SCIENCE · INVESTIGATE AND EXPLAIN

Repair an unfair comparison

Separate treatment effects from group composition.

Learning outcome: Compute group means and diagnose a confounded headline difference.

Starting recap

A weighted mean multiplies each stratum's mean by its fraction of the group.

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Try

The treatment adds 4 points to both baseline groups, yet the headline difference can be much larger.

Open the experiment

Predict

Can the treatment appear harmful even though it adds 4 in both groups?

Your prediction and reason

Experiment

Swap treatment and comparison high-baseline shares, then make them equal.

Repair a group comparison

Worked teaching model · synthetic values · High-baseline outcomes are 80 points; low-baseline outcomes are 40. The programmed treatment adds 4 in both strata. No noise or unmeasured confounder is included.

High-baseline share in treatment group: 80 % High-baseline share in comparison group: 20 %

Reset keeps your prediction, captures and practice history.

Calculated values on the same fixed scale
Row	Quantity	Value (points)
1	Treatment mean	76
2	Comparison mean	48
3	High stratum treatment	84
4	High stratum comparison	80
5	Low stratum treatment	44
6	Low stratum comparison	40

Headline difference: 28.0 points. Within each stratum: 4 points.

Treatment mean = 84 × treated high share + 44 × treated low share. The headline gap equals 4 + 40 × (treated high share − comparison high share). Matching composition repairs this particular bias, not every possible confounder.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Group composition can imitate an effect

The lab has high-baseline outcomes of 80 points and low-baseline outcomes of 40 points. Treatment adds 4 points in either group.

If the treated group is 80% high-baseline, its mean is 0.8 × 84 + 0.2 × 44 = 76. The comparison group, with 20% high-baseline, averages 0.2 × 80 + 0.8 × 40 = 48. The headline gap is 28 points, although the programmed within-group treatment effect is only 4.

The extra 24 points come from composition: 40 × (0.8 − 0.2). This is a confounded comparison because group membership changes with treatment status.

Repair this comparison without claiming too much

Set both groups to the same composition. At 50% high-baseline, treatment averages 64 and comparison averages 60. Their difference is 4.

Matching this observed factor repairs the bias in this specific model. It does not prove that an observational comparison in the world has no unmeasured differences. Random assignment, suitable pairing and careful measurement each address different threats.

Swap the original compositions. Treatment now averages 52 and comparison 72: a negative headline gap despite a positive within-group effect. Explain the reversal using the displayed strata, not simply the color of a bar.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Use the stratum values to explain the headline gap and the repaired comparison.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Correct an offset without hiding uncertainty Next: Choose controls and independent repetitions

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