When a Pooled PK Comparison Reverses
A pooled mean can tell a different story from the means within clinically relevant groups. In pharmacokinetics, this can happen when the groups being compared contain different proportions of subjects from strata with different typical concentrations. The resulting reversal is called Simpson’s paradox: a direction of comparison seen within every stratum changes when data are aggregated.
The reversal is not a contradiction in arithmetic. The pooled means answer a composition-dependent question: what is the mean among the subjects actually in each group? The within-stratum means answer a different question: how do the groups compare among subjects in the same stratum? Keeping those questions separate is essential before interpreting a pooled concentration or exposure summary.
Set Up the Comparison Carefully
Suppose each subject contributes one prespecified concentration summary, such as concentration at the same postdose time, measured in the same units. Let \(Y\) denote that summary, \(T\) the group or treatment, and \(G\) a categorical stratum. For example, \(G\) might identify lower- versus higher-clearance subjects. Write \(\bar{Y}_{tg}\) for the mean among subjects in group \(t\) and stratum \(g\), and \(n_{tg}\) for the number of such subjects.
The pooled mean for group \(t\) is the subject-weighted average of its stratum means:
Thus, the weights are the proportions of subjects in each stratum within that group. Unless those proportions are the same across groups, the pooled comparison combines differences in stratum-specific means with differences in composition. This formula concerns subject-level summaries: if subjects contribute unequal numbers of concentration observations and all observations are pooled directly, the resulting weights can differ from subject weights.
Worked Example: A Reversal in Concentration at Four Hours
Consider a hypothetical comparison of two formulations, A and B, at the same dose and at four hours after dosing. Each subject contributes one \(C_4\) value in \(\mu\text{g}/\text{L}\). Subjects are classified as lower- or higher-clearance. The table gives the number of subjects and their mean \(C_4\) in each cell.
| Formulation | Stratum | Number of subjects | Mean \(C_4\) (\(\mu\text{g}/\text{L}\)) |
|---|---|---|---|
| A | Lower clearance | 90 | 12 |
| A | Higher clearance | 10 | 4 |
| B | Lower clearance | 10 | 13 |
| B | Higher clearance | 90 | 5 |
Within the lower-clearance stratum, B has a higher mean: \(13-12=1\ \mu\text{g}/\text{L}\). Within the higher-clearance stratum, B also has a higher mean: \(5-4=1\ \mu\text{g}/\text{L}\). Yet the groups have very different compositions. Formulation A is 90% lower-clearance subjects, while formulation B is 90% higher-clearance subjects.
For A, the pooled mean is
For B, it is
The pooled difference is \(5.8-11.2=-5.4\ \mu\text{g}/\text{L}\), so the pooled means make B appear lower than A. Within both strata, B is higher by \(1\ \mu\text{g}/\text{L}\). The reversal occurs because the pooled means give much more weight to the higher-concentration lower-clearance stratum for A, and to the lower-concentration higher-clearance stratum for B.
Why the Reversal Occurs
The pooled difference can be separated into a within-stratum component and a composition component. For two groups, B and A, write \(p_{tg}=n_{tg}/n_t\) for the proportion in stratum \(g\). Add and subtract \(\sum_g p_{Bg}\bar{Y}_{Ag}\) in the difference between pooled means:
The first term averages the within-stratum differences using B’s stratum proportions. The second term reflects the difference in composition, using A’s stratum means as a reference. This decomposition is an identity; choosing the other group as the reference gives a different allocation of the total difference between the two terms, but the total remains the same.
For the hypothetical data, order the strata as lower then higher clearance. The within-stratum term is
The composition term is
Adding the terms gives \(1.0-6.4=-5.4\ \mu\text{g}/\text{L}\), exactly the pooled difference. The within-stratum advantage for B is outweighed by the composition term.
Standardizing to a Shared Composition
A standardized comparison uses the same chosen stratum weights for both groups. If \(w_g\geq 0\) and \(\sum_g w_g=1\), define the standardized mean as \(\bar{Y}^{\,*}_t=\sum_g w_g\bar{Y}_{tg}\). These weights describe the target composition for the summary; they are not automatically the composition of either observed group.
A useful consequence follows directly. The difference between standardized means is
If B’s mean is higher in every stratum and every stratum receives a nonnegative weight, this weighted difference is positive. Shared weights therefore prevent this particular composition-driven reversal. They do not, by themselves, prove a treatment effect or settle which target composition is scientifically appropriate.
