Continuous-Outcome Estimands for Means and Quantiles

Why the Summary Scale Matters

A continuous clinical outcome may be strongly skewed: most patients might have modest symptom improvement while a smaller group has very large improvement, or most hospital stays may be short while a few are prolonged. In such settings, a treatment contrast in means answers a different question from a contrast in medians or upper quantiles. Both can be useful, but neither is a universal summary of the entire outcome distribution.

The estimand must first specify the outcome, target population, treatment conditions, time point, and strategy for intercurrent events. Those choices determine which outcome distribution is being summarized. Here, the focus is on defining and interpreting mean and quantile contrasts from the resulting treatment-specific marginal distributions. The distributions are marginal over the target population, rather than conditional on a particular covariate profile.

A negative contrast means the active-treatment mean or selected quantile is lower than the corresponding control value. Whether that is desirable depends on the outcome: lower symptom scores might be better, whereas a higher functional score might be better. A quantile contrast compares two locations in two population distributions. It does not, in general, describe the effect for an identifiable group of individuals ranked at the \(p\)-th position.

Conditions for a Meaningful Contrast

The distributions \(F_A\) and \(F_C\) must refer to the same target population and to well-defined treatment conditions. In a randomized trial, randomization supports comparison of the assigned treatment groups; consistency connects observed outcomes to the specified treatment conditions. If the target is a broader population or the analysis adjusts for covariates, the model and standardization must also support that target. The outcome handling after intercurrent events must follow the chosen estimand strategy; changing that strategy changes the outcome distributions and therefore changes the estimand.

For a mean contrast, each treatment-specific mean must be finite. This is more than a technical detail: a distribution with a sufficiently heavy tail may have no finite mean, or may yield a mean so sensitive to a small number of extreme values that it is not a stable clinical summary. Quantiles can still be defined when a mean is not finite, although their interpretation and estimation require care. For a quantile to be uniquely and stably located, it is helpful for the distribution to be continuous with positive density near that quantile. Sparse information in a tail can make an upper quantile especially uncertain.

In Bayesian analysis, a model represents uncertainty about the treatment-specific distributions. For each posterior draw, calculate the two treatment-specific means or quantiles and then subtract control from active treatment. The resulting draws form the posterior distribution of the estimand. The treatment distributions and their dependence within each draw matter: uncertainty in a contrast should be calculated from the paired treatment-specific quantities in that draw, not by treating their marginal posterior summaries as independent.

How Skewness Separates Means and Quantiles

A short calculation shows why the mean and quantile can tell different stories. Suppose \(Y\) is lognormal: \(\log(Y)=Z\), where \(Z\sim N(\mu,\sigma^2)\). Since exponentiation is increasing, the \(p\)-quantile of \(Y\) is the exponential of the \(p\)-quantile of \(Z\). Also, the normal moment-generating function gives the mean of \(Y\).

The \(\sigma^2/2\) term in the mean, absent from the median, shows that a larger spread on the log scale raises the arithmetic mean even when the median is unchanged. Thus, a mean contrast can reflect both location and the distribution’s upper tail. A quantile contrast focuses on a specified point in the distribution, but says nothing by itself about the rest of that distribution.

Posterior Contrasts: Calculate Within Each Draw

Suppose the posterior distribution is represented by \(M\) draws. In draw \(m\), let \(\mu_A^{(m)}\) and \(\mu_C^{(m)}\) be the model-implied marginal means. The draw for the mean-difference estimand is \(\Delta_\mu^{(m)}=\mu_A^{(m)}-\mu_C^{(m)}\). For a quantile estimand, calculate \(q_{A,p}^{(m)}-q_{C,p}^{(m)}\) in each draw. Posterior probabilities about a contrast are then proportions of draws satisfying the stated condition. For example, the estimated posterior probability that the mean difference is below zero is the number of draws with \(\Delta_\mu^{(m)}<0\), divided by \(M\).

Quantiles and the Shape of the Distribution

Quantiles make it possible to ask targeted questions about different parts of a skewed distribution. The median describes its center in the sense of the 50th percentile; a 90th-percentile contrast describes the difference at a high outcome value. These contrasts need not be similar. A treatment may have a modest median difference but a larger upper-tail difference, which could matter when unusually high outcomes carry substantial clinical burden.

The quantile level should be selected for a clinical reason, not because it looks favorable after inspecting results. The same applies to the mean: it is often valuable because it incorporates all outcome values and can connect to total burden, but it can be sensitive to a long tail. Neither summary replaces examination of the outcome distributions themselves.

Diagnostics, Alternatives, and Limitations

Before relying on a posterior contrast, check whether the fitted model represents the parts of the outcome distribution that the estimand uses. Posterior predictive checks should compare relevant features of replicated and observed outcomes, including skewness, extreme values, and upper-tail behavior. For a high quantile, inspect whether the data contain enough information near and beyond that part of the distribution. A credible interval can be precise under a restrictive model and still fail to reflect uncertainty about model misspecification.

For outcomes with many extreme values, examine whether a small number of observations dominate the mean. For quantiles, check sensitivity to the assumed distribution and to sparse tail data; a high quantile is often less stable than the median. Also verify that reported contrasts use the intended marginal target population. A conditional model coefficient is not automatically the marginal mean or quantile contrast: if the scientific question is marginal, obtain the treatment-specific marginal distributions by averaging over the target population before calculating the contrasts.

The main alternative is not a single summary but a different estimand. A mean difference is natural when average outcome burden is the clinical target and the mean is well-defined. A quantile difference is useful when a particular distributional location is clinically important or when the mean is unrepresentative of a skewed distribution. Ratios or contrasts on a transformed scale may answer other questions, but they change the scale and interpretation and should be specified as such. Reporting several summaries can describe different features, but selecting whichever one appears most favorable after seeing the results creates a misleading account.

Finally, marginal quantile differences should not be interpreted as effects on the same individuals at a particular rank. The patient at the active-treatment 75th percentile need not be the same patient who would occupy the control 75th percentile. Likewise, a posterior probability that a contrast is negative is conditional on the likelihood, prior, model, and data used; it does not remove uncertainty about whether those choices adequately represent the trial and its target population.