Why the Order of Pooling and Prediction Matters
A fitted regression model can be used to predict an outcome for a specified covariate profile. With multiple imputations, there are several fitted models—one for each completed data set—and therefore several predictions for that same profile. One option is to calculate a prediction from every model and average those predictions. Another is to pool the model coefficients first and then calculate a prediction from the pooled coefficients. For an identity link these procedures agree; for a nonlinear link they generally do not.
The difference is not merely a computational detail. A predicted probability, for example, is a nonlinear function of the regression coefficients. Averaging probabilities and converting an average linear predictor to a probability are distinct operations. To interpret either result, first state the prediction target and then keep that target the same across imputations.
These quantities refer here to a model's predicted mean response at a specified profile, such as an expected outcome or a probability. They are not, by themselves, predictions of an individual's realized outcome: a realized outcome also includes outcome variability beyond uncertainty about the fitted mean.
When the Two Predictions Agree
For an identity-link model, \(g(\eta)=\eta\). At the same fixed profile in every imputation, averaging the imputation-specific predictions is exactly equivalent to predicting from the averaged coefficient vector:
The equality follows because multiplication by a fixed vector and averaging are linear operations. It requires the same \(\mathbf{x}\) and the same coefficient definitions in every imputation. For a nonlinear inverse link, however, taking an average and applying \(g\) do not generally commute:
The two sides can be close, particularly when imputation-specific estimates are similar, but closeness is not an identity and should not be assumed. The discrepancy depends on the spread of the linear predictors and on the curvature of the inverse link over the range of those predictors.
Worked Example: A Linear-Model Prediction
Suppose two hypothetical completed data sets produce linear regression coefficients \((\hat{\beta}_{0,1},\hat{\beta}_{1,1})=(1,2)\) and \((\hat{\beta}_{0,2},\hat{\beta}_{1,2})=(3,4)\). Predict the mean outcome at \(x=2\), using the model \(\hat{y}=\hat{\beta}_0+\hat{\beta}_1x\).
The two predictions are \(1+2(2)=5\) and \(3+4(2)=11\). Their average is \((5+11)/2=8\). The pooled coefficients are \(\bar{\beta}_0=(1+3)/2=2\) and \(\bar{\beta}_1=(2+4)/2=3\), so the prediction from those coefficients is \(2+3(2)=8\). The results agree because this is an identity-link model and the profile \(x=2\) is held fixed.
Nonlinear Links: Logistic-Regression Predictions
In logistic regression, the linear predictor is converted to a probability by the inverse-logit function \(g(\eta)=1/(1+\exp(-\eta))\). The nonlinearity means that an average of fitted probabilities need not equal the probability calculated from an average linear predictor. In particular, averaging coefficients and then applying the inverse link is not generally a substitute for calculating the probability in each imputation and averaging those probabilities.
Worked Example: Two Logistic Predictions at One Profile
Consider a hypothetical binary-outcome analysis. At the same covariate profile, suppose the two imputation-specific fitted linear predictors are \(\hat{\eta}_1=0\) and \(\hat{\eta}_2=2\). Calculate both approaches.
The inverse-logit predictions are \(g(0)=1/(1+\exp(0))=0.5000\) and \(g(2)=1/(1+\exp(-2))\approx0.8808\). Averaging them gives
The average linear predictor is \((0+2)/2=1\). Predicting from that average gives \(g(1)=1/(1+\exp(-1))\approx0.7311\). Thus the averaged imputation-specific probability is 0.6904, while the probability from the average linear predictor is 0.7311. They differ by about 0.0407, or 4.07 percentage points. Neither arithmetic is wrong; they are different summaries of the imputation-specific model results.
A short Taylor expansion explains why the difference depends on both spread and curvature. Write \(\bar{\eta}=M^{-1}\sum_m\hat{\eta}_m\). When the imputation-specific predictors are near \(\bar{\eta}\), expand \(g(\hat{\eta}_m)\) around \(\bar{\eta}\). The first-order terms average to zero because \(\sum_m(\hat{\eta}_m-\bar{\eta})=0\). Keeping the second-order term gives the approximation
This is a local approximation, not an exact identity. It shows that, to second order, the gap depends on the curvature \(g''\) and the dispersion of the predictors. For the logistic inverse link, \(g''(\eta)=p(1-p)(1-2p)\), where \(p=g(\eta)\). The function is convex when \(\eta<0\) and concave when \(\eta>0\); accordingly, the direction of the difference is not fixed for every set of imputations.
Worked Example: Curvature Can Change the Direction
Suppose three hypothetical imputations give the linear predictors \(-3\), \(-1\), and \(1\) at one fixed profile. The corresponding logistic probabilities are approximately \(0.0474\), \(0.2689\), and \(0.7311\). Their average is
The average linear predictor is \((-3-1+1)/3=-1\). Its inverse-logit is \(g(-1)=1/(1+\exp(1))\approx0.2689\). Here the average probability is greater than the probability from the average predictor, by about 0.0802, or 8.02 percentage points. In the previous example, the average probability was smaller. Nonlinearity alone does not determine the direction; the range of predictors relative to the link's curvature matters.