Worked Example: Comparing Formulations in an Equal-Stratum Population
Use the same hypothetical stratum means, but give each stratum equal weight: \(w_{\text{lower}}=0.50\) and \(w_{\text{higher}}=0.50\). For A, the standardized mean is
For B, it is
The equal-stratum standardized difference is \(9-8=1\ \mu\text{g}/\text{L}\), matching the \(1\ \mu\text{g}/\text{L}\) difference in each stratum. The reversal in the observed pooled means is absent because both formulations are evaluated with the same 50:50 composition. This summary answers a hypothetical target-population question; it is not either formulation’s observed pooled mean.
Worked Example: Choosing a Lower-Clearance-Enriched Target
Suppose the scientific target instead has 80% lower-clearance and 20% higher-clearance subjects. These shared weights are \(0.80\) and \(0.20\). For A:
For B:
The standardized difference is \(11.4-10.4=1.0\ \mu\text{g}/\text{L}\). It is positive because both stratum-specific differences are \(1\ \mu\text{g}/\text{L}\). The standardized means differ from the equal-weight results because the target composition differs. A standardization is interpretable only when its weights and target population are stated.
Diagnostics, Alternatives, and Limits
When a pooled PK comparison seems surprising, check the data structure before treating the pooled result as a general pattern. A concise diagnostic sequence is:
Confirm that the compared values refer to the same time point or exposure measure, dose context, units, and subject-level calculation.
For each group-by-stratum cell, show \(n_{tg}\), \(\bar{Y}_{tg}\), and the group’s stratum proportions. Empty or very small cells make stratum comparisons unstable or unavailable.
Calculate each pooled mean from the cell counts and means, then check whether the pooled direction differs from all or some stratum-specific directions.
If using common weights, name the target stratum proportions and explain why they are relevant to the question.
The main alternatives answer different questions. The observed pooled mean describes the subjects in that group with their actual composition. Stratum-specific means compare groups within each stated stratum. A standardized mean compares groups under a shared, explicitly chosen composition. None is universally correct without regard to the scientific question.
Several limitations remain. Strata can be too broad: subjects with meaningfully different characteristics may be combined, leaving important composition differences hidden within a stratum. Conversely, dividing data into many small cells can make means imprecise or impossible to calculate. A reversal may also arise from sampling variability, so descriptive means alone do not quantify uncertainty. Formal inference requires an analysis suited to the design and dependence structure.
Finally, the labels “within-stratum” and “pooled” do not establish causation. If groups were not assigned comparably, other differences may remain even after stratifying on the displayed variable. A standardized descriptive comparison is not automatically a causal treatment contrast.
A Common Mistake
A common mistake is to conclude that one set of means must be wrong because the pooled ordering contradicts the ordering in every stratum. The correction is to calculate the weights: pooled means are weighted by each group’s own stratum proportions, while standardized means use common weights. In the example, B’s \(5.8\ \mu\text{g}/\text{L}\) pooled mean is arithmetically correct for its observed composition, but it does not describe a comparison with A under the same composition.
Check Your Understanding
Try each question, then compare your work with the solutions.
- In a two-stratum comparison, group A has stratum means 10 and 4, and group B has means 11 and 5. Is B higher within both strata?
- Using the means in Question 1, calculate the standardized difference if the shared weights are 25% for the first stratum and 75% for the second.
- Why can pooled means reverse the within-stratum direction even when each group’s stratum means are calculated correctly?
- If the stratum proportions are identical in groups A and B, can differing proportions cause a Simpson reversal in their pooled means?
Solutions. (1) Yes: \(11-10=1\) and \(5-4=1\), so B is higher by 1 unit in each stratum. (2) The standardized difference is \(0.25(11-10)+0.75(5-4)=0.25+0.75=1\) unit, so B remains higher. (3) Each pooled mean uses its group’s own stratum proportions; if those proportions differ and the stratum means differ, the composition contribution can outweigh the within-stratum differences. (4) No. With common proportions \(p_g\), the pooled difference is \(\sum_g p_g(\bar{Y}_{Bg}-\bar{Y}_{Ag})\); although it can have a direction different from some individual strata when stratum differences vary in sign, differing composition cannot be the cause of a reversal. If B is higher in every stratum, the common-weight pooled difference must also be positive.