Example: Pearson's Chi-squared Test Versus An Exact Test
A simple example of the occasion for this concept may be seen by observing that Pearson's chi-squared test is an approximate test. Suppose Pearson's chi-squared test is used to ascertain whether a six-sided die is "fair", i.e. gives each of the six outcomes equally often. If the die is thrown n times, then one "expects" to see each outcome n/6 times. The test statistic is
where Xk is the number of times outcome k is observed. If the null hypothesis of "fairness" is true, then the probability distribution of the test statistic can be made as close as desired to the chi-squared distribution with 5 degrees of freedom by making the sample size n big enough. But if n is small, then the probabilities based on chi-squared distributions may not be very close approximations. Finding the exact probability that this test statistic exceeds a certain value then requires combinatorial enumeration of all outcomes of the experiment that result in such a large value of the test statistic. Moreover, it becomes questionable whether the same test statistic ought to be used. A likelihood-ratio test might be preferred as being more powerful, and the test statistic might not be a monotone function of the one above.
Read more about this topic: Exact Test
Famous quotes containing the words pearson, test and/or exact:
“Education is considered the peculiar business of women; perhaps for that very reason it is one of the worst-paid businesses in the world ...”
—Katharine Pearson Woods (1853–1923)
“The first reading of a Will, where a person dies worth anything considerable, generally affords a true test of the relations’ love to the deceased.”
—Samuel Richardson (1689–1761)
“Neither Aristotelian nor Russellian rules give the exact logic of any expression of ordinary language; for ordinary language has no exact logic.”
—Sir Peter Frederick Strawson (b. 1919)