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Unformatted text preview: STA 100 Lecture 23 Paul Baines Department of Statistics University of California, Davis March 2nd, 2011 Admin for the Day I Homework 6 due today at 5pm! I Project Proposals feedback later today (very short) I Optional status report can be submitted on Monday I Practical Significance: do the results actually matter References for Friday: Rosner, Ch 10.110.6 (7th Ed.) References for Wednesday: Rosner, Ch 11 (7th Ed.) χ 2 Goodness of Fit Test Last time we saw how we can compare observed counts to expected expects (under certain assumptions), and use the χ 2test to see if the assumptions (e.g., independence) are plausible. The idea of comparing what we observe to what we expect under a specific model is very general. It allows us to test the ‘Goodness of Fit’ of our model i.e., are our assumptions plausible? I H : Our model assumptions are correct, vs., I H 1 : Our model assumptions are not correct. Cholera Example Example: From lecture 9, the number of cases of Cholera in each household in an Indian village were recorded. The data look like this: > cholera house cholera_cases 1 1 2 2 3 1 4 1 ... etc ... > table(cholera$cholera_cases) 1 2 3 4 35 32 16 6 1 1 2 3 4 Indian Cholera Cases Cases per household Frequency 5 10 15 20 25 30 35 Cholera: Poisson? Let X i be the number of Cholera cases in house i . Then suppose X i ∼ Poisson ( λ ), with each house being independent. The sample mean of the data was 0.93 cases/household. I Q: Estimate λ . We estimate ˆ λ = 0 . 93. I Q: What is the probability of a house having no cases of Cholera? P ( X i = 0) = e . 93 = 0 . 394 I Q: How many of the 90 households might we expect to have no cases of Cholera? 90 * . 394 = 35 . 5 Cholera: Poisson or not? Cholera Cases per Household: 1 2 3 4 5+ Observed: 35 32 16 6 1 Expected if Pois(0.93): 35.5 33.0 15.4 4.8 1.1 0.2 It looks like the Poisson assumption is pretty good for this data. Later in the course we will formally answer whether it is good enough. . . (Today is the ‘later in the course’. . . ) GoodnessofFit Tests Just as for contingency tables, the χ 2 goodnessoffit test requires approximations. Each ‘cell’ or ‘bin’ must not have an expected value less than 5. We can combine cells to achieve this. . . Cholera: Poisson or not? The combined version: Cholera Cases per Household: 1 2 3+ Observed: 35 32 16 7 Expected if Pois(0.93): 35.5 33.0 15.4 6.1 X 2 = ( O 1 E 1 ) 2 E 1 + ( O 2 E 2 ) 2 E 2 + ( O 3 E 3 ) 2 E 3 + ( O 4 E 4 ) 2 E 4 = (35 35...
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