8.2 Simpson’s Paradox in the 2 × 2 Table 129In general, Karl Pearson ’ s chi - square test determines the goodness
of fi t of the observed data to the expected value under a model. This is
a general asymptotic result that applies to a wide variety of problems,
including testing for independence between two variables, as in the
current example. The asymptotic distribution in the general case of an
R × C contingency table is the central chi - square distribution with
( R − 1)( C − 1) degrees of freedom. So in the 2 × 2 table, R = 2 and
C = 2, and hence the degrees of freedom is 1. The chi - square statistic
is given by the formula
χ^22 =−∑(),OE Eii i
where O i is the observed total in for cell I , and E i is the expected total
for cell i.
χ^222
249 39 5 39 5 30 39 5 39 5
51 60 5 60 5 70 60
=− +− +
−+−
(.)/.(.)/.
(.)/.( .5 5 )/.^2 605 755=..
For α = 0.05, the critical value for a chi - square random variable
with 1 degree of freedom is 3.84. So, since 7.55 >> 3.84, the choice of
type of medical care does differ between men and women.8.2 SIMPSON ’ S PARADOX IN THE 2 × 2 TABLE
Sometimes, as for example in a meta - analysis, it may be reasonable to
combine results from two or more experiments, each of which produces
a 2 × 2 table. We simple cumulate for the corresponding cells in each
table the sum of the counts over all the tables.
However, this creates an apparent paradox, known as Simpson ’ s
paradox. Basically, Simpson ’ s paradox occurs when we see an apparent
association in each of the individual tables, but not in the combined
table or the association reverses!
To understand this better, we take the following example from
Lloyd ( 1999 , pp. 153 – 154). For this analysis (a fi ctitious example used
to illustrate the issue), a new cancer treatment is given, experimentally,
to the patients in hospital A, who have been categorized as either ter-
minal or nonterminal.
Before we analyze the patients based on their terminal/nonterminal
category, we naively think that we can see a difference in survival