Statistical Methods for Psychology

was motivated by a question sent to me by Jennifer Mahon at the University of Leicester,

England, who has graciously allowed me to use her data for this example. Ms Mahon was

interested in the question of whether the likelihood of dropping out of a study on eating

disorders was related to the number of traumatic events the participants had experienced in

childhood.

The data from this study are shown below. I have taken the liberty of altering them very

slightly so that I don’t have to deal with the problem of small expected frequencies at the

same time that I am trying to show how to make use of the ordinal nature of the data. The

altered data are still a faithful representation of the effects that she found.

Number of Traumatic Events

012341 Total

Dropout 25 13 9 10 6 63

Remain 31 21 6 2 3 63

Total 56 34 15 12 9 126

At first glance we might be tempted to apply a standard chi-square test to these data,

testing the null hypothesis that dropping out of treatment is independent of the number of

traumatic events the person experienced during childhood. If we do that we find a chi-

square of 9.459 on 4 df,which has an associated probability of .051. Strictly speaking, this

result does not allow us to reject the null hypothesis, and we might conclude that traumatic

events are not associated with dropping out of treatment. However, that answer is a bit too

simplistic.

Notice that Trauma represents an ordered variable. Four traumatic events are more than

3, 3 traumatic events are more than 2, and so on. If we look at the percentage of partici-

pants who dropped out of treatment as a function of the number of traumatic events they

had experienced as children, we see that there is a general, though not a monotonic,

increase in dropouts as we increase the number of traumatic events. However, this trend

was not allowed to play any role in our calculated chi-square. What we want is a statistic

that does take order into account.

A Correlational Approach

There are several ways we can accomplish what we want, but they all come down to as-

signing some kind of ordered metric to our independent variables. Dropout is not a prob-

lem because it is a dichotomy. We could code dropout as 1 and remain as 2, or dropout as 1

and remain as 0, or any other two values we like. The result will not be affected by our

choice of values. When it comes to the number of traumatic events, we could simply use

the numbers 0, 1, 2, 3, and 4. Alternatively, if we thought that 3 or 4 traumatic events would

be much more important than 1 or 2, we might use 0, 1, 2, 4, 6. In practice, as long as we

chose numbers that are monotonically increasing, and are not very extreme, the result will

not change much as a function of our choice. I will choose to use 0, 1, 2, 3, and 4.

Now that we have established a metric for each independent variable, there are several

different ways that we could go. We’ll start with one that has good intuitive appeal. We will

simply correlate our two variables.^3 Each participant will have a score of 0 or 1 on Dropout,

and a score between 0 and 4 on Trauma. The standard Pearson correlation between those

Section 10.4 Analysis of Contingency Tables with Ordered Variables 307

(^3) Many articles in the literature refer to Maxwell (1961) as a source for dealing with ordinal data. With one minor
exception, Maxwell’s approach is the one advocated here, though it is difficult to tell that from his description
because his formulae were selected for computational ease.