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# note16 - STAT5044 Regression and Anova Inyoung Kim 1 37...

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STAT5044: Regression and Anova Inyoung Kim 1 / 37

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Outline 1 Test of goodness- of- ﬁt 2 Test of independence 3 Test of homogeneity 2 / 37
Test of goodness-of-ﬁt: Test whether the data come from a multinomial (or binomial) distribution Test of independence: Test whether two categoricl variables, measured from a randpm sample, are independence of each other. Test of homogeneity: Test whether two random samples of categorical data have the same distribution. 3 / 37

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Introduction To test whether a coin is biased, we ﬂip a coin 100 times and observe the number of heads. Denoting π = Pr ( Head ) , we can write H 0 : π = 0 . 5 vs H a : π 6 = 0 . 5. Test statistic Z = ˆ π - π q π ( 1 - π ) n N ( 0 , 1 ) under H 0 asymotitically. Decision Rule: reject H 0 if | Z | > Z α / 2 4 / 37
Introduction Consider the distribution of Z 2 which has an approximately a chi-square dist with 1 df. Rewrite Z 2 in the following form: Z 2 = ( n ˆ π - n π ) 2 n π ( 1 - π ) = ( n ˆ π - n π ) 2 n π + ( n ˆ π - n π ) 2 n ( 1 - π ) and by letting π 2 = 1 - π , ˆ π 2 = 1 - ˆ π and recognizing ( n ˆ π - n π ) 2 = ( n ˆ π 2 - n π 2 ) , Z 2 = ( n ˆ π - n π ) 2 n π + ( n ˆ π 2 - n π 2 ) 2 n π 2 χ 2 1 5 / 37

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Introduction More generally, we can think of an experiment having k ( 2 ) mutually exclusive and exhaustive outcomes A 1 ,..., A k which form a partitaion of the sample space S For i = 1 ,..., k , let π i = Pr ( A i ) . If we repreat the experiment n times and denote the number of observed outcome A i by Y i . Then it can be shown that Z 2 = k i = 1 ( Y i - n π i ) 2 n π i χ 2 k - 1 6 / 37
Introduction Example Suppose we observed 49 heads and 51 tails in 100 tossings. Under H 0 : π = 0 . 5, the expected numbers of head and tail are n π = 100 × 0 . 5 = 50. The observed statistic is χ 2 obs = ( 49 - 50 ) 2 50 + ( 51 - 50 ) 2 50 = 2 50 = 0 . 25 If we observed 40 heads and 60 tails instead, then the observed statistic is χ 2 obs = ( 40 - 50 ) 2 50 + ( 60 - 50 ) 2 50 = 200 50 = 4 Siince χ 2 has an approximate chi-square distribution with 1 df, the critical value for an α = 0 . 05 test is χ 2 1 ( 0 . 05 ) = 3 . 84 The ﬁrst result would accept H 0 while the second one would reject H 0 at 5% level. 7 / 37

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Remark The minimum sample size n for the χ 2 test is n π i 5 for all i = 1 ,..., k This is to ensure that the chi-square approximation work. In practice this recommendation is not met quite often, and statisticians tend to combine two or more catgories for analysis. 8 / 37
Goodness of ﬁt test Example Does the sex of successive children in a family behave like independent Bernoulli trials? If that is the case, then the number of boys in a family of a given size has a binomial distribution with a ﬁxed “success” probability π . The following table summarizes the data on the number of boys in

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