Summing Predictions Across Groups

From Group Predictions to an Expected Total

A fitted count model often produces a predicted mean for each of several groups: for example, expected admissions in each hospital or expected events in each age band. If the question concerns the expected number across all the groups together, the group means must be summed. The uncertainty of that sum also matters. Because each group prediction usually uses the same estimated regression coefficients, their errors are generally correlated.

This tutorial develops the sum and its coefficient-based variance. The target is the total of the groups’ conditional expected counts, not the realized count that might be observed later. The distinction is important: adding fitted means answers an expectation question, while predicting a future observed count requires additional distributional variation.

Each \(\boldsymbol{x}_g\) describes the covariate profile used for that group. The offset can, for example, represent a known exposure on the log scale. It is treated as fixed here. If the target is a weighted total, with fixed weights \(w_g\), replace the definition by \(T(\boldsymbol{\beta})=\sum_g w_g\mu_g(\boldsymbol{\beta})\); the same derivation applies with \(w_g\) included in each term.

Deriving the Variance of the Sum

Write \(\widehat{\boldsymbol V}\) for the estimated covariance matrix of \(\widehat{\boldsymbol{\beta}}\). In a large-sample calculation, the delta method propagates this coefficient uncertainty through a differentiable function of the coefficients. For a fitted mean, differentiating the exponential gives \(\nabla_{\boldsymbol{\beta}}\mu_g(\boldsymbol{\beta})=\mu_g(\boldsymbol{\beta})\boldsymbol{x}_g\). Differentiation is linear, so the gradient of the total is the sum of these group gradients.

The result follows by applying the multivariate delta method to \(T\). To state the approximation precisely, suppose that for the true coefficient vector \(\boldsymbol{\beta}_0\), \(\sqrt n(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)\xrightarrow{d}N(\boldsymbol{0},\boldsymbol{\Sigma})\), with \(\widehat{\boldsymbol V}\) consistently estimating \(\boldsymbol{\Sigma}/n\). The function \(T\) is differentiable, and its gradient at \(\boldsymbol{\beta}_0\) is \(\boldsymbol{a}_0=\sum_g\mu_g(\boldsymbol{\beta}_0)\boldsymbol{x}_g\). The delta method therefore gives \(\sqrt n\{T(\widehat{\boldsymbol{\beta}})-T(\boldsymbol{\beta}_0)\}\xrightarrow{d}N(0,\boldsymbol{a}_0^\mathsf{T}\boldsymbol{\Sigma}\boldsymbol{a}_0)\). Replacing the unknown gradient and covariance by their estimates yields the stated variance estimate.

The same calculation makes the covariance structure explicit. Let \(\widehat{\boldsymbol{\mu}}\) be the vector of fitted means. The Jacobian matrix of the map from \(\boldsymbol{\beta}\) to this vector has row \(g\) equal to \(\widehat\mu_g\boldsymbol{x}_g^\mathsf{T}\). Thus the delta-method covariance estimate for the vector of means is \(\widehat{\boldsymbol J}\widehat{\boldsymbol V}\widehat{\boldsymbol J}^\mathsf{T}\). Its \((g,h)\) entry is \(\widehat\mu_g\widehat\mu_h\boldsymbol{x}_g^\mathsf{T}\widehat{\boldsymbol V}\boldsymbol{x}_h\).

To verify the equivalent form, set \(\widehat{\boldsymbol a}=\sum_g\widehat\mu_g\boldsymbol{x}_g\). Expanding the matrix product gives \(\widehat{\boldsymbol a}^\mathsf{T}\widehat{\boldsymbol V}\widehat{\boldsymbol a} =\sum_g\sum_h\widehat\mu_g\widehat\mu_h\boldsymbol{x}_g^\mathsf{T}\widehat{\boldsymbol V}\boldsymbol{x}_h\). This is precisely the sum of the entries of the estimated covariance matrix of the fitted means, as required for the variance of their sum. In particular, the off-diagonal terms need not all be positive; their signs depend on the coefficient covariance and the group design vectors. The overall variance estimate remains nonnegative when \(\widehat{\boldsymbol V}\) is positive semidefinite, because it is a quadratic form in that matrix.

Worked Calculations

Why the Covariance Cannot Usually Be Dropped

A frequent mistake is to calculate a standard error for each fitted mean, square those standard errors, and add them as if the predictions were independent. In the first example, that would give \(4+33=37\), rather than \(57\), because it omits twice the covariance of \(10\). Predictions for groups sharing an intercept or other coefficients are typically dependent as estimates, even when the groups themselves are distinct.

The fitted total is a sum of functions of one common coefficient estimate. Its variance must reflect the uncertainty in that common estimate. In some cases covariances are negative, and omitting them can instead overstate the variance. The safe calculation is to form the total gradient first and evaluate its quadratic form with the coefficient covariance matrix, or to sum all entries of the fitted-means covariance matrix.

This calculation concerns uncertainty in the model-based expected total arising from estimating the coefficients. It does not add the random variation of a future count around its conditional mean. Nor does it automatically account for uncertainty in offsets, group definitions, or model specification. Those require additional assumptions and calculations beyond this coefficient-based variance.