Why Subgroup Reporting Needs Care
A trial’s average treatment effect can conceal variation across clinically relevant groups. For example, treatment benefit might differ by disease severity or age. Reporting subgroup results can help readers understand where the data are informative, but subgroup estimates are often less precise than the overall estimate. A larger-looking estimate in one subgroup does not, by itself, establish that treatment works differently across groups.
The central reporting distinction is between two questions: What is the estimated treatment effect within each subgroup? and How much evidence is there that the treatment effects differ between subgroups? The first is answered by subgroup-specific posterior distributions. The second requires a posterior distribution for a contrast between effects. Separate intervals or probabilities do not answer that second question.
This notation requires a consistent endpoint, effect scale, and direction across subgroups. For example, if \(\theta_g\) is the reduction in event risk under treatment compared with control, then it can be reported in percentage points and positive values indicate fewer events with treatment. A contrast of \(3\) percentage points means the estimated risk reduction is 3 percentage points greater in subgroup 1 than in subgroup 2. It does not mean that every individual in subgroup 1 benefits more.
What a Subgroup Summary Should Communicate
For each subgroup, report the estimate on a clinically interpretable scale and quantify its uncertainty. A posterior mean or median can summarize the location of the posterior distribution; a credible interval communicates its spread. If a posterior probability is reported, state the event whose probability is being calculated, such as \(P(\theta_g>0\mid\text{data})\). The posterior probability is conditional on the data and the specified model, including its assumptions.
Also identify the subgroup definitions and give the relevant denominators, such as the number of randomized participants or outcome events in each group. These figures help readers see where information is limited. They do not replace the posterior uncertainty: a subgroup’s precision also depends on the outcome, model, and prior. Describe whether subgroup analyses were planned or exploratory, and avoid presenting an exploratory pattern as a confirmed treatment difference.
For a normal posterior approximation, if \(\theta_g\mid\text{data}\) has mean \(m_g\) and standard deviation \(s_g\), an approximate central 95% credible interval is \(m_g\pm1.96s_g\). If the posterior is skewed or multimodal, report appropriate posterior quantiles or other summaries that represent its shape rather than relying on this symmetric approximation. The earlier material on Reporting Skewed and Multimodal Posterior Effects is relevant when a single symmetric interval would be misleading.
Worked Example: Reporting Benefit Within Each Subgroup
Suppose a hypothetical randomized trial compares a new therapy with control for a binary clinical event. Define \(\theta_g\) as the absolute event-risk reduction in subgroup \(g\), in percentage points, so positive values favor the new therapy. A model-based analysis gives approximately normal posterior distributions: subgroup A has mean \(4.8\) percentage points and standard deviation \(2.0\) percentage points; subgroup B has mean \(1.2\) percentage points and standard deviation \(3.0\) percentage points.
For subgroup A, the approximate central 95% credible interval is \(4.8\pm1.96(2.0)=4.8\pm3.92\), or \(0.88\) to \(8.72\) percentage points. For subgroup B, it is \(1.2\pm1.96(3.0)=1.2\pm5.88\), or \(-4.68\) to \(7.08\) percentage points.
Under the normal approximation, the posterior probability of a positive risk reduction is \(P(\theta_A>0\mid\text{data})=\Phi(4.8/2.0)=\Phi(2.4)\approx0.9918\) for subgroup A. For subgroup B it is \(P(\theta_B>0\mid\text{data})=\Phi(1.2/3.0)=\Phi(0.4)\approx0.6554\). Here \(\Phi\) denotes the standard normal cumulative distribution function.
A calibrated report would say that the posterior estimate of risk reduction is larger and more precise in subgroup A, while the subgroup B estimate is more uncertain. These summaries do not establish that the treatment effects differ: that requires the posterior distribution of \(\theta_A-\theta_B\), including the dependence between the subgroup estimates. The interval for subgroup B includes zero, but this is not evidence by itself that subgroup B has no treatment benefit.
