Introduction to Categorical Data Analysis

2.6 EXACT INFERENCE FOR SMALL SAMPLES 47
2, 3, 4). The observed table, three correct guesses of the four cups having milk added
first, has probability
P(3) =
� �� �
4 4
3 1
� �
8
4
= [4!/(3!)(1!)][4!/(1!)(3!)]
[8!/(4!)(4!)]
= 16
= 0.229
70
For Ha: θ>1, the only table that is more extreme consists of four correct guesses.
It has n11 = n22 = 4 and n12 = n21 = 0, and a probability of
� �� ��� �
4 4 8
P(4) =
= 1/70 = 0.014
4 0 4
Table 2.9 summarizes the possible values of n11 and their probabilities.
The P -value for Ha: θ>1 equals the right-tail probability that n11 is at least as
large as observed; that is, P = P(3) + P(4) = 0.243. This is not much evidence
against H0: independence. The experiment did not establish an association between
the actual order of pouring and the guess, but it is difficult to show effects with such
a small sample.
For the potential n11 values, Table 2.9 shows P -values for Ha: θ>1. If the tea
taster had guessed all cups correctly (i.e., n11 = 4), the observed result would have
been the most extreme possible in the right tail of the hypergeometric distribution.
Then, P = P(4) = 0.014, giving more reason to believe her claim.
2.6.3 P -values and Conservatism for Actual P (Type I Error)
The two-sided alternative Ha: θ �= 1 is the general alternative of statistical dependence,
as in chi-squared tests. Its exact P -value is usually defined as the two-tailed
sum of the probabilities of tables no more likely than the observed table. For Table 2.8,
summing all probabilities that are no greater than the probability P(3) = 0.229 of
Table 2.9. Hypergeometric Distribution for Tables with
Margins of Table 2.8
n11 Probability P -value X 2
0 0.014 1.000 8.0
1 0.229 0.986 2.0
2 0.514 0.757 0.0
3 0.229 0.243 2.0
4 0.014 0.014 8.0
Note: P -value refers to right-tail hypergeometric probability for one-sided
alternative.