Like our test for a population proportion, the chi-square test uses some approximations that become more accurate as we take more observations. Here is a rough rule for when it is safe to use this test.

The cocaine study easily passes this test: all the expected cell counts are either 8 or 16. Here is a concluding example that outlines the examination of a two-way table.

The first step is to write the data as a two-way table by adding the counts of subjects who did not suffer from heart disease. We also add the row and column totals, which we need to find the expected counts.

We can now follow the steps for a significance test, familiar from Chapter 22.

The hypotheses. The chi-square method tests these hypotheses:

H₀: no association between anger and CHD

H_a: some association between anger and CHD

The sampling distribution. We will see that all the expected cell counts are larger than 5, so we can safely apply the chi-square test. The two-way table of anger versus CHD has two rows and three columns. We will use critical values from the chi-square distribution with degrees of freedom df = $(2-1\left)\right(3-1)=2$.

The data. First find the expected cell counts. For example, the expected count of high-anger people with CHD is

$$\begin{array}{ccc}\mathrm{expected}\mathrm{count}& =& \frac{\text{row}1\text{total}\times \text{column}3\text{total}}{\text{table}\text{total}}\\ & =& \frac{\left(190\right)\left(633\right)}{8474}=14.19\end{array}$$

Here is the complete table of observed and expected counts side by side:

Looking at these counts, we see that the high-anger group has more CHD than expected and the low-anger group has less CHD than expected. This is consistent with what the percentages in Example 6 show. The chi-square statistic is

$$\begin{array}{lll}{\displaystyle {\chi }^2}\hfill & {\displaystyle =}\hfill & {\displaystyle \frac{{\left(53-69.73\right)}^2}{69.73}+\frac{{\left(110-106.08\right)}^2}{106.08}+\frac{{\left(27-14.19\right)}^2}{14.19}}\hfill \\ \hfill & \hfill & {\displaystyle +\frac{{\left(3057-3040.27\right)}^2}{3040.27}+\frac{{\left(4621-4624.92\right)}^2}{4624.92}+\frac{{\left(606-618.81\right)}^2}{618.81}}\hfill \\ \hfill & {\displaystyle =}\hfill & {\displaystyle 4.014+0.145+11.564+0.092+0.003+0.265=16.083}\hfill \end{array}$$

In practice, statistical software can do all this arithmetic for you. Look at the six terms that we sum to get ${\chi }^2$. Most of the total comes from just one cell: high-anger people have more CHD than expected.

Significance? Look at the df = 2 line of Table 24.1. The observed chi-square ${\chi }^2=16.083$ is larger than the critical value 13.82 for $ɑ=0.001$. We have highly significant evidence ($P<0.001$) that anger and heart disease are related. Statistical software can give the actual $P$-value. It is$P=0.0003$.

The conclusion. Can we conclude that proneness to anger causes heart disease? This is an observational study, not an experiment. It isn’t surprising to find that some lurking variables are confounded with anger. For example, people prone to anger are more likely than others to be those who drink and smoke. The study report used advanced statistics to adjust for many differences among the three anger groups. The adjustments raised the $P$-value from $P=0.0003$ to $P=0.02$ because the lurking variables explain some of the heart disease. This is still good evidence for a relationship if a significance level of 0.05 is used. Because the study started with a random sample of people who had no CHD and followed them forward in time, and because many lurking variables were measured and accounted for, it does give some evidence for causation. The next step might be an experiment that shows anger-prone people how to change. Will this reduce their risk of heart disease?