Background

Simpson's paradox occurs when a bivariate association is reversed in a multivariate model. This example uses simulated data from hospitals (Wang et al., 2018).

Setup

. clear all

. use "https://github.com/agrogan1/newstuff/blob/master/categorical/simpsons-paradox-hospital-data/hospitaldata.dta?raw=true"

. list

     ┌──────────────────────────────────────────┐
     │ hospital      severity   outcome   count │
     ├──────────────────────────────────────────┤
  1. │   better   less severe   success      18 │
  2. │   better   less severe   failure       2 │
  3. │   better   more severe   success      32 │
  4. │   better   more severe   failure      48 │
  5. │   normal   less severe   success      64 │
     ├──────────────────────────────────────────┤
  6. │   normal   less severe   failure      16 │
  7. │   normal   more severe   success       4 │
  8. │   normal   more severe   failure      16 │
     └──────────────────────────────────────────┘

Outcome By Hospital Type

It appears as though patients do better at the normal hospital, as opposed to the better hospital.

. tabulate hospital outcome [fweight = count], row 

┌────────────────┐
│ Key            │
├────────────────┤
│   frequency    │
│ row percentage │
└────────────────┘

           │        outcome
  hospital │   failure    success │     Total
───────────┼──────────────────────┼──────────
    better │        50         50 │       100 
           │     50.00      50.00 │    100.00 
───────────┼──────────────────────┼──────────
    normal │        32         68 │       100 
           │     32.00      68.00 │    100.00 
───────────┼──────────────────────┼──────────
     Total │        82        118 │       200 
           │     41.00      59.00 │    100.00 

. graph bar (count) [fweight = count], over(outcome) over(hospital) blabel(bar) title("Hospital Outcomes") scheme(michigan)

. graph export bivariategraph.png, width(1000) replace
file bivariategraph.png saved as PNG format

{width=50%}

Outcome By Hospital Type by Severity

When we factor in the severity of the illness, we arrive at the reverse conclusion. Patients do better at the better hospital.

. generate severity_hospital = severity + " " + hospital // concatenate severity + hospital type

. tabulate severity_hospital outcome [fweight=count], row 

┌────────────────┐
│ Key            │
├────────────────┤
│   frequency    │
│ row percentage │
└────────────────┘

                   │        outcome
 severity_hospital │   failure    success │     Total
───────────────────┼──────────────────────┼──────────
less severe better │         2         18 │        20 
                   │     10.00      90.00 │    100.00 
───────────────────┼──────────────────────┼──────────
less severe normal │        16         64 │        80 
                   │     20.00      80.00 │    100.00 
───────────────────┼──────────────────────┼──────────
more severe better │        48         32 │        80 
                   │     60.00      40.00 │    100.00 
───────────────────┼──────────────────────┼──────────
more severe normal │        16          4 │        20 
                   │     80.00      20.00 │    100.00 
───────────────────┼──────────────────────┼──────────
             Total │        82        118 │       200 
                   │     41.00      59.00 │    100.00 

. graph bar [fweight = count] if severity == "less severe", ///
> title(Less Severe) ///
> over(outcome) ///
> blabel(bar) ///
> by(hospital) ////
> scheme(michigan) ///
> name(lesssevere, replace)

. graph bar [fweight = count] if severity == "more severe", ///
> title(More Severe) ///
> over(outcome) ///
> blabel(bar) ///
> by(hospital) ///
> scheme(michigan) ///
> name(moresevere, replace)

. graph combine lesssevere moresevere, title(Hospital Outcomes) scheme(michigan)

. graph export multivariategraph.png, width(1000) replace
file multivariategraph.png saved as PNG format

{width=50%}

Reference

Wang, B., Wu, P., Kwan, B., Tu, X. M., & Feng, C. (2018). Simpson’s Paradox: Examples. Shanghai Archives of Psychiatry, 30(2), 139–143. https://doi.org/10.11919/j.issn.1002-0829.218026