Why Prespecify Subgroups?
An average treatment effect can conceal variation across people, but a subgroup effect is meaningful only after specifying which people belong to the subgroup. “The effect for older adults” is incomplete unless “older” has a defined rule, such as age 65 or above at enrollment. That rule determines the population whose average effect is being described.
Prespecification means fixing the subgroup definition before examining outcome results. It helps distinguish a scientifically motivated question from a pattern selected because it looked striking in the observed outcomes. Prespecification does not itself establish a causal effect, remove sampling uncertainty, or guarantee that the subgroup is useful in a different population. Those conclusions depend on the design, assumptions, and target population.
For example, \(G=1\) might mean that age at enrollment is at least 65 years. The definition should use a characteristic measured before treatment, and the rule should be explicit enough that another analyst could classify the same people without seeing their outcomes. A subgroup can also be defined using several baseline characteristics, but each added condition changes who the estimand describes.
From Subgroup Definition to Causal Interpretation
Potential outcomes define a causal contrast, but observed data reveal only the outcome under the treatment actually received. To interpret a subgroup contrast causally, the design or assumptions must justify comparisons between observed treatment groups within that subgroup.
In a randomized trial, the central requirements are that treatment assignment is randomized with positive probability of each assignment within the subgroup, the observed outcome equals the potential outcome under the assigned treatment (consistency), and one person's assignment does not change another person's outcome (no interference). Under these conditions, the difference in mean observed outcomes between assigned treatment and control within \(G\) identifies \(\tau_G\) in expectation. In an observational study, randomization is replaced by conditional exchangeability: given measured baseline covariates \(X\), treatment assignment is independent of the potential outcomes. Positivity requires both treatment options to have positive probability at the relevant covariate values, in addition to consistency and no interference.
A subgroup definition must not depend on post-treatment variables, such as whether a participant adhered to treatment or experienced an outcome-related event. Conditioning on such variables can select people based on consequences of treatment and generally changes the causal question. Prespecification is most useful when the eligibility rule, measurement time, and cutpoints are all stated before outcomes are analyzed.
Worked Example: Age-Group Effects in a Randomized Trial
Consider a hypothetical randomized trial of a symptom-reducing treatment. The prespecified age groups are younger than 65 and 65 or older. The outcome is a symptom score, for which a larger value is better. Within each group, treatment assignment is randomized, each arm has positive probability, and outcomes are recorded for all assigned participants. Assume consistency and no interference.
| Prespecified age group | Mean outcome, treatment | Mean outcome, control | Share of target population |
|---|---|---|---|
| Age < 65 | 7.2 points | 5.0 points | 0.60 |
| Age ≥ 65 | 5.8 points | 5.1 points | 0.40 |
For each age group, the randomized difference in means estimates that group’s average treatment effect. For participants younger than 65, $$ \widehat{\tau}_{<65}=7.2-5.0=2.2 \text{ points}. $$ For participants aged 65 or older, $$ \widehat{\tau}_{\geq 65}=5.8-5.1=0.7 \text{ points}. $$ The estimates describe average improvements of 2.2 and 0.7 symptom-score points, respectively, in these groups. They do not imply that every younger participant benefits by 2.2 points or that every older participant benefits by 0.7 points.
To summarize the effect for the stated target population, weight the two subgroup estimates by that population’s age-group shares: $$ \widehat{\tau}_{\mathrm{target}} =0.60(2.2)+0.40(0.7) =1.32+0.28 =1.60 \text{ points}. $$ This is a weighted average for a target population that is 60% younger than 65 and 40% aged 65 or older. Changing those shares would change the overall average even if the two subgroup effects stayed the same.
How Target-Population Weights Enter
Suppose fixed subgroup definitions divide a target population into \(K\) mutually exclusive and exhaustive groups. Let \(q_g\) be the proportion of that target population in group \(g\), where \(q_g\geq 0\) and \(\sum_{g=1}^{K}q_g=1\). If \(\tau_g\) is the average treatment effect in group \(g\), then the overall target-population effect is
The weights are proportions of people, not proportions of treatment and control observations in the trial unless those proportions happen to match the target population. This identity follows by averaging the individual treatment effect over the groups: the target population’s mean is the sum of each group’s mean multiplied by the probability that a randomly selected target-population member belongs to that group.
The identity also clarifies a distinction between a subgroup estimand and an estimator. The subgroup estimand says which group’s average potential-outcome contrast is of interest. A sample calculation estimates that contrast from observed data; changing a sample’s subgroup counts does not, by itself, change the target estimand. Conversely, when the goal is to describe a different target population, its group proportions may require different weights.
