Why Misspecification Changes Inference
An intensity model is a statistical description of how events occur over time, conditional on the observed history. In practice, a chosen model may simplify the true process: its covariate effects may be inadequate, its time pattern may be too restrictive, or its event-count variability may differ from what the model implies. Score equations can still produce a useful estimate in this situation, but the estimate need not describe a true parameter in the data-generating intensity. It instead targets a value defined by the working model and the population score.
This distinction matters for both interpretation and uncertainty. Under misspecification, the estimator may converge to a well-defined pseudo-true parameter, while the usual model-based standard error can be wrong. Sandwich variance estimation uses the observed variability of subject-level scores rather than assuming that the working model correctly supplies that variability. It does not, however, make the working model true or change the target.
Working and True Intensities
For clarity, suppose \(n\) independent subjects are followed over a fixed interval \([0,\tau]\). Let \(N_i(t)\) count events for subject \(i\), let \(Y_i(t)\) indicate whether that subject is under observation and at risk, and let \(X_i(t)\) be a predictable covariate vector. A common working model is \(\lambda_i(t;\beta)=Y_i(t)\exp\{\beta^\mathsf{T}X_i(t)\}\). Here the working intensity is the rate specified by the model, conditional on the observed history. Write \(\lambda_i^0(t)\) for the true intensity under that history. Misspecification means that, in general, \(\lambda_i^0(t)\ne\lambda_i(t;\beta)\) for every fixed \(\beta\).
For this log-linear working model, the subject-level score is \[ U_i(\beta)=\int_0^\tau X_i(t)\{dN_i(t)-\lambda_i(t;\beta)\,dt\}. \] The full score is \(\sum_{i=1}^n U_i(\beta)\), and the estimator \(\hat\beta\) solves its sample score equation. The score remains a computable estimating function even if the working intensity is wrong. Its expectation is taken under the true data-generating process, so \(\beta^*\) is generally a compromise that makes the average working-model score zero, rather than a parameter that reproduces the true intensity at every time and covariate value.
The standard interpretation of this setup requires independent subjects, a well-defined observation process, and predictable covariates for the stated intensity formulation. For large-sample score inference, also require a finite population score mean and covariance near \(\beta^*\), differentiability of the expected score, a nonsingular sensitivity matrix, and sufficient control of extreme subject contributions. The estimator must be an interior solution near \(\beta^*\); boundary estimates or a nearly singular sensitivity matrix require additional care. The form of the score and the observation scheme must also reflect censoring or entry appropriately.
How the Sandwich Variance Arises
Define the sensitivity matrix \(A\) and score-variability matrix \(B\) at the pseudo-true parameter by \[ A=-E\left\{\frac{\partial U_i(\beta^*)}{\partial\beta^\mathsf{T}}\right\}, \qquad B=\operatorname{Var}\{U_i(\beta^*)\}. \] The sign convention makes \(A\) positive in the usual well-behaved log-linear example. If \(A\) is nonsingular, a first-order expansion of the sample score around \(\beta^*\) gives the basis for the covariance formula.
Since \(\sum_i U_i(\hat\beta)=0\), a Taylor expansion around \(\beta^*\) gives \(0\approx\sum_i U_i(\beta^*)+\{\sum_i\partial U_i(\beta^*)/\partial\beta^\mathsf{T}\}(\hat\beta-\beta^*)\).
The average derivative converges to \(-A\), so \(\sqrt n(\hat\beta-\beta^*)\) is approximately \(A^{-1}n^{-1/2}\sum_i U_i(\beta^*)\).
Independent subject scores with mean zero and covariance \(B\) have an average that is asymptotically normal with covariance \(B\). Transforming by \(A^{-1}\) yields \(A^{-1}BA^{-\mathsf{T}}\).
Thus, under the regularity conditions above and a suitable central limit theorem, \[ \sqrt n(\hat\beta-\beta^*)\ \overset{d}{\longrightarrow}\ N\left(0,A^{-1}BA^{-\mathsf{T}}\right). \] Equivalently, the approximate covariance of \(\hat\beta\) is \(A^{-1}BA^{-\mathsf{T}}/n\). The two matrices play different roles: \(A\) describes how rapidly the expected score changes with \(\beta\), while \(B\) describes how much the subject scores vary. Under misspecification, there is no general reason for \(A=B\).
