Adjusted Means for an External Target Population

Why Specify an External Target?

An adjusted treatment mean is not determined by the fitted ANCOVA model alone. It also depends on the covariate values over which predictions are averaged. Standardizing over the study participants’ covariate distribution answers a question about that distribution. Sometimes the intended question concerns a different, specified population: for example, eligible patients in a region or a group described by an external registry. In that case, use the external population’s covariate values and weights to define the adjusted means.

The external population supplies the target covariate distribution, not additional treatment outcomes. For each treatment, the fitted model predicts an outcome at each target covariate value. A weighted average of those predictions is the standardized mean for that treatment. Comparing two such averages gives an adjusted comparison targeted to that external population.

Definition: Let $g$ denote a treatment group, and let the external target population have covariate values $x_1,\ldots,x_J$ with weights $w_1,\ldots,w_J$, where $w_j\geq 0$ and $\sum_{j=1}^{J}w_j=1$. If $\hat m_g(x_j)$ is the fitted mean outcome for treatment $g$ at covariate value $x_j$, the standardized treatment mean for the external target is $$ \hat\mu_g^{\,T}=\sum_{j=1}^{J}w_j\hat m_g(x_j). $$ The weights describe the target population’s covariate distribution. They are not the treatment-group proportions in the study.

The superscript $T$ labels the target population. If the target distribution is described by a continuous distribution $F_T$ rather than a finite list of values, the corresponding definition is $\hat\mu_g^{\,T}=\int \hat m_g(x)\,dF_T(x)$. In either form, the calculation averages model-predicted outcomes, not observed outcomes from the external population.

For a comparison between treatments $g$ and $h$, the target-standardized contrast is $\hat\mu_g^{\,T}-\hat\mu_h^{\,T}$. It is expressed in the outcome’s units. Its interpretation is tied to the specified target distribution: it compares the model-predicted mean outcomes under the two treatments after averaging over that population’s covariate values.

Model and Conditions for Interpretation

The predictions must come from a model appropriate for the treatment groups and covariate. With treatment-specific slopes, a useful linear ANCOVA mean model is

Y_{ig}=\alpha_g+\beta_g(x_{ig}-c)+\varepsilon_{ig}, \qquad E(\varepsilon_{ig}\mid x_{ig},g)=0.

Here $Y_{ig}$ is the outcome for participant $i$ in treatment group $g$, $x_{ig}$ is that participant’s covariate, $c$ is a chosen reference value, and $\alpha_g,\beta_g$ are the treatment-specific intercept and slope. The fitted conditional mean is $\hat m_g(x)=\hat\alpha_g+\hat\beta_g(x-c)$. If the groups are adequately described by a common slope, the model can instead use a shared slope. The choice must be appropriate to the data; a model with treatment-specific slopes allows the treatment difference to vary with the covariate.

For the standardized means to represent the intended target, several conditions matter. The covariate must have the same meaning and measurement scale in the study and target population. The fitted conditional mean should describe the outcome at relevant covariate values, including any treatment-by-covariate pattern that matters. Study observations should be independent for the usual individual-level ANCOVA analysis; normal, constant-variance errors are also commonly assumed for exact small-sample inference. Those error-distribution assumptions are not needed merely to perform the weighted-average calculation, but they matter to standard inferential procedures.

The target weights must accurately describe the population the analysis intends to address. Also check whether target covariate values are supported by study data in each treatment group. If many target values lie beyond the study data, predictions rely on extrapolation and may be unreliable. This calculation does not by itself establish a causal effect: a causal interpretation requires an appropriate study design and additional assumptions beyond the weighted averaging.

Calculating Target-Standardized Means

The practical calculation has three parts: obtain the target values and weights, predict the outcome for every treatment at each target value, and average those predictions using the target weights. Use the same target values and weights for every treatment in a comparison. This ensures that differences in standardized means do not simply reflect averaging each treatment over a different covariate distribution.

Worked Example: A Registry Target With Three Covariate Values

Suppose a hypothetical ANCOVA study compares treatments A and B on an outcome measured in points. Its fitted treatment-specific mean equations are $\hat m_A(x)=120+0.8(x-50)$ and $\hat m_B(x)=128+1.2(x-50)$. An external registry defines the target covariate distribution: 20% have $x=40$, 50% have $x=50$, and 30% have $x=70$. Calculate both standardized means and their contrast.

Assume the study observations are independent, the fitted linear mean equations adequately describe the treatment-specific covariate relationships, and the target values are within the covariate range represented in both treatment groups. The registry weights are nonnegative and sum to one: $0.20+0.50+0.30=1$.

For treatment A, the predictions at $x=40,50,70$ are $112,120,136$ points. Thus,

\hat\mu_A^{\,T} =0.20(112)+0.50(120)+0.30(136) =22.4+60+40.8 =123.2\text{ points}.

For treatment B, the corresponding predictions are $116,128,152$ points, giving

\hat\mu_B^{\,T} =0.20(116)+0.50(128)+0.30(152) =23.2+64+45.6 =132.8\text{ points}.

The target-standardized contrast is $132.8-123.2=9.6$ points. Under these fitted models, treatment B’s predicted mean is 9.6 points higher than treatment A’s when both are averaged over the registry’s covariate distribution. This is a model-based comparison for that target, not a claim that every person would have an outcome 9.6 points higher.

