Statistical Methods for Psychology

Degrees of Freedom

Before we can compare our value of to the value in Appendix , we must know the de-

grees of freedom. For the analysis of contingency tables, the degrees of freedom are given by

df 5 (R 2 1)(C 2 1)

where

R 5 the number of rows in the table

and

C 5 the number of columns in the table

For our example we have R 5 2 and C 5 2; therefore, we have (2 2 1)(2 2 1) 51 df.

Evaluation of x^2

With 1 dfthe critical value of , as found in Appendix , is 3.84. Because our value of

7.71 exceeds the critical value, we will reject the null hypothesis that the variables are in-

dependent of each other. In this case we will conclude that whether a death sentence is im-

posed is related to the race of the defendant. When the victim was white, nonwhite

defendants were more likely to receive the death penalty than white defendants.^3

2 3 2 Tables are Special Cases

There are some unique features of the treatment of 2 3 2 tables, and the example that we

have been working with offers a good opportunity to explore them.

Correcting for Continuity

Many books advocate that for simple 2 3 2 tables such as Table 6.4 we should employ

what is called Yates’ correction for continuity,especially when the expected frequencies

are small. (The correction merely involves reducing the absolute value of each numerator

by 0.5 units before squaring.) There is an extensive literature on the pros and cons of Yates’

correction, with firmly held views on both sides. However, the common availability of

Fisher’s Exact Test, to be discussed next, makes Yates’ correction superfluous.

Fisher’s Exact Test

Fisher introduced what is called Fisher’s Exact Test in 1934 at a meeting of the Royal

Statistical Society. (Good (2001) has pointed out that one of the speakers who followed

Fisher referred to Fisher’s presentation as “the braying of the Golden Ass.” Statistical de-

bates at that time were far from boring, and no doubt Fisher had something equally kind to

say about that speaker.)

Without going into details, Fisher’s proposal was to take all possible 2 3 2 tables that

could be formed from the fixed set of marginal totals. He then determined the proportion

of those tables whose results are as extreme, or more so, than the table we obtained in our data.

x^2 x^2

x^2 x^2

Section 6.3 Two Classification Variables: Contingency Table Analysis 147

(^3) If the victim was nonwhite there was no significant relationship between race and sentence, although that has
been found in other data sets. The authors point out that when the victim was non white the prosecutor was more
likely to plea bargain, and the overall proportion of death sentences are much lower.
Yates’ correction
for continuity