In practice, estimate \(A\) by the negative average derivative of the subject scores at \(\hat\beta\), and estimate \(B\) by their empirical covariance. If \(\bar U=n^{-1}\sum_i U_i(\hat\beta)\), a centered version is \(\hat B=n^{-1}\sum_i\{U_i(\hat\beta)-\bar U\}\{U_i(\hat\beta)-\bar U\}^{\mathsf{T}}\). At an exact score root, \(\bar U=0\). The estimated sandwich covariance is \(\hat A^{-1}\hat B\hat A^{-\mathsf{T}}/n\). This subject-level construction presumes independent subjects; with dependence within clusters, the variability calculation must use independent clusters as its units.
Worked Examples
Worked Example: A Misspecified Poisson Rate
Suppose each subject contributes one unit of observation time and an event count \(N_i\). Hypothetically, the population has \(P(N_i=0)=0.5\) and \(P(N_i=4)=0.5\). Fit the working Poisson model \(N_i\sim\operatorname{Poisson}(\beta)\), even though this two-point count distribution is not Poisson.
The working log-likelihood score for one subject is \(U_i(\beta)=N_i/\beta-1\). Since \(E(N_i)=0(0.5)+4(0.5)=2\), its population score is \(E\{U_i(\beta)\}=2/\beta-1\). Setting this equal to zero gives \(\beta^*=2\) events per unit of observation time. The estimator is \(\hat\beta=\bar N\), so it estimates the mean count, not a rate parameter that makes the true distribution Poisson.
At \(\beta^*=2\), the sensitivity is \(A=E(N_i)/(\beta^*)^2=2/4=0.5\). The true score variance is \(B=\operatorname{Var}(N_i)/(\beta^*)^2=4/4=1\), because \(E(N_i^2)=0.5(0^2)+0.5(4^2)=8\), so \(\operatorname{Var}(N_i)=8-2^2=4\). Therefore the sandwich variance for \(\hat\beta\) is \(A^{-1}BA^{-1}/n=2(1)2/n=4/n\), and its asymptotic standard error is \(2/\sqrt n\) events per unit time.
The working Poisson information instead gives variance \(1/(nA)=2/n\), with standard error \(\sqrt{2/n}\). That model-based variance is too small here because it relies on the Poisson variance relationship, which this hypothetical count distribution violates. The robust calculation supports inference about the mean count \(\beta^*=2\), not about a fully correct Poisson model.
Worked Example: Computing a Two-Parameter Sandwich Estimate
Suppose a hypothetical analysis has \(n=8\) independent subjects and reports the following estimated sensitivity and score-variability matrices: \[ \hat A=\begin{pmatrix}3&0\\0&2\end{pmatrix}, \qquad \hat B=\begin{pmatrix}2&1\\1&4\end{pmatrix}. \] Compute the estimated robust covariance and compare the first coefficient’s standard error with its model-based counterpart.
First, \(\hat A^{-1}=\hat A^{-\mathsf{T}}=\begin{pmatrix}1/3&0\\0&1/2\end{pmatrix}\). The sandwich covariance is \[ \frac{1}{8}\hat A^{-1}\hat B\hat A^{-\mathsf{T}} =\frac{1}{8}\begin{pmatrix}2/9&1/6\\1/6&1\end{pmatrix} =\begin{pmatrix}0.02778&0.02083\\0.02083&0.1250\end{pmatrix}. \] Thus the robust standard error for the first coefficient is \(\sqrt{0.02778}=0.1667\), and for the second it is \(\sqrt{0.1250}=0.3536\), in the respective coefficient units.
If the working-information covariance is used instead, it is \(\hat A^{-1}/8=\operatorname{diag}(1/24,1/16)\). The model-based standard error for the first coefficient is therefore \(\sqrt{(1/3)/8}=\sqrt{1/24}=0.2041\). In this example the robust first-coefficient standard error is smaller, not larger. Robust variance is not automatically an inflation factor; it replaces the model’s variance assumption with an estimate based on score variability.
Worked Example: Interpreting a Pseudo-True Constant Rate
Imagine two types of subjects observed for different total amounts of time. In a hypothetical population, type A contributes 80 units of exposure at a true rate of 1 event per unit, and type B contributes 20 units at a true rate of 3 events per unit. Fit a working model with one constant rate \(\beta\) and no type effect.