Why the Target Mean Can Be Enough Under a Linear Model

For a linear treatment-specific mean model, the weighted average has a useful simplification. Define the target’s weighted covariate mean as $\bar{x}_T=\sum_jw_jx_j$. Then, for treatment $g$,

\begin{aligned} \hat\mu_g^{\,T} &=\sum_{j=1}^{J}w_j\{\hat\alpha_g+\hat\beta_g(x_j-c)\}\\ &=\hat\alpha_g\sum_{j=1}^{J}w_j +\hat\beta_g\left(\sum_{j=1}^{J}w_jx_j-c\sum_{j=1}^{J}w_j\right)\\ &=\hat\alpha_g+\hat\beta_g(\bar{x}_T-c), \end{aligned}

because the weights sum to one. Thus, when the conditional mean is linear in this single covariate, averaging predictions over all target values gives the same result as predicting at the target’s weighted mean covariate value. The target distribution still defines the estimand; its weighted mean happens to be sufficient for this particular linear calculation.

Subtracting the corresponding results for two treatments shows how the standardized contrast is obtained:

\hat\mu_g^{\,T}-\hat\mu_h^{\,T} =(\hat\alpha_g-\hat\alpha_h) +(\hat\beta_g-\hat\beta_h)(\bar{x}_T-c).

This expression includes both differences in fitted intercepts and differences in fitted slopes. If slopes differ, the standardized contrast depends on the target’s covariate mean. If the fitted slopes are equal, the slope-difference term is zero. This identity is a calculation rule, not a reason to ignore the model’s adequacy or the source of the target values.

Worked Example: Using the Target’s Weighted Mean

A hypothetical study’s fitted mean equations for two treatments are $\hat m_C(x)=10+0.5(x-50)$ and $\hat m_D(x)=13+0.2(x-50)$, with outcomes measured in units. An external target has covariate values $30,50,80$ and respective weights $0.50,0.30,0.20$. Assume independent study observations, adequate treatment-specific linear mean fits, and that these target values are supported by observations in both groups.

First calculate the target’s weighted mean covariate value:

\bar{x}_T=0.50(30)+0.30(50)+0.20(80)=15+15+16=46.

The standardized mean for treatment C can be found by predicting at $x=46$:

\hat\mu_C^{\,T}=10+0.5(46-50)=10-2=8\text{ units}.

For treatment D,

\hat\mu_D^{\,T}=13+0.2(46-50)=13-0.8=12.2\text{ units}.

Therefore, the target-standardized contrast $D-C$ is $12.2-8=4.2$ units. The shortcut works because each fitted conditional mean is linear in $x$, the same target weights apply to both treatments, and the weights sum to one. The result is not obtained by averaging the two treatment equations’ intercepts alone; their covariate slopes also contribute.

Working With Target Weights

External populations are often summarized using counts or sampling weights rather than proportions. Convert counts to proportions before taking the weighted average. If the target contains $N_j$ people at covariate value $x_j$, then $w_j=N_j/\sum_kN_k$. If supplied weights are survey weights $a_j$, normalize them as $w_j=a_j/\sum_ka_k$. The normalized weights must sum to one so that the calculation returns an average outcome in the original outcome units.

Worked Example: Normalizing External Population Counts

Suppose a hypothetical external population is divided into three covariate strata with counts 60, 30, and 10 at $x=2,6,10$, respectively. A study’s fitted mean for treatment P is $\hat m_P(x)=40+3x$, and for treatment Q it is $\hat m_Q(x)=44+2x$. Outcomes are measured in millimeters. Assume the study’s observations are independent, both fitted linear mean equations are adequate, and the three target covariate values are represented in both treatment groups.

The total target count is $60+30+10=100$, so the normalized weights are $0.60,0.30,0.10$. For treatment P, the predicted means at the three values are $46,58,70$ millimeters. Therefore,

\hat\mu_P^{\,T} =0.60(46)+0.30(58)+0.10(70) =27.6+17.4+7 =52.0\text{ mm}.

For treatment Q, the predictions are $48,56,64$ millimeters, so

\hat\mu_Q^{\,T} =0.60(48)+0.30(56)+0.10(64) =28.8+16.8+6.4 =52.0\text{ mm}.

The standardized means are equal, and the target-standardized contrast is $0.0$ mm. To verify the shortcut, the target mean covariate is $0.60(2)+0.30(6)+0.10(10)=4$. Substitution gives $40+3(4)=52$ mm for P and $44+2(4)=52$ mm for Q. The equality is specific to these fitted equations and this target; the separate predictions across covariate values are not all equal.

Common Mistake and Practical Interpretation

A common mistake is to use the study’s observed covariate distribution automatically, even when the question specifies an external target. The resulting number may be a valid standardized mean, but it answers a different question. Another mistake is to average raw outcomes from the external population: the target supplies covariate values and weights, while the study model supplies treatment-specific predictions. Keep these sources separate.

A standardized mean is an average of model-predicted outcomes for a clearly stated target distribution. Report the target population, the outcome units, and the covariate distribution used. If reporting a contrast, identify its direction—for example, “treatment B minus treatment A”—and state that it refers to the same external target distribution for both treatments. Uncertainty in fitted model coefficients can affect the estimate; a confidence interval should account for that model uncertainty rather than treating the predictions as known constants.

Key takeaway: To target an external population, predict each treatment’s mean outcome at the target covariate values and average those predictions using normalized external weights. The resulting means and contrasts describe that specified target, conditional on the fitted model being suitable for it.

Check Your Understanding

Answer each question using the definitions and calculations above.

An external target has covariate values $x_1,x_2$ with weights $0.25$ and $0.75$. Write the formula for the target-standardized mean under treatment A.
Why should the same target weights be used to standardize both treatments in a contrast?
For a linear treatment-specific mean model, explain why predicting at the target’s weighted mean covariate value gives the same standardized mean as averaging predictions over all target values.
An external source gives counts of 40 and 60 in two covariate strata. What normalized weights should be used?
What concern arises if many target covariate values are outside the range represented in a treatment group’s study observations?