fluctuations alone if

is really true? You need some kind of measurement unit by which to judge how

unlikely those difference values are. Recall from Chapter 10 that the standard error (SE) expresses

the general magnitude of random sampling, so looking at the SE as a type of measurement unit is a

good way for judging the size of the differences you may expect to see from random fluctuations alone.

It turns out that it is easy to approximate the SE of the differences because this is approximately equal

to the square root of the expected counts. The rigorous proof behind this is too complicated for most

mathophobes (as well as some normal people) to understand. Nevertheless, a simple informal

explanation is based on the idea that random event occurrences typically follow the Poisson

distribution for which the SE of the event count equals the square root of the expected count (as

discussed in Chapter 10).

Summarizing and combining scaled differences

For the upper-left cell in the cross-tab (CBD–treated participants who experience pain relief), you see

the following:

The observed count (Ob) is 33.

The expected count (Ex) is 25.8.

The difference (Diff) is

, or

.

The SE of the difference is

or

You can “scale” the Ob-Ex difference (in terms of unit of SE) by dividing it by the SE measurement

unit, getting the ratio

, or 1.42. This means that the difference between the

observed number of CBD-treated participants who experience pain relief and the number you would

have expected if the CBD had no effect on survival is about 1.42 times as large as you would have

expected from random sampling fluctuations alone. You can do the same calculation for the other three

cells and summarize these scaled differences. Figure 12-4 shows the differences between observed

and expected cell counts, scaled according to the estimated standard errors of the differences.

© John Wiley & Sons, Inc.

FIGURE 12-4: Differences between observed and expected cell counts.

The next step is to combine these individual scaled differences into an overall measure of the

difference between what you observed and what you would have expected if the CBD or NSAID use

really did not impact pain relief differentially. You can’t just add them up because the negative and