ON THE STATISTICAL PROBLEMS EMERGED FROM
THE INVESTIGATION OF THE 2´2 CONTINGENCY TABLES

Róbert Rajkó

Department of Unit Operations and Environmental Engineering
College Faculty of Food Engineering University of Szeged
H-6701 Szeged, POB 433
E-mail: rajko@sol.cc.u-szeged.hu

Evaluating many scientific experiments leads to the investigation of 2´2 contingency tables, e.g., studying biological, agrotechnical, trophological, clinical, toxicological, environmental damage, medicinal etc. effects of drugs, chemicals, compounds etc. ("materials"). In these cases there are two groups: one of them is given from the investigated material(s), the other is not (this latter one is called as control group). After that the effects in the two groups are compared, i.e., comparing the effects of the materials in the first group with the unprompted effects in the control group. Statistical decisions are made based on the difference between the effects in the two groups. The decision might be based on the following 2´2 contingency table:
 

 
effects exist
no effects
sum:
1. group (treated)
a
c
a+c=m
2. group (control)
b
d
b+d=n
sum:
a + b = r
c + d = s
N

The outcomes a and b, when there are effects, can be considered as random values derived from two binomial distributions with parameters (m, p1) and (n, p2). For the statistical decision the following null and alternative hypotheses can be chosen:

In case of (2) and (3) the tests are one-sided, while for (4) it is two-sided. The test statistics can be given conditional or unconditional, as well as exact or approximation forms. The well-known and used conditional Fisher's exact test was presented in [1] as the selection criterion for pair-correlation method (PCM).

The aim of this presentation is to introduce the unconditional exact test, which is unknown in chemometrics yet, its advantages and disadvantages, and the first and second kind of errors.

[1] R. Rajkó, K. Héberger: Conditional Fisher's exact test as a selection criterion for pair-correlation method. Type I and Type II errors. Chemometrics and Intelligent Laboratory Systems, 57/1, 1-14, 2001.