Worked Example: Reweighting Risk Groups to a Target Clinic
A hypothetical randomized trial prespecifies low- and high-baseline-risk groups. The outcome is days alive and out of hospital during a fixed follow-up period, so a positive treatment effect is favorable. Suppose the estimated within-group treatment effects are 1.0 day for low-risk participants and 3.0 days for high-risk participants. The trial is 75% low risk and 25% high risk, while a clinic that will use the results is 40% low risk and 60% high risk.
Under random assignment within each risk group, consistency, no interference, and positive assignment probabilities, the trial’s within-group contrasts estimate trial-population subgroup effects. To use those effects for the clinic, additionally assume that the treatment effects within each risk group apply to the clinic’s members. This transport assumption is not guaranteed by randomization within the trial.
The trial-weighted average is $$ \widehat{\tau}_{\mathrm{trial}} =0.75(1.0)+0.25(3.0) =0.75+0.75 =1.50 \text{ days}. $$ For the clinic’s target composition, the standardized average is $$ \widehat{\tau}_{\mathrm{clinic}} =0.40(1.0)+0.60(3.0) =0.40+1.80 =2.20 \text{ days}. $$ The 0.70-day difference between these averages results from the target clinic having a larger share of the high-risk group, whose estimated effect is larger. It does not mean that the treatment effect changed within either risk group.
Subgroup Definitions Are Part of the Estimand
Choosing a cutpoint is not merely a way to organize a table. It determines membership, and therefore the population over which potential outcomes are averaged. Two sensible age definitions can produce different subgroup effects because they refer to different people. The resulting estimates are not automatically competing answers to the same question.
A threshold should be scientifically interpretable and fixed before outcome analysis when the goal is confirmatory interpretation. Other prespecified rules may define age bands, a clinical risk category, or a combination of baseline attributes. More detailed groups can answer more specific questions, but they may also contain fewer people and produce less precise estimates. The estimand remains defined even when its estimate is uncertain.
Worked Example: Two Age Thresholds, Two Subgroup Estimands
In a hypothetical randomized trial, age is recorded at enrollment and the outcome is a functional score, with higher values better. Consider three baseline age bands and suppose their estimated average treatment effects are as follows:
| Age band | Estimated effect | Share among everyone aged 60 or older in the target population |
|---|---|---|
| 40–59 years | 2.4 points | Not included |
| 60–69 years | 1.2 points | 0.70 |
| 70 years or older | 0.3 points | 0.30 |
For the prespecified subgroup aged 60 or older, the target-population effect is the weighted average of the two included bands: $$ \widehat{\tau}_{\geq 60} =0.70(1.2)+0.30(0.3) =0.84+0.09 =0.93 \text{ points}. $$ For the different subgroup aged 70 or older, the estimated effect is 0.3 points because this subgroup contains only the 70-or-older band. These estimates answer different questions: the first averages over everyone aged at least 60, while the second averages over the older subgroup alone.
The arithmetic uses the stated target shares among those aged 60 or older; the 40–59 group receives no weight in that subgroup estimate. As with the previous example, a causal interpretation requires valid within-group causal contrasts, and applying the estimates to a target population requires that the relevant group effects apply there. Neither the age threshold nor the target weights should be chosen after looking for the most appealing estimate.
Prespecification and Interpretation
A protocol can make a subgroup question reproducible by specifying the baseline variable, its measurement time, the group rule, the outcome, and the target population. It should also distinguish a planned primary subgroup analysis from an exploratory analysis. A definition fixed in advance limits outcome-driven selection, but it does not remove the need to consider the number of analyses, uncertainty, or whether the subgroup is large enough to support a useful estimate.
One common error is to infer treatment-effect heterogeneity because an estimated effect is “statistically significant” in one subgroup and not in another. That reasoning compares two separate significance decisions rather than directly assessing how the effects differ; different precision alone can produce different decisions. This tutorial’s central point is about the estimands: subgroup effects are averages for defined populations, and comparing them requires a comparison of those effects with appropriate uncertainty. A later treatment of heterogeneity tests can address that inferential question.
Another error is to report a trial-wide average as if it automatically applied to every clinic or policy population. The overall effect is a weighted average, so differences in subgroup composition matter when subgroup effects vary. Report the subgroup definitions and the weights used for any overall target-population summary. If the target group proportions or the assumptions needed to transport subgroup effects are unknown, say so rather than presenting a reweighted value as established.
Check Your Understanding
For each question, state the target group or population explicitly.
- Define the subgroup average treatment effect for participants whose baseline age is at least 70 years.
- Two prespecified age groups have effects of 1.5 and 3.0 days. What additional information is needed to calculate their overall effect for a target population?
- A trial is 80% low risk, but a target clinic is 50% low risk. Why might the trial-wide average differ from the clinic’s target-population average?
- Explain why an age-60-or-older effect and an age-70-or-older effect are different estimands, even when calculated from the same trial.
- Why can defining a subgroup using a post-treatment event undermine a causal interpretation?