Selecting the Right Objective Measure for Association Analysis*

where
and
∆ 1 = log(ad − bc) − log(pq − rs)
∆ 2 = log(a + b)(a + c)(b + c)(b + d) − log(p + q)(p + r)(q + s)(r + s).
If the marginal totals for both tables are identical, then any observed difference
between log(φ X ) and log(φ Y ) comes from the first term, ∆ 1 . Conversely, if the
marginals are not identical, then the observed difference in φ can be caused by
either ∆ 1 , ∆ 2 , or both.
The problem of non-uniform marginals is somewhat analogous to using accuracy
for evaluating the performance of classification models. If a data set
contains 99% examples of class 0 and 1% examples of class 1, then a classifier
that produces models that classify every test example to be class 0 would have
a high accuracy, despite performing miserably on class 1 examples. Thus, accuracy
is not a reliable measure because it can be easily obscured by differences
in the class distribution. One way to overcome this problem is by stratifying the
data set so that both classes have equal representation during model building. A
similar “stratification” strategy can be used to handle contingency tables with
non-uniform support, i.e., by standardizing the frequency counts of a contingency
table.
6.2 IPF Standardization
Mosteller presented the following iterative standardization procedure, which is
called the Iterative Proportional Fitting algorithm or IPF [5], for adjusting the
cell frequencies of a table until the desired margins, fi+ ∗ and f +j ∗ , are obtained:
Row scaling : f (k)
ij
Column scaling : f (k+1)
ij
= f (k−1)
ij × f ∗ i+
f (k−1)
i+
= f (k)
ij × f ∗ +j
f (k)
+j
(3)
(4)
An example of the IPF standardization procedure is demonstrated in Figure 6.
Theorem 1. The IPF standardization procedure is equivalent to multiplying the
contingency matrix M = [a b; c d] with
[ ] [ ] [ ]
k1 0 a b k3 0
0 k 2 c d 0 k 4
where k 1 , k 2 , k 3 and k 4 are products of the row and column scaling factors.
Proof. The following lemma is needed to prove the above theorem.
Lemma 1. The product of two diagonal matrices is also a diagonal matrix.