Biostatistics For Dummies, 2nd Edition (Monika Wahi, John Pezzullo)

CHAPTER 12 Comparing Proportions and Analyzing Cross-Tabulations 163

» ^{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.

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 2

2 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 H0 your

observed data are. The question remains: Are those difference values larger than

what may have arisen from random fluctuations alone if H0 is really true? You

need some kind of measurement unit by which to judge how unlikely those differ-

ence 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 mea-

surement 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

FIGURE 12-3:

Differences

between

observed and

expected cell

counts if the null

hypothesis is

true.