Recovery at two hospitals (Simpson's paradox)

Patient-level outcomes for 1 000 patients treated for the same condition at two fictional hospitals (Hospital A, a regional referral centre, and Hospital B, a community hospital). Each row is one patient with hospital, severity on admission, age, length of stay and whether they recovered. Hospital A has the higher recovery rate within both mild and severe cases yet the lower rate overall, because it receives most of the severe cases — Simpson's paradox. The aggregated counts are in hospital_outcomes_summary.

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Rows: 1,000
Columns: 6
One row is: one patient admission
Version: v1

Data dictionary

Column	Type	Unit	Summary	Description
`patient_id` Patient	id			Anonymous patient code.
`hospital` Hospital	category		2 levels: Hospital A, Hospital B	Treating hospital (fictional).
`severity` Severity	ordinal		2 levels: mild, severe	Severity on admission.
`age` Age	integer	years	mean 57.16, median 58, SD 15.37, range 18–95	Age on admission.
`length_of_stay_days` Length of stay	integer	days	mean 6.615, median 5, SD 5.917, range 1–41	Days in hospital.
`recovered` Recovered	boolean		2 levels	True if the patient recovered by 30 days after admission.

What it is good for

Compute recovery rates overall and by severity and observe the reversal (Simpson's paradox).
Explain the reversal through the different case mix (severity is a confounder).
Compute a severity-standardised (adjusted) recovery rate for each hospital.
Discuss which comparison answers the question 'where should I be treated?'.

Methods: contingency tables, conditional proportions, stratification, direct standardisation, Mantel–Haenszel, logistic regression.

Data quality notes

Constructed teaching data: exact cell counts guarantee the paradox (A better within each severity, worse overall).
Fictional hospitals and patients; no missing values.

Provenance and licence

The counts in each hospital × severity × outcome cell are fixed exactly (A mild 95/100, A severe 300/400, B mild 360/400, B severe 70/100). Age and length of stay are simulated: severe patients are older (mean ≈66 vs ≈48) and stay longer (gamma, mean ≈9 vs ≈3 days); non-recovered patients stay longer. Row order is shuffled.

Synthetic data · CC0-1.0 (synthetic data generated by Statistics Quest) · generated deterministically (seed 20241021). It does not describe real people or organisations; any resemblance is coincidental.

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Dataset

Recovery at two hospitals (Simpson's paradox)

1,000 rows · one row = one patient admission · synthetic data

About this dataset Open in the lab

Patient	Hospital	Severity	Age (years)	Length of stay (days)	Recovered
PT0001	Hospital B	mild	54	5	true
PT0002	Hospital A	severe	77	8	true
PT0003	Hospital B	mild	61	5	true
PT0004	Hospital B	mild	50	3	true
PT0005	Hospital A	mild	38	7	true
PT0006	Hospital B	severe	42	6	true
PT0007	Hospital A	severe	57	13	false
PT0008	Hospital A	severe	78	7	true
PT0009	Hospital A	mild	40	1	true
PT0010	Hospital A	severe	78	9	true
PT0011	Hospital B	mild	69	1	false
PT0012	Hospital A	severe	84	12	true

Used in

Contingency tables · Comparing Groups and Categorical Data
Chi-square test of independence · Comparing Groups and Categorical Data
Choosing a test · Comparing Groups and Categorical Data
Simpson's paradox · Resampling and Statistical Reasoning
Rates and proportions in public health · Applied Statistics for Professionals