Estimating a Difference Between Subgroup Effects
To assess whether two subgroup effects differ, calculate the posterior distribution of \(\Delta=\theta_1-\theta_2\) from the joint posterior distribution of \((\theta_1,\theta_2)\). If this joint posterior is approximately bivariate normal, with marginal posterior means \(m_1,m_2\), standard deviations \(s_1,s_2\), and correlation \(\rho\), then the contrast has posterior mean \(m_1-m_2\) and variance \(\operatorname{Var}(\Delta\mid\text{data})=s_1^2+s_2^2-2\rho s_1s_2\).
The variance identity follows by expanding the variance of a difference: for any two random variables \(U\) and \(V\), \(\operatorname{Var}(U-V)=\operatorname{Var}(U)+\operatorname{Var}(V)-2\operatorname{Cov}(U,V)\). If the joint posterior is not well approximated by a bivariate normal, the contrast can instead be calculated for each draw from the joint posterior, \(\Delta^{(k)}=\theta_1^{(k)}-\theta_2^{(k)}\), and summarized across draws. In either case, marginal subgroup summaries alone do not generally provide the uncertainty in the contrast.
Worked Example: Reporting the Posterior Contrast
In a hypothetical trial, suppose the posterior treatment effects are approximately jointly normal on the same percentage-point risk-reduction scale. Subgroup A has posterior mean \(5.0\) and standard deviation \(2.0\); subgroup B has posterior mean \(2.0\) and standard deviation \(2.5\). The posterior correlation is \(0.30\). Define \(\Delta=\theta_A-\theta_B\), so positive values indicate a larger risk reduction in subgroup A.
The posterior mean contrast is \(5.0-2.0=3.0\) percentage points. Its posterior variance is \(2.0^2+2.5^2-2(0.30)(2.0)(2.5)=4.0+6.25-3.0=7.25\) square percentage points. Thus its posterior standard deviation is \(\sqrt{7.25}\approx2.693\) percentage points.
The approximate central 95% credible interval for \(\Delta\) is \(3.0\pm1.96(2.693)=3.0\pm5.277\), or approximately \(-2.277\) to \(8.277\) percentage points. The normal approximation gives \(P(\Delta>0\mid\text{data})=\Phi(3.0/2.693)\approx\Phi(1.114)\approx0.867\).
A suitable interpretation is that the posterior probability that the risk reduction is greater in subgroup A than in subgroup B is about 0.867, under this model; the 95% credible interval for the difference extends from a modestly greater reduction in B to a substantially greater reduction in A. The interval includes zero, so the direction and size of the difference remain uncertain. Neither summary proves equality or a difference. The correlation matters: omitting it would change the posterior uncertainty in the contrast.
Assumptions, Diagnostics, and Limits
Subgroup reporting is meaningful only when the comparison is well defined. State how subgroup membership was determined, how the treatment effect is oriented, and whether the same outcome definition, follow-up, and effect scale apply across groups. In a randomized trial, randomization supports treatment comparisons within baseline-defined subgroups, subject to the trial design and adequate representation. It does not randomize people to subgroup membership. Therefore, an estimated difference across subgroups is not evidence that subgroup membership itself causes different outcomes.
Check whether each subgroup has enough observations and events to support the degree of detail being reported. Sparse data can leave a posterior broad or make it sensitive to modeling assumptions. Look for unstable estimates, very wide intervals, and results that change materially under reasonable model specifications. A small subgroup is not automatically uninformative, but its uncertainty and the assumptions used to estimate its effect should be visible. Posterior computation problems also matter; a reported summary is not reliable if the computation has not adequately represented the posterior.
