is a widely used nursing practice rooted in mysticism but alleged to have a scientific basis.
Practitioners of TT claim to treat many medical conditions by using their hands to manipu-
late a ‘human energy field’ perceptible above the patient’s skin.” Emily recruited 21 practi-
tioners of therapeutic touch, blindfolded them, and then placed her hand over one of their
hands. If therapeutic touch is a real phenomenon, the principles behind it suggest that the
participant should be able to identify which of their hands is below Emily’s hand. Out of
280 trials, the participant was correct on 123 of them, which is an accuracy rate of 44%.
By chance we would expect the participants to be correct 50% of the time, or 140 times.
Although we can tell by inspection that participants performed even worse that chance
would predict, I have chosen this example in part because it raises an interesting question
of the statistical significance of a test. We will return to that issue shortly. The first ques-
tion that we want to answer is whether the data’s departure from chance expectation is sta-
tistically significantly greater than chance. The data follow in Table 6.1.
Even if participants were operating at chance levels, one category of response is likely
to come out more frequently than the other. What we want is a goodness-of-fit testto ask
whether the deviations from what would be expected by chance are large enough to lead us
to conclude that responses weren’t random.
The most common and important formula for involves a comparison of observed
and expected frequencies. The observed frequencies,as the name suggests, are the fre-
quencies you actually observed in the data—the numbers in row two of the table above.
The expected frequenciesare the frequencies you would expect if the null hypothesis were
true. The expected frequencies are shown in row 3 of Table 6.1. We will assume that par-
ticipants’ responses are independent of each other. (In this use of “independence,” I mean
that what the participant reports on trial kdoes not depend on what he or she reported on
trial k 2 1, though it does not mean that the two different categories of choice are equally
likely, which is what we are about to test.)
Because we have two possibilities over 280 trials, we would expect that there would be
140 correct and 140 incorrect choices. We will denote the observed number of choices with
the letter “O” and the expected number of choices with the letter “E.” Then our formula for
chi-square iswhere summation is taken over both categories of response.
This formula makes intuitive sense. Start with the numerator. If the null hypothesis is
true, the observed and expected frequencies (Oand E) would be reasonably close together
and the numerator would be small, even after it is squared. Moreover, how large the differ-
ence between Oand Ewould be ought to depend on how large a number we expected. If
we were taking about 140 correct, a difference of 5 choices would be a small difference.
But if we had expected 10 correct choices, a difference of 5 would be substantial. To keep
the squared size of the difference in perspective relative to the number of observations we
expect, we divide the former by the latter. Finally, we sum over both possibilities to com-
bine these relative differences.x^2 = a(O 2 E)^2
E
x^2142 Chapter 6 Categorical Data and Chi-Square
Table 6.1 Results of experiment on therapeutic touch
Correct Incorrect Total
Observed 123 157 280
Expected 140 140 280goodness-of-fit
test
observed
frequencies
expected
frequencies