© John Wiley & Sons, Inc.

FIGURE 12-2: Expected cell counts if the null hypothesis is true (there is no association between either drug and the outcome).

The reason you need these expected counts is that they represent what would happen under the null

hypothesis (meaning if the null hypothesis were true). If the null hypothesis were true:

In the CBD-treated group, you’d expect about 25.8 participants to experience pain relief (43

percent of 60), with the remaining 34.2 reporting no pain relief.

In the NSAIDs-treated group, you’d expect about 17.2 participants to feel pain relief (43 percent of

40) with the remaining 22.8 reporting no pain relief.

As you can see, this expected table assumes that you still have the overall pain relief rate of 43

percent, but that you also have the pain relief rates in each group equal to 43 percent. This is what

would happen under the null hypothesis.

Now that you have observed and expected counts, you’re no doubt curious as to how each cell in the

observed table differs from its companion cell in the expected table. To get these numbers, you can

subtract each expected count from the observed count in each cell to get a difference table (observed –

expected), as shown in Figure 12-3.

© John Wiley & Sons, Inc.

FIGURE 12-3: Differences between observed and expected cell counts if the null hypothesis is true.

As you review Figure 12-3, because you know the observed and expected tables in Figures 12-1 and

12-2 always have the same marginal totals by design, you should not be surprised to observe that the

marginal totals in the difference table are all equal to zero. All four cells in the center of this

difference table have the same absolute value (7.2), with a plus and a minus value in each row and

each column.

The pattern just described is always the case for

tables. For larger tables, the difference

numbers aren’t all the same, but they always sum up to zero for each row and each column.

The values in the difference table in Figure 12-3 show how far off from

your observed data are.

The question remains: Are those difference values larger than what may have arisen from random