Lecture 21 Comparing Counts - Chi-square test

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1 Lecture 21 Comparing Counts - Chi-square test Thais Paiva STA Summer 2013 Term II August 5, / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

2 Lecture Plan 1 Goodness of fit 2 Independence 3 Homogeneity 2 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

3 CEOs and the Zodiac Fortune magazine collected the zodiac signs of 256 heads of the 400 largest companies: Sign Count Aries 23 Taurus 20 Gemini 18 Cancer 23 Leo 20 Virgo 19 Libra 18 Scorpio 21 Sagittarius 19 Capricorn 22 Aquarius 24 Pisces 29 3 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

4 CEOs and the Zodiac There are more Pisces than anything else Are there enough of them to constitute evidence that CEOs are more likely to be some signs rather than others? If the distribution were uniform, we would expect = births in each sign How closely to the observed births fit this null model? 4 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

5 Goodness of fit We will be testing whether the null model fits the observed data. Since there is no parameter to estimate, CIs do not make sense If we were only interested in Pisces, we could use a one-proportion z-test with H 0 : p = 1 12 H A : p > 1 12 But we are interested in the distribution of births. We need a test that takes all signs into account 5 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

6 Goodness of fit Assumptions and conditions: The data are counts for two or more categories Each unit is independent of the other units Sample size: We should expect to see at least 5 in each cell (some authors say at least 1) In the zodiac example, we would need n such that n 12 > 5 6 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

7 Goodness of fit Calculations: In each cell, we expect a certain count under the null ( n 12 ) If the null model fits the data, the expected and observed counts should be similar Just like with OLS regression, we will square the differences: p χ 2 (Obs i Exp i ) 2 = Exp i i=1 χ 2 p 1, where p is the number of cells 7 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

8 The χ 2 distribution Like the t, the χ 2 is indexed by degrees of freedom Unlike the t, the χ 2 is skewed and always positive density density x x χ 2 4 χ / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

9 The χ 2 test for goodness of fit p χ 2 (Obs i Exp i ) 2 = Exp i i=1 χ 2 p 1 density α 1 α χ α 2 x 9 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

10 One-sided or two-sided? If the data does not match the null model, then χ 2 will always be big It works like a one-sided test 10 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

11 Back to the zodiac H 0 : p Aries = p Taurus =... = p Pisces H A : not all p i equal Exp i = = for each i = 1,..., 12 χ 2 = 12 i=1 = (Obs i ) χ 2 11 which gives a p-value > 0.25 From the back of the book (p. 676): P[χ 2 11 > 13.7] = 0.25, so > 5.094] > 0.25 P[χ 2 11 There is virtually no evidence of a non-uniform distribution of zodiac signs among executives 11 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

12 Independence A study at the University of Texas Southwestern examined whether the risk of hepatitis C was related to whether people had tattoos and to where they got their tattoos. Hep. C No hep C Total Tattoo, parlor Tattoo, elsewhere No tattoo Total This type of data is different we have a single group that is categorized according to two variables This is called a contingency table 12 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

13 Independence We could convert the counts to proportions, and see whether P(Parlor and hep C) = P(Parlor) P(hep C), for example But even if the variables were independent, the equality would never turn out to hold exactly How close does it have to be? We can use a χ 2 test for independence 13 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

14 Independence χ 2 = R,C i=1,j=1 (Obs ij Exp ij ) 2 Exp ij χ 2 (R 1)(C 1) Hep. C No hep C Total Tattoo, parlor Tattoo, elsewhere No tattoo Total Assuming independence, Exp 11 = = 3.9 Number of degrees of freedom is now (R 1)(C 1) R: # of rows and C: # of columns 14 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

15 Independence: tattoos and hepatitis H 0 :Tattoo status and hepatitis status are independent H A :Tattoo status and hepatitis status are not independent χ 2 = 3,2 i=1,j=1 = (Obs ij Exp ij ) 2 Exp ij χ 2 2 which gives a p-value < (from the back of the book) There is strong evidence towards tattoo status and hepatitis C status not being independent 15 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

16 Homogeneity Many high schools survey graduating classes to determine their plans. We might wonder whether plans have stayed roughly the same through the decades Total College Employment Military Travel Total / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

17 Homogeneity Assumptions and conditions: The data are counts for two or more categories, with two or more groups to compare Each unit that is counted is independent of the other units Sample size: We should expect to see at least a 5 in each cell In the class plans example, we would need n such that n 4 > 5 for each year 17 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

18 Homogeneity χ 2 = R,C i=1,j=1 (Obs ij Exp ij ) 2 Exp ij χ 2 (R 1)(C 1) Total College Employment Military Travel Total Assuming independence, Exp 21 = = Calculation is exactly the same as in the independence test 18 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

19 Back to high school H 0 : The post-high school choices have the same distribution in 1980, 1990, and 2000 H A : The post-high school choices in 1980, 1990, and 2000 do not have the same distribution χ 2 = 4,3 i=1,j=1 = (Obs ij Exp ij ) 2 Exp ij χ 2 6 which gives a p-value < (from the back of the book) There is strong evidence of inhomogeneity 19 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

20 Summary 1 Goodness of fit 2 Independence 3 Homogeneity 20 / 20 Thais Paiva STA Summer 2013 Term II Lecture 21, 08/05/2013

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