The Pooled Point Estimate Across Imputations

Why Pool Point Estimates?

Multiple imputation produces several completed datasets, each reflecting a different plausible realization of the missing values. Applying the same analysis to each dataset gives several estimates of the quantity of interest. The pooled point estimate summarizes those estimates in one number. It is not the estimate from a specially selected “best” completed dataset, and it is not an average of the completed data values themselves.

The central requirement is that each analysis estimates the same target on the same scale. If one estimate is a mean in milligrams and another is a mean in grams, they cannot be averaged as written. More subtly, estimates from different model specifications or of different parameters cannot be made comparable just by taking their average. The estimand and notation introduced in the earlier material on estimands and notation for multiple-imputation pooling provide the reference point: each completed-data estimate must correspond to the same $Q$.

Definition: Let $M$ be the number of completed datasets. Let $Q_m$ be the estimate of a fixed target quantity $Q$, obtained by applying the same analysis to completed dataset $m$, for $m=1,\ldots,M$. The pooled point estimate is $$ \bar{Q}=\frac{1}{M}\sum_{m=1}^{M}Q_m. $$ Each estimate receives equal weight in this point-estimate calculation.

The formula is simple, but using it responsibly requires checking what each $Q_m$ represents. The arithmetic mean always returns a number; it does not guarantee that the number answers a meaningful statistical question. The completed datasets must be analyzed consistently, with the same outcome, predictor definitions, model, contrast, and units. For inferential results to be valid, the imputation procedure must also adequately represent uncertainty about the missing values under the assumptions of the analysis. The arithmetic formula itself cannot repair a poor imputation model.

Calculating the Equal-Weight Mean

Suppose five completed datasets yield five estimates of a population mean. The pooled point estimate is their sum divided by five. No estimate is given more weight because its completed dataset happens to produce a smaller standard error. The point-estimate rule is the arithmetic mean across imputations; the uncertainty calculations are separate parts of multiple-imputation pooling.

Worked Example: A Mean Outcome Across Five Imputations

In a hypothetical analysis, the outcome is a clinical measurement in milligrams per deciliter. Applying the same mean-estimation procedure to five completed datasets gives $Q_1=68.0$, $Q_2=70.0$, $Q_3=67.0$, $Q_4=71.0$, and $Q_5=69.0$ mg/dL. Assume each dataset contains the same analysis units and the procedure estimates the same mean.

The pooled point estimate is $$ \bar{Q}=\frac{1}{5}(68.0+70.0+67.0+71.0+69.0) =\frac{345.0}{5}=69.0\text{ mg/dL}. $$

Thus the pooled point estimate of the mean is 69.0 mg/dL. This is a summary of the five estimates. It does not mean that any one completed dataset necessarily had a mean of exactly 69.0 mg/dL, nor does it quantify the uncertainty of the pooled estimate.

The imputation index $m$ labels completed datasets, not people, variables, or repeated measurements on the same participant. In this calculation, each completed dataset supplies one estimate of $Q$. If the analysis is a regression, for example, $Q_m$ might be one specified coefficient from the fitted model on dataset $m$. The same coefficient must be selected each time.

Worked Example: Pooling a Regression Slope

Suppose five hypothetical completed datasets are analyzed with the same linear regression model. The predictor is age in years, and the outcome is systolic blood pressure in mmHg. The coefficient of age is the target in every fit. The resulting slopes are 0.42, 0.38, 0.47, 0.43, and 0.40 mmHg per year. Assume the outcome, predictor coding, included covariates, and model specification are identical across fits.

The pooled slope is $$ \bar{Q}=\frac{0.42+0.38+0.47+0.43+0.40}{5} =\frac{2.10}{5}=0.420\text{ mmHg per year}. $$

The pooled point estimate is 0.420 mmHg per year. Under this model, the estimated conditional mean systolic blood pressure is higher by 0.420 mmHg for each additional year of age, holding the other included covariates fixed. This is an association described by the regression model; the calculation alone does not establish a causal effect.

If one fit instead used age in decades, its coefficient would be on a different numerical scale: a coefficient per decade is ten times the coefficient per year. Convert it to the common per-year scale before averaging, or consistently express all five slopes per decade. Averaging coefficients with mixed units would have no coherent interpretation.

Why the Same Scale Matters

A parameter can be represented on more than one scale. For example, a logistic regression coefficient is expressed in log-odds units, while its exponentiation is an odds ratio. Both describe the same modeled contrast through a transformation, but they are not interchangeable numbers. The scale chosen for $Q$ must be fixed before averaging. If the analysis targets a log odds ratio, average the log-coefficient estimates. If a summary on the odds-ratio scale is desired, transform the pooled log estimate afterward.

Worked Example: Pooling Log Odds Ratios

Consider a hypothetical binary-outcome analysis using the same logistic regression specification on three completed datasets. The coefficient in each fit estimates the same log odds ratio for a specified predictor contrast. Suppose the exponentiated coefficients, or odds ratios, are 1.20, 1.50, and 1.10. To pool on the log-odds scale, first take logarithms: $$ Q_1=\log(1.20)=0.1823,\quad Q_2=\log(1.50)=0.4055,\quad Q_3=\log(1.10)=0.09531. $$

Average the coefficients on their common log scale: $$ \bar{Q}=\frac{0.1823+0.4055+0.09531}{3} =\frac{0.68311}{3}=0.2277. $$

Exponentiating this pooled log estimate gives $\exp(0.2277)\approx1.256$. Thus the point estimate reported on the odds-ratio scale is approximately 1.256. It is the exponentiated arithmetic mean of the log odds ratios, equivalently their geometric mean on the odds-ratio scale. It is not their arithmetic mean, which would be $(1.20+1.50+1.10)/3=1.267$. The two calculations differ because exponentiation is nonlinear.

