Statistical Methods for Psychology

of the direction in which the totals will be ordered. I referred to this problem a few pages

back when discussing a problem raised by Jennifer Mahon. A solution will be given in

Chapter 10 (Section 10.4), where I discuss creating a correlational measure of the relation-

ship between the two variables.

6.9 Likelihood Ratio Tests

An alternative approach to analyzing categorical data is based on likelihood ratios.

(Exhibit 6.1b included the likelihood ratio along with the standard Pearson chi-square.) For

large sample sizes the two tests are equivalent, though for small sample sizes the standard

Pearson chi-square is thought to be better approximated by the exact chi-square distribu-

tion than is the likelihood ratio chi-square (Agresti, 1990). Likelihood ratio tests are heav-

ily used in log-linear models, discussed in Chapter 17, for analyzing contingency tables,

because of their additive properties. Such models are particularly important when we want

to analyze multi-dimensional contingency tables. Such models are being used more and

more, and you should be exposed to such methods, at least minimally.

Without going into detail, the general idea of a likelihood ratio can be described quite

simply. Suppose we collect data and calculate the probability or likelihood of the data

occurring given that the null hypothesis is true. We also calculate the likelihood that the

data would occur under some alternative hypothesis (the hypothesis for which the data

are most probable). If the data are much more likely for some alternative hypothesis than

for , we would be inclined to reject. However, if the data are almost as likely under

as they are for some other alternative, we would be inclined to retain. Thus, the

likelihood ratio (the ratio of these two likelihoods) forms a basis for evaluating the null

hypothesis.

Using likelihood ratios, it is possible to devise tests, frequently referred to as “maxi-

mum likelihood ,” for analyzing both one-dimensional arrays and contingency tables.

For the development of these tests, see Agresti (2002) or Mood and Graybill (1963).

For the one-dimensional goodness-of-fit case,

where and are the observed and expected frequencies for each cell and “ln” denotes

the natural logarithm (logarithm to the base e). This value of can be evaluated using the

standard table of on C 2 1 degrees of freedom.

For analyzing contingency tables, we can use essentially the same formula,

where and are the observed and expected frequencies in each cell. The expected fre-

quencies are obtained just as they were for the standard Pearson chi-square test. This statis-

tic is evaluated with respect to the distribution on (R 2 1)(C 2 1) degrees of freedom.

Death Sentence

Defendant’s Race Yes No Total

Nonwhite 33 251 284

White 33 508 541

Total 66 759 825

x^2

Oij Eij

x^2 (R 2 1)(C 2 1)= (^2) aOijlna
Oij
Eij
b
x^2
x^2
Oi Ei
x^2 (C 2 1)= (^2) aOilna
Oi
Ei
b
x^2

H 0 H 0

H 0 H 0

156 Chapter 6 Categorical Data and Chi-Square

likelihood ratios