By Tang, Wan; He, Hua; Tu, Xin M
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Extra info for Applied categorical and count data analysis
05. 2 Inference for Multinomial Variable A categorical variable x with more than two levels is said to follow the multinomial distribution. Let j index the possible levels of x (j = 1, 2, . . , k). , k Pr (x = j) = pj , pj ≥ 0 j = 1, 2, . . , k, pj = 1. 6) j=1 k Note that since j=1 pj = 1, only (k − 1) of the pj ’s are free parameters.
Thus, we only need to consider the following one-sided test: H0 : p = p0 vs. Ha : p > p0 . Similarly, if p < p0 , we only consider the one-sided alternative: Ha : p < p0 . 5) n k p (1 − k 0 p0 )n−k . Sometimes, the two-sided p-value is also defined by p = 2 min(τl , τu ). These two definitions are consistent for large sample size. For small samples, there may be some difference between the p-values obtained from the two methods. Note that because of the discrete nature of the distribution of K, the test is conservative.
5. By the estimate p and its asymptotic distribution, we can readily carry out the test by computing the probability of the occurrence of outcomes that are as or more extreme than the value of p based on the observed data. The meaning of “extreme” depends on the alternative hypothesis Ha , which the null is against. 5. 5) may be considered. More precisely, suppose the null is H0 : p = p0 , and let p0 denote the value of the statistic p computed based on the observed data. If the alternative is Ha : p = p0 , then the values of p based on data from all potential samples drawn from a population with p = p0 will generally deviate more from p0 than p0 does.