This example illustrates a general rule: choose the scale that matches the analysis and the intended target, pool estimates consistently on that scale, and only then apply a transformation if needed. Transforming each estimate first and then taking an arithmetic average generally produces a different result. Which transformed summary is appropriate depends on the target and the pooling procedure, so the scale should not be changed casually.

Justification for the Arithmetic Mean

The equal-weight arithmetic mean has a direct interpretation. Imagine selecting one of the $M$ completed datasets at random, with probability $1/M$ for each, and then reporting its estimate. The mean of this finite distribution of estimates is

E_{\text{finite}}(Q)=\sum_{m=1}^{M}\frac{1}{M}Q_m =\frac{1}{M}\sum_{m=1}^{M}Q_m=\bar{Q}.

Thus $\bar{Q}$ is the average estimate across the equally represented imputations. In multiple imputation, each completed dataset represents a draw that accounts for uncertainty about missing values. Averaging the estimates summarizes the center of those repeated analyses. If the imputations adequately represent the relevant uncertainty and the analyses target the same $Q$, this average serves as the multiple-imputation point estimate.

This justification does not say that $\bar{Q}$ is automatically unbiased, nor that it is precise. Its statistical properties depend on the imputation and analysis models, as well as on the data and target. It also does not mean that every possible estimate should receive equal weight in every statistical problem. The equal-weight rule here is the standard point-estimate pooling rule across the $M$ imputations; it is not a claim that all other forms of weighting are invalid for unrelated purposes.

Consistency Checks and Failure Modes

Before calculating $\bar{Q}$, verify that the values being averaged truly are estimates of the same quantity. A practical check is to write down the target in words and then compare the analysis that produced each estimate. Ask whether the same outcome, population or analysis sample, model terms, contrast, coefficient coding, and measurement units were used. The completed datasets will differ in imputed values, so their estimates need not be identical; that variation is expected. The question is whether the analysis definition stayed fixed.

Different targets: an adjusted coefficient and an unadjusted coefficient usually estimate different quantities. Averaging them does not produce a pooled estimate of either one.
Different units: a mean expressed in grams cannot be averaged directly with one expressed in kilograms. Convert estimates to a common unit first.
Different parameterizations: coefficients for a binary predictor coded 0/1 and 1/2 represent the same contrast only after accounting for the coding change. Verify the contrast and scale before combining them.
Different model fits: if a predictor, covariate set, or outcome definition changes between imputations, the resulting estimates may no longer target the same $Q$.
One estimate per imputation: do not mistakenly treat the $M$ completed datasets as $M$ new independent samples of participants. The index $m$ labels imputations, and the pooling formula averages the corresponding analysis estimates.

A warning sign is a large difference among estimates that cannot be explained by the intended imputation variability. Check the analysis code and saved results for inconsistent sample restrictions, recoding, model convergence, or missing variables in a completed dataset. A conspicuous estimate is not automatically an error: imputed values can legitimately affect results. But unexplained changes in model specification or units are not legitimate sources of between-imputation variation.

Pooling the point estimates is only one part of inference. The arithmetic mean does not provide a standard error, confidence interval, or test by itself. Those require combining information about uncertainty within and between imputations, which is handled in the subsequent pooling calculations. Reporting $\bar{Q}$ alone is appropriate only when a point estimate is the requested summary, not when the goal is a complete inferential statement.

Common Mistake

Common mistake: “The estimate from one completed dataset has the smallest standard error, so it should count more in the pooled point estimate.” That reasoning replaces the standard equal-weight point estimate with a different weighted average and does not follow the multiple-imputation pooling rule. First calculate $\bar{Q}=M^{-1}\sum_{m=1}^{M}Q_m$. The uncertainty information from the completed-data analyses is used in the separate variance-pooling step; it does not change this point-estimate formula.

Another tempting shortcut is to average whatever numbers each software fit reports under the same label. A label such as “coefficient” or “mean” is not enough to establish comparability. Inspect the model, the exact parameter, and its scale. If the definitions differ, resolve the discrepancy before pooling rather than treating the arithmetic output as meaningful.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Four completed datasets give estimates of the same mean: 8.2, 7.8, 8.5, and 8.3 units. Calculate $\bar{Q}$.
Three completed datasets give slopes of 2.0, 2.4, and 2.2 outcome units per year. A fourth fit reports 0.23 outcome units per month. Can these four values be averaged as written? If not, convert and calculate the common-scale mean.
Two logistic models estimate a coefficient for the same predictor contrast. Their odds ratios are 0.80 and 1.25. Calculate the pooled log coefficient and its exponentiated value. What relationship do you notice between the two odds ratios?
Across five imputations, an analyst fits an adjusted model in three datasets and an unadjusted model in two, then averages the five coefficients. Explain the problem and state a correction.

Solutions. (1) $\bar{Q}=(8.2+7.8+8.5+8.3)/4=32.8/4=8.20$ units. (2) No. Convert 0.23 outcome units per month to a yearly slope: $0.23\times12=2.76$ outcome units per year. The common-scale mean is $(2.0+2.4+2.2+2.76)/4=9.36/4=2.34$ outcome units per year. (3) The log coefficients are $\log(0.80)\approx-0.2231$ and $\log(1.25)\approx0.2231$. Their mean is approximately 0, and exponentiating gives $\exp(0)=1.00$. The odds ratios are reciprocals, so their log coefficients cancel. (4) The analyses generally estimate different targets because their covariate adjustment differs. Refit all completed datasets using the same prespecified adjusted model (or the same unadjusted model, if that is the intended target), extract the same coefficient, and average those comparable estimates.