Statistical Methods in Medical Research 4ed

516 Empirical methods for categorical data
Table 15.5 Frequencies (and percentages) of ABO blood groups in patients with peptic ulcer, patients
with gastric cancer and controls (Snedecor & Cochran, 1989, Ex. 11.10.2).
Blood
group
Peptic
ulcer (%)
Gastric
cancer (%) Controls (%) Total
O 983 (55) 383 (43) 2892 (48) 4258
A 679 (38) 416 (47) 2625 (43) 3720
B 134(7) 84(10) 570 (9) 788
Total 1796 (100) 883 (100) 6087 (100) 8766
As noted earlier, the total of the X 2 1 statistics is a little different from the X 2 4 value of
40 54for the whole table, but the discrepancy is slight. The value of X 2 1 for (c) gives
P ˆ 0 021, giving rise to some doubt about the lack of association of blood groups and
gastric cancer. The outstanding contrast is that between the proportions of group O in the
peptic ulcer patients and in the other subjects.
The process of collapsing rows and columns could have been speeded up by combining
some of the 1 DF contrasts into 2 DF contrasts. For example, (a) and (b) could have been
combined in a 2 3 table, (e). This gives an X 2 …2† of 1 01, scarcely different from the sum of
0 32 and 0 68, representing the overall association of blood groups A and B with the two
disease groups and controls. Or, to provide an overall picture of the association of blood
groups with gastric cancer, (a) and (c) could have been combined in a 3 2 table, (f). This
gives an X 2 …2† of 5 64(very close to 0 32 ‡ 5 30), which is, of course, less significant than
of 5 30 from (c).
the X 2 …1†
There are many ways of subdividing a contingency table. In Example 15.5,
the elementary table (a) could have been chosen as any one of the 2 2 tables
forming part of the whole table. The choice, as in that example, will often be
data-dependentÐthat is, made after an initial inspection of the data. There is,
therefore, the risk of data-dredging, and this should be recognized in any interpretation
of the analysis. In Example 15.5, of course, the association between
group O and peptic ulcer is too strong to be explained away by data-dredging.
15.6 Combination of 2 3 2tables
Sometimes a number of 2 2 tables, all bearing on the same question, are
available, and it seems natural to combine the evidence for an association
between the row and column factors. For example, there may be a number of
retrospective studies, each providing evidence about a possible association
between a certain disease and a certain environmental factor. Or, in a multicentre
clinical trial, each centre may provide evidence about a possible difference
between the proportions of patients whose condition is improved with treatment
A and treatment B. These are examples of stratification. In the clinical trial, for