Simulation of Simpson’s Paradox With Hospital Data

1 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).

2 Setup

clear all

use "hospitaldata.dta", clear
    
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 |
     +------------------------------------------+

3 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 
    
graph bar (count) [fweight = count], over(outcome) over(hospital) blabel(bar) title("Hospital Outcomes") scheme(michigan)
        
quietly: graph export bivariategraph.png, width(2000) replace

| 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 

Bivariate Relationships

4 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 
    
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)

quietly: graph export multivariategraph.png, width(2000) replace

| 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 

Multivariate Relationships

5 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