The main alternative summaries answer different questions. An overall treatment effect describes the trial population as a whole and can be more precise, but it may not show subgroup variation. Separate subgroup estimates describe effects within each group, but do not directly quantify their difference. A treatment-effect contrast directly addresses heterogeneity between two specified groups, while still depending on the model, data, and chosen scale. No one summary removes the need to state the estimand and its uncertainty.
| Summary | Question it addresses | What it does not establish |
|---|---|---|
| Overall posterior effect | What is the treatment effect for the trial population? | That the effect is identical across all subgroups |
| Subgroup-specific posterior effect | What is the estimated effect within this subgroup? | That it differs from the effect in another subgroup |
| Posterior contrast \(\Delta\) | How different are the effects in the specified subgroups? | That the observed pattern generalizes beyond the model and population studied |
A Common Overclaim
A frequent mistake is to call a subgroup difference “significant” because one subgroup has a high posterior probability of benefit while another does not. This compares two separate judgments against zero, not the posterior distribution of the difference between effects. Uncertainty can be substantial in both subgroups, and a difference between their posterior probabilities is not the posterior probability that their treatment effects differ.
Instead, report the subgroup estimates and intervals, then report a contrast and its uncertainty if the scientific question concerns heterogeneity. If a contrast was not estimated, say that the subgroup summaries do not establish a difference. Also avoid saying an effect is absent merely because its interval includes zero or its posterior probability of benefit is not high. Describe the estimated magnitude and remaining uncertainty.
Worked Example: A Small Subgroup and an Uncertain Pattern
Consider a hypothetical trial with 24 participants in subgroup A and 180 in subgroup B. On the same risk-reduction scale, the fitted analysis gives subgroup A a posterior mean of \(2.0\) percentage points and standard deviation \(4.0\); subgroup B has a posterior mean of \(5.0\) percentage points and standard deviation \(1.5\). These are model-based summaries, not raw event-rate calculations.
For subgroup A, the approximate 95% credible interval is \(2.0\pm1.96(4.0)=2.0\pm7.84\), or \(-5.84\) to \(9.84\) percentage points. Its posterior probability of a positive risk reduction is \(\Phi(2.0/4.0)=\Phi(0.5)\approx0.6915\). For subgroup B, the interval is \(5.0\pm1.96(1.5)=5.0\pm2.94\), or \(2.06\) to \(7.94\) percentage points. The corresponding probability is \(\Phi(5.0/1.5)=\Phi(3.333)\approx0.9996\).
The subgroup A estimate is much less precise, and its posterior probability of benefit is lower. It would be an overclaim to conclude that treatment benefits only subgroup B. The contrast has posterior mean \(2.0-5.0=-3.0\) percentage points, but its standard deviation and interval cannot be calculated from these marginal summaries alone: the joint posterior covariance is also needed. A report should state that the two subgroup estimates differ in precision and that these summaries do not, by themselves, establish effect heterogeneity.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- On a scale where positive values indicate benefit, subgroup A has an estimated effect of 6 percentage points and subgroup B has an estimated effect of 2 percentage points. Does this alone establish that effects differ?
- A subgroup’s 95% credible interval includes zero. What can and cannot be concluded from that fact?
- For a contrast \(\Delta=\theta_A-\theta_B\), why is the covariance between subgroup estimates relevant to its uncertainty?
- What additional posterior information is needed to assess \(P(\theta_A>\theta_B\mid\text{data})\) if only marginal means and standard deviations have been reported?
- Why should subgroup sample sizes and event counts accompany posterior summaries?
Solutions. (1) No. The point estimates differ by 4 percentage points, but uncertainty in the contrast is needed to assess evidence for a treatment-effect difference. (2) The interval indicates that the posterior assigns substantial uncertainty to values on both sides of zero under the stated model and interval construction. It does not prove no effect or equivalence. (3) Because \(\operatorname{Var}(\theta_A-\theta_B\mid\text{data})=s_A^2+s_B^2-2\operatorname{Cov}(\theta_A,\theta_B\mid\text{data})\); ignoring covariance can give the wrong uncertainty. (4) The joint posterior, or at least its dependence structure, is needed to obtain the posterior distribution of the difference. (5) They show how much information each subgroup contributes and help readers interpret differences in precision, while not replacing the posterior uncertainty itself.