For a constant-rate model, the score over all observations is \(U(\beta)=N_{\mathrm{tot}}/\beta-E_{\mathrm{tot}}\), where \(N_{\mathrm{tot}}\) is the total event count and \(E_{\mathrm{tot}}\) is total exposure. The expected events are \(80(1)+20(3)=140\), and the total exposure is \(80+20=100\). Consequently the population score equation \(140/\beta-100=0\) gives \(\beta^*=1.4\) events per unit of exposure.
This value is the exposure-weighted average of the two rates: \(\{80(1)+20(3)\}/100=1.4\). It is not the rate for either type, and it does not describe the differing rates exactly. Changing the population mix or exposure distribution can change this pseudo-true target even if the type-specific rates remain fixed. A confidence interval based on the sandwich therefore concerns the constant-rate score target for the specified population and observation scheme; it does not establish that the two types share a common rate.
What Robust Inference Does—and Does Not—Provide
For a scalar component, a large-sample Wald interval has the form \(\hat\beta_j\pm z_{1-\alpha/2}\widehat{SE}_{\mathrm{robust}}(\hat\beta_j)\). With regularity conditions and a consistent sandwich estimator, repeated-sample coverage approaches the stated nominal level for the corresponding component of \(\beta^*\). The interval is not a probability statement that the fixed parameter lies within this particular realized interval. For a nonlinear contrast, such as \(\exp(\beta_j^*)\), use the delta method or an appropriate transformation of the interval and state the resulting scale and units clearly.
The main alternative is model-based information variance, which is usually more efficient when the complete working intensity model is correct and its assumptions hold. Under correct specification, the score is a martingale integral under the modeled intensity, and the information identity gives matching sensitivity and score variability under standard conditions. Under misspecification, that identity generally fails. Robust variance is designed to handle the resulting difference, but it cannot repair a biased or scientifically unsuitable target, a wrong dependence unit, unaccounted informative observation, or a lack of independent sampling units.
| Approach | Variance basis | What it supports | Main caution |
|---|---|---|---|
| Model-based information | Working-model information, commonly \(\hat A^{-1}/n\) | Efficient inference when the model is correctly specified | May misstate uncertainty if the working variance structure fails |
| Robust sandwich | Empirical subject-score variability, \(\hat A^{-1}\hat B\hat A^{-\mathsf{T}}/n\) | Large-sample inference for the pseudo-true score target | Needs enough independent units and does not make the target scientifically appropriate |
Useful warning signs include a poorly conditioned \(\hat A\), a very small number of independent subjects, a few unusually influential subject scores, and sensitivity of \(\hat\beta\) to reasonable changes in the working model or follow-up window. These do not by themselves prove misspecification, but they can make the approximation unstable or the target hard to interpret. Examine the score contributions and compare estimates under scientifically plausible working specifications. The purpose is to assess stability and meaning, not to infer that a robust standard error validates the intensity model. Detailed residual-based diagnostics are a separate tool.
Common Mistake
A frequent error is to say, “The sandwich standard error is robust, so the coefficient estimates the true intensity effect even if the model is wrong.” That conclusion does not follow. The score equation determines the estimand: under misspecification, it is generally the pseudo-true parameter \(\beta^*\), which depends on the model family and the observed population. The sandwich changes the estimated sampling variance around that target; it does not replace the score equation, eliminate misspecification bias relative to another scientific parameter, or justify a causal interpretation.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In one sentence, distinguish the pseudo-true parameter from a true intensity parameter.
- If \(A=2\), \(B=8\), and \(n=50\) in a scalar model, calculate the sandwich variance and standard error.
- Why can a robust standard error be smaller than a model-based standard error?
- Name two conditions or design features that can invalidate the usual subject-level sandwich calculation.
Solutions. (1) The pseudo-true parameter is the value that makes the expected working-model score zero under the actual data distribution; it need not parameterize the true intensity. (2) The variance is \(A^{-1}BA^{-1}/n=(1/2)(8)(1/2)/50=2/50=0.0400\), so the standard error is \(\sqrt{0.0400}=0.200\) in coefficient units. (3) The empirical score variability can be smaller than the variability implied by the working-model information, so the sandwich need not inflate the model-based result. (4) Examples include dependent subjects treated as independent, too few independent units for reliable large-sample approximation, an incorrectly handled informative observation process, or a singular sensitivity matrix.