note20_GOF_print(4).pdf - Topic 20 Pearson Chi-Square...

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Topic 20. Pearson Chi-Square Goodness of Fit Test 1/9
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Pearson Chi-Square Goodness-of-fit Test Pearson’s chi-squared test is used to assess two types of comparison: I tests of independence/association (in the notes on Categorical Data Analysis) I tests of goodness of fit (this note set). A test of goodness of fit establishes whether or not an observed frequency distribution differs from a theoretical distribution. 2/9
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I Suppose { Y 1 , Y 2 , . . . , Y n } is a random sample from some distribution F in a Euclidean space. I We want to test H 0 : F = F 0 , where F 0 is a completely specified distribution. I Let Ω be the support of F 0 . Also let { A 1 , A 2 , . . . , A J } be a partition of Ω, that is, J j =1 A j = Ω and A j A = for 1 j 6 = J . I We further let n j = # { i : Y i A j } and p 0 j = P F 0 ( Y i A j ) for j = 1 , 2 , . . . , J . I Therefore, under H 0 , E ( n j ) = np 0 j for j = 1 , 2 , . . . , J . I The Pearson chi-square statistic is given by Q P = J X j =1 ( n j - np 0 j ) 2 np 0 j .
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