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

CHAPTER 12 Comparing Proportions and Analyzing Cross-Tabulations 165

Instead of summing the differences, statisticians prefer to sum the squares of dif-

ferences, because the squares are always positive. This is exactly what’s done in

the chi-square test. Figure 12-5 shows the squared scaled differences, which are

calculated from the observed and expected counts in Figures 12-1 and 12-2 using

the formula Ob

/Ex

–

(rather than by squaring the rounded-off numbers in

Figure 12-4, which would be less accurate).

You then add up these squared scaled differences: 2 01

1 52

3 01

2 27

8 81

to get the chi-square test statistic. This sum is an excellent test statistic to mea-

sure the overall departure of your data from the null hypothesis:»

» ^{If the null hypothesis is true (use of CBD or NSAID does not impact pain relief}

status), this statistic should be quite small.»

» ^{If one of the levels of treatment has a disproportionate association with the}

outcome (in either direction), it will affect the whole table, and the result will

be a larger test statistic.

Determining the p value

Now that you calculated the test statistic, the only remaining task before interpre-

tation is to determine the p value. The p value represents the probability that ran-

dom fluctuations alone, in the absence of any true effect of CBD or NSAIDs on pain

relief, could lead to a value of 8.81 or greater for this test statistic. (We introduce p

values in Chapter 3.) Once again, the rigorous proof is very complicated, so we

present an informal explanation:

When the expected cell counts are very large, the Poisson distribution becomes

very close to a normal distribution (see Chapter 24 for more on the Poisson distri-

bution). If the H0 is true, each scaled difference should be an approximately nor-

mally distributed random variable with a mean of zero and a standard deviation

of 1. The mean is zero because you subtract the expected value from the observed

value, and the standard deviation is 1 because it is divided by the SE. The sum of

the squares of one or more normally distributed random numbers is a number

FIGURE 12-5:

Components of

the chi-square

statistic: squares

of the scaled

differences.