The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

8.2 Simpson’s Paradox in the 2 × 2 Table 131

Here, the picture is quite different. The survival rate is much lower
in the terminal patients (as we should expect), and many more terminal
patients got the new treatment compared with the old 118 versus 38.
In the terminal group, the survival rate for those getting the new treat-
ment is 14.4% compared with only 5.2% for patients on the old treat-
ment. For the nonterminal patients, the new treatment group has a
survival rate of 97.1%, slightly higher than the 95.6% for the old treat-
ment group.
So here is the paradox. The new treatment is apparently better in
both subgroup analyses (although probably not statistically signifi -
cantly better for the nonterminal patients). So based on the subgroup
analysis, the new treatment might get regulatory approval. However, if
we only had the combined results, we would be convinced that the new
treatment is inferior to the old treatment.
So why does Simpson ’ s paradox occur and how do we resolve it?
First, notice the imbalance between the subgroups. Only 156 patients
were terminal compared with 286 in the nonterminal group. Also, in
the terminal group, there were far more patients getting the new treat-
ment (118), while only 38 patients got the old treatment. These imbal-
ances mask the benefi t of the new treatment when the data is combined.
Also notice that the survival rates are so drastically different for termi-
nal and nonterminal patients.
A total of 275 out of 286 nonterminal patients survived (96.15%),
whereas only 19 out of 156 survived among the terminal patients only
(12.18%). So the combination of the two groups makes no sense. It is
like adding apples with oranges. In reality, the combined table is mean-
ingless and presents a distorted picture.
In such cases, we would not combine these two studies in a meta -
analysis, as they are estimating radically different success probabilities.

Table 8.4

Nonterminal Patients Only: Survival Versus

Treatment

Treatment Survived 2 years Died within 2 years Total

New 100 3 103

Old 175 8 183

Total 275 11 286