chapter10 - Chapter 10 Categorical Data Analysis Inference...

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Unformatted text preview: Chapter 10 Categorical Data Analysis Inference for a Single Proportion ( π ) • Goal: Estimate proportion of individuals in a population with a certain characteristic ( π ). This is equivalent to estimating a binomial probability • Sample: Take a SRS of n individuals from the population and observe y that have the characteristic. The sample proportion is y / n and has the following sampling properties: ) 5 ) 1 ( , : thumb of (Rule samples large for normal ely approximat : Shape 1 : Error Standard Estimated ) 1 ( : on distributi sampling of Dev. Std. and Mean : proportion Sample ^ ^ ^ ^ ^ ^ ≥- - =- = = = π π π π π π σ π μ π π π π n n n SE n n y Large-Sample Confidence Interval for π • Take SRS of size n from population where π is true (unknown) proportion of successes. – Observe y successes – Set confidence level (1- α ) and obtain z α /2 from z-table m C z m n n y ± = - = = ^ 2 / ^ ^ ^ : for interval confidence % SE : error of Margin 1 SE : Error Standard Estimated : Estimate Point ^ ^ π π π π π π α π Example - Ginkgo and Azet for AMS • Study Goal: Measure effect of Ginkgo and Acetazolamide on occurrence of Acute Mountain Sickness (AMS) in Himalayan Trackers • Parameter: π = True proportion of all trekkers receiving Ginkgo&Acetaz who would suffer from AMS. • Sample Data: n= 126 trekkers received G&A, y =18 suffered from AMS ) 204 ,. 082 (. 061 . 143 . : for CI % 95 061 . ) 031 (. 96 . 1 : %) 95 % 100 ) 1 (( error of Margin 031 . 126 ) 86 )(. 14 (. SE 143 . 126 18 ^ ^ ≡ ± = = =- = = = = π α π π m Wilson’s “Plus 4” Method • For moderate to small sample sizes, large-sample methods may not work well wrt coverage probabilities • Simple approach that works well in practice ( n ≥ 10) : – Pretend you have 4 extra individuals, 2 successes, 2 failures – Compute the estimated sample proportion in light of new “data” as well as standard error: m z m n n y ±- = + - = + + = ~ 2 / ~ ~ ~ : for interval confidence % 100 ) 1 ( SE : error of Margin 4 1 SE : Error Standard Estimated 4 2 : Estimate Point ~ ~ π π α π π π π α π Example: Lister’s Tests with Antiseptic • Experiments with antiseptic in patients with upper limb amputations (John Lister, circa 1870) • n =12 patients received antiseptic y =1 died ) 40 ,. ( ) 3988 ,. 0038 . ( 1913 . 1875 . : for CI % 95 1913 . ) 0976 (. 96 . 1 : %) 95 )100%- 1 ( error( of Margin 0976 . 16 ) 8125 (. 1875 . SE 1875 . 16 3 4 12 2 1 ~ ~ 2245- ≡ ± = = = = = = + + = π α π π Significance Test for a Proportion • Goal test whether a proportion ( π ) equals some null value π H : π= π ) ( 2 value- : : ) ( value- : : ) ( value- : : ) 1 ( : Statistic Test 2 / ^ obs obs a obs obs a obs obs a o obs z Z P P z z RR H z Z P P z z RR H z Z P P z z RR H n z ≥ = ≥ ≠ ≤ =- ≤ < ≥ = ≥-- = α α α π π π π π π π π π π Large-sample test works well when n π and n (1- π ) ≥ 5 Ginkgo and Acetaz for AMS...
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This note was uploaded on 06/04/2011 for the course STA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.

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chapter10 - Chapter 10 Categorical Data Analysis Inference...

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