Pool Predictions and Their Uncertainty
When the target is the predicted mean response at a common profile, a direct multiple-imputation analysis can define \(q_m\) as that prediction in completed data set \(m\), and \(u_m\) as its estimated within-imputation variance. The usual scalar Rubin pooling quantities are
Here \(\bar{U}\) represents average estimated variance within completed data sets, \(B\) represents variation among the imputation-specific estimates, and \(T\) is the estimated total variance of the pooled estimate. The standard error is \(\sqrt{T}\). The \(u_m\) must be variances for the same prediction estimand and scale as \(q_m\); a standard error is not a variance. For a logistic predicted mean, for instance, \(u_m\) may be estimated on the probability scale using the model's coefficient covariance and the derivative of the inverse link. These variances represent uncertainty in the estimated mean probability, not the full variability of a future binary outcome.
Worked Example: Pooling a Predicted Mean and Its Variance
For a hypothetical continuous-outcome prediction, suppose \(M=2\) completed data sets give predicted means \(q_1=10\) and \(q_2=12\) outcome units, with within-imputation variances \(u_1=4\) and \(u_2=9\) outcome-units-squared. The pooled estimate is \(\bar{q}=(10+12)/2=11\) outcome units, and \(\bar{U}=(4+9)/2=6.5\) outcome-units-squared. The between-imputation variance is
Therefore \(T=6.5+(1+1/2)(2)=9.5\) outcome-units-squared, and the standard error is \(\sqrt{9.5}\approx3.082\) outcome units. This calculation pools the prediction as the scalar estimand. It is not the same operation as pooling coefficients and then transforming them, and the point estimate here happens to be on a continuous outcome scale.
Choosing and Checking the Target
For an inferential report, choose the target before choosing the order of calculation. If the target is the average of the completed-data estimates of a predicted mean at one profile, calculate that predicted mean in each imputation and pool those estimates with their variances. If instead the target is explicitly a function evaluated at the pooled coefficient vector, report that quantity as such; it is not automatically the same estimand as the average imputation-specific prediction. The uncertainty calculation must match the chosen target and transformation.
In ordinary use, the following conditions are important for a meaningful comparison and pooling calculation:
- The same outcome model, link, coefficient parameterization, and covariate profile are used in every imputation.
- Each prediction refers to the same estimand, outcome definition, and scale. For example, do not average probabilities in some imputations and log odds in others.
- The imputation and analysis procedures support valid multiple-imputation inference under the applicable missing-data and model assumptions. Rubin pooling does not repair a misspecified outcome model or incompatible estimands.
- The within-imputation variance \(u_m\) correctly represents uncertainty in the estimate being pooled. For nonlinear predictions, it must account for the transformation, not simply reuse an untransformed coefficient variance.
Useful checks include tabulating \(\hat{\eta}_m\) and \(q_m\) across imputations, verifying that all profiles use the intended reference categories and transformations, and confirming that the reported result is on the promised scale. Large variation in \(\hat{\eta}_m\), predictions near probability boundaries, or predictions outside a scientifically plausible range can flag a need to inspect the fitted models and imputed covariates. A numerical difference between \(\bar q\) and \(g(\mathbf{x}^{\mathsf T}\bar{\boldsymbol{\beta}})\) is expected under a nonlinear link; by itself it is not evidence of a fitting failure.
How the Approaches Compare
| Approach | Calculation | Interpretation and caution |
|---|---|---|
| Average imputation-specific predictions | Calculate \(q_m=g(\mathbf{x}^{\mathsf T}\hat{\boldsymbol{\beta}}_m)\), then average across imputations | Directly pools estimates of the prediction-scale target; pair with the corresponding \(u_m\) values. |
| Predict from pooled coefficients | Calculate \(g(\mathbf{x}^{\mathsf T}\bar{\boldsymbol{\beta}})\) | Evaluates the inverse link at the pooled linear predictor; generally differs from the first approach for nonlinear \(g\). |
| Average linear predictors, then apply the link | Calculate \(g(M^{-1}\sum_m\hat{\eta}_m)\) | At a fixed profile this equals prediction from the averaged coefficient vector, but not generally the average of imputation-specific predictions. |
No method removes the need to state the estimand. Also, a prediction for one fixed profile is not a population-average prediction over a distribution of covariates. Averaging across profiles introduces an additional averaging operation and a different target; it should not be confused with averaging across imputations.
Check Your Understanding
Answer each question, then compare with the solutions.
- For an identity-link model at a fixed profile, why does averaging the imputation-specific predictions equal predicting from the average coefficient vector?
- Two imputations give logistic linear predictors \(-1\) and \(1\). Calculate the average probability and the probability from the average linear predictor.
- State the roles of \(\bar{U}\) and \(B\) in pooling a scalar prediction estimate.
- A researcher uses the same coefficient vector in every imputation but changes the covariate profile from one imputation to another. Can the fixed-profile equality be applied directly? Explain.
Solutions. (1) With an identity link, each prediction is \(\mathbf{x}^{\mathsf T}\hat{\boldsymbol{\beta}}_m\); linearity gives \(M^{-1}\sum_m\mathbf{x}^{\mathsf T}\hat{\boldsymbol{\beta}}_m=\mathbf{x}^{\mathsf T}\bar{\boldsymbol{\beta}}\). (2) The probabilities are \(g(-1)\approx0.2689\) and \(g(1)\approx0.7311\), so their average is \(0.5000\). The average predictor is zero, and \(g(0)=0.5000\); they happen to agree by symmetry, not because the logistic link is linear. (3) \(\bar{U}\) is the average within-imputation variance; \(B\) is the sample variance of the imputation-specific estimates and captures between-imputation variation. Rubin's total variance combines them as \(T=\bar{U}+(1+1/M)B\). (4) No. The proof requires the same fixed \(\mathbf{x}\). With changing profiles, differences may reflect both the nonlinear link and the changed prediction target, so first define a common target and calculate it consistently.