DAO1704 Chap.5 Grouped Comparisons and Simpson's Paradox
Grouped Comparisons and Simpson's Paradox
Overall rates carry the observed mix
An aggregate rate is a weighted average of rates inside subgroups. When compared groups face different proportions of easy and hard cases, the overall comparison combines performance with allocation.
Simpson's paradox is the striking form in which the aggregate direction reverses, but confounding matters even when only the magnitude changes.
A confounder needs an operational role
The third variable must relate to the outcome, differ across the compared groups and not merely be a consequence of the group label. Stratification then shows like-with-like rates.
Denominators remain visible because a small stratum should not look as stable as a large one.
Standardisation creates a common comparison
Applying one stated reference mix to each group's stratum-specific rates asks what the totals would be under comparable workloads. The raw rate remains useful for planning actual volume, while the standardised rate supports a fairer process comparison.
Neither alone establishes causality, so the conclusion names both views and the factors that remain unresolved.
The reference mix is another input
A combined historical mix, an equal mix and a forecast workload answer different questions. Test more than one defensible set of weights and report whether the ranking or only the size of the gap changes.
Preserve the observed rate beside the standardised result: operations need the actual workload, while comparison needs a common workload. Showing both prevents a fairness adjustment from being mistaken for a forecast and makes the management decision easier to audit.
What this chapter covers
- 01
Aggregate rates as weighted averages
- 02
Case mix and denominator visibility
- 03
Confounder conditions
- 04
Stratified like-with-like comparisons
- 05
Standardisation to a common mix
- 06
Operational and fair-comparison conclusions
Worked example · free
Standardising two support teams
- 2Calculate the four stratum rates.
- 2Calculate both observed overall rates.
- 2Apply the common reference mix.
- 2Interpret workload and process separately.
Key terms
- Aggregate Rate
- An overall proportion formed across all subgroups under their observed weights.
- Stratum
- A subgroup defined by a variable relevant to the comparison.
- Case Mix
- The proportions of easier, harder or otherwise distinct cases assigned to a group.
- Confounder
- A third variable associated with the outcome and distributed unevenly across compared groups.
- Standardisation
- Applying one common subgroup distribution to several sets of stratum rates.
- Simpson's Paradox
- A reversal between an aggregate comparison and the corresponding comparisons within subgroups.
Grouped Comparisons and Simpson's Paradox FAQ
Can case mix matter without a full reversal?
Yes. Unequal workloads can compress or exaggerate an overall difference even when the winner remains the same. Show the subgroup rates and apply a common mix when the size of the process difference affects the decision.
How should a meaningful grouping variable be chosen?
Use process knowledge to identify a variable linked to the outcome and unevenly distributed across groups. Do not try many arbitrary cuts until the preferred ranking appears. State the variable's role before computing the grouped comparison.
Does standardisation prove that one group causes better outcomes?
No. It removes the arithmetic effect of the chosen case mix under a stated reference distribution. Training, tools, selection and unmeasured differences may remain, so the result is a fairer descriptive comparison rather than a causal estimate.
Exam move
Build a two-by-two table with group, difficulty, successes and attempts. Calculate raw totals, stratum rates and a standardised rate under two different reference mixes. Explain why each view answers a different management question. Then create a second table where the aggregate direction reverses and identify the weight change that caused it. Keep denominators beside every percentage throughout the exercise.
Plot the standardised difference as the routine-case weight moves from zero to one and locate any crossover. Finish by stating which workload forecast would make each group preferable and which unmeasured factors prevent a causal conclusion. This turns one paradox example into a sensitivity method. Repeat the explanation without using the word paradox.
If you can describe the differing weights, subgroup rates and management purpose plainly, the reasoning will transfer to a less dramatic comparison.
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