z\), their product has units of outcome squared per unit \(A\) per unit \(Z\). That product may be mathematically well defined without being a directly interpretable causal effect.

Deriving the Within-Imputation Variance

The delta method approximates the variability of a smooth function of estimated coefficients by linearizing that function near the estimates. For \(g(\beta_1,\beta_2)=\beta_1\beta_2\), the gradient is \((\beta_2,\beta_1)\). Applying the covariance-matrix formula for the variance of a linear combination gives the product variance estimate.

The covariance contribution is zero if \(U_{12,i}=0\) or if either estimated coefficient is zero. Otherwise it must be included. A positive covariance increases the estimated variance when the coefficient product is positive, and decreases it when the product is negative; a negative covariance has the reverse effect. The whole expression remains nonnegative when the estimated covariance matrix is positive semidefinite.

Pooling Product Estimates Across Imputations

Assume there are \(m\) imputations, each estimating the same product estimand on the same scale. For each imputation, calculate \(Q_i\) and its delta-method variance \(U_i\). Rubin’s scalar pooling quantities are the mean estimate \(\bar{Q}\), the average within-imputation variance \(\bar{U}\), the between-imputation variance \(B\), and the total variance \(T\):

The pooled standard error is \(\sqrt{T}\). Under the usual scalar Rubin degrees-of-freedom approximation with effectively infinite complete-data degrees of freedom, the degrees of freedom are \(df=(m-1)\{1+\bar{U}/[(1+1/m)B]\}^2\), when \(B>0\). A \(t\)-based interval is \(\bar{Q}\pm t_{1-\alpha/2,df}\sqrt{T}\). This approximation relies on the usual multiple-imputation conditions: the imputations adequately represent uncertainty under the adopted missing-data and imputation models, the complete-data estimates and variance estimates are suitable for Rubin pooling, and the product’s sampling distribution is adequately approximated by the first-order delta method. In particular, the same coefficients and estimand must be defined comparably in every imputation.

Contrasts Between Products

Sometimes the target is a difference between product-defined quantities, such as a product in one group minus the corresponding product in another. Define \(C=g_1-g_0=\beta_{11}\beta_{21}-\beta_{10}\beta_{20}\). Its within-imputation delta variance must use the joint covariance matrix of all four coefficients. The gradient, in the order \((\beta_{11},\beta_{21},\beta_{10},\beta_{20})\), is \((\beta_{21},\beta_{11},-\beta_{20},-\beta_{10})\). If \(V_i\) is their full estimated covariance matrix, the variance is the gradient transposed, multiplied by \(V_i\), and multiplied by the gradient. Cross-group covariance terms therefore matter whenever the estimates are correlated.

Assumptions, Diagnostics, and Alternatives

The delta method is a local, first-order approximation. It is most defensible when the coefficient estimates are reasonably concentrated, the product function changes smoothly over their uncertainty region, and the joint covariance estimates are reliable. It can be poor when coefficients are highly uncertain, when a product distribution is strongly skewed, or when estimates lie near a region where the linear approximation does not describe the function well. A confidence interval spanning zero is not by itself evidence that the approximation failed; inspect the coefficient uncertainties and the implied product distribution as well.

Useful checks include confirming that every imputation uses the same coefficient definitions, units, coding, and target population; checking that the covariance matrix is symmetric and positive semidefinite; and independently verifying each product and its variance. For a product contrast, retain the full covariance among all coefficients, not only the two variances within each product. Examine whether a few imputations produce extreme products or unusually large variances. Such values may signal model instability, separation in a fitted model, a coding mismatch, or genuine sensitivity to missing-data uncertainty.

The main alternative is to obtain uncertainty for the nonlinear estimand by a method that better captures its sampling distribution, such as a suitable bootstrap carried out within each imputation and combined under a defensible procedure. Such alternatives require care: pooling standard errors or averaging test statistics is not a substitute for pooling estimates and variances. A different option is to report the underlying coefficients and their covariance as well as the product, so readers can see which estimates drive it. No method removes the need for a well-defined estimand and adequate imputation and outcome models.

A frequent mistake is to multiply the separately pooled coefficients and treat the result as the pooled product. In general, \(\frac{1}{m}\sum_i\hat{\beta}_{1i}\hat{\beta}_{2i}\ne (\frac{1}{m}\sum_i\hat{\beta}_{1i})(\frac{1}{m}\sum_i\hat{\beta}_{2i})\). Another mistake is to omit the covariance term because coefficients appear in separate rows of a software table. Separate reporting does not imply zero covariance. Use the covariance matrix, and omit the contribution only when the covariance estimate is zero or either estimated coefficient is zero.