Exact Test - Example: Pearson's Chi-squared Test Versus An Exact Test

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

\sum \frac{(\text{observed}-\text{expected})^2}{\text{expected}} = \sum_{k=1}^6 \frac{(X_k - n/6)^2}{n/6},

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.

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