In the case of testing the association between two or more variables, the data are portrayed as contingency tables.

**UNDERSTANDING CHI-SQUARE**

In chapters 8 and 9, we covered the *t* test and *Z* test, which use interval or ratio mea-sures. Now we turn to the
chi-square test, which is appropriate for nominal and or-dinal measurement. The
chi-square test may be used for two specific applications: (1) to assess
whether an observed proportion agrees with expectations; (2) to deter-mine
whether there is a statistically significant association between two variables
(such as variables that represent nominal level measurement or, in some cases,
ordi-nal level measurement).

In the case of testing the association between two
or more variables, the data are portrayed as contingency tables. these tables
are also known as cross-tabulation ta-bles. For example, the investigator might
cross-tabulate the results for a study of gender and smoking status. A
chi-square test could be used to determine the associ-ation between these two
variables. Later, we will give an example of how to set up a contingency table
and perform a chi-square test.

The formula for many test statistics with
approximate chi-square distributions is:

where

O = observed frequency

*E *= expected frequency

As an example of one of the simplest uses of the
foregoing formula, let us per-form the chi-square test for a single proportion.
(We will see that in some instances, the chi-square test may be used as an
alternative to tests of proportion discussed in Chapter 10.) The chi-square
test that we will use in this example shall be called a test with an a priori
theoretical hypothesis, because the expected frequency of the outcome is known
theoretically.

Suppose we run a coin toss experiment with 100
trials and find 70 heads; is this a biased outcome? That is, we want to know
whether this is a very unusual event for a fair coin toss. If so, we may decide
that the alternative—that the coin is loaded in favor of heads—may be more
plausible. The data may be portrayed as shown in Table 11.1.

We would expect a fair coin toss to produce 50%
heads and 50% tails in the long run (the theoretical a priori expectation).
Table 11.1 lists all of the elements re

This value is shown at the intersection of the last
column and last row. Substituting it in the chi-square formula, we obtain:

In order to evaluate whether this is a significant
chi-square value—i.e., whether the coin toss is unfair—we need to compare the
result we have obtained with the value in a chi-square table. We need to know
the number of degrees of freedom as-sociated with the coin toss experiment.
Degrees of freedom (the term means “free to vary”) are denoted by the symbol *df*. In this case, *df* = 1. (You may surmise that in a given number of coin tosses,
once the number of heads is known, then the number of tails is fixed; only one
value is free to vary. Let us say that in a small trial of 10 coin tosses, we
find six heads; the number of tails *must*
be four.)

In our example, we need to do a table lookup to
determine the chi-square critical value. As with other statistical tests, the
level of significance may be set to *p*
< 0.05 or 0.01 or 0.001. We know from a chi-square table that the chi-square
critical value is 3.84 for *df* = 1 at *p* < 0.05.

Therefore, the null hypothesis that the coin toss
is unbiased would be rejected, as we obtained a chi-square of 16. The coin toss
seems to be favoring heads. By the way, it is helpful to memorize this
particular chi-square value as it comes up in many situations that have one
degree of freedom, such as the 2 × 2 tables (shown in Sections 11.3 and 11.6).

One of the best statistical texts that deals
explicitly with categorical data is Agresti (1990). Refer to it if you are
interested in more details or aspects of the theory.

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