Agreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
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1 Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log as the two methods use equivalet levels of sigificace / cofidece ad the SEs are the same. A two sided hypothesis with threshold of α is equivalet to a cofidece iterval with CL = 1 α. A oe sided hypothesis with threshold of α is equivalet to a cofidece iterval with CL = 1 (2 α). If H 0 is rejected, a cofidece iterval that agrees with the result of the hypothesis test should ot iclude the ull value. If H 0 is failed to be rejected, a cofidece iterval that agrees with the result of the hypothesis test should iclude the ull value. Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sigificace level vs. cofidece level Sigificace level vs. cofidece level Two sided Oe sided Sigificace level vs. cofidece level Example - Waitig Times A 95% cofidece iterval for the average waitig time at a emergecy room is (128 miutes, 147 miutes). Determie if the followig statemets are true or false, (a) a hypothesis test of H A : µ 120 mi at α = 0.05 is equivalet to this CI. (b) a hypothesis test of H A : µ > 120 mi at α = is equivalet to this CI. (c) This iterval does ot support the claim that the average wait time is 120 miutes (d) The claim that the average wait time is 120 miutes would ot be rejected usig a 90% cofidece iterval Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
2 Sigle populatio proportio Example - Experimetal Desig Two scietists wat to kow if a certai drug is effective agaist high blood pressure. The first scietist wats to give the drug to 1000 people with high blood pressure ad see how may of them experiece lower blood pressure levels. The secod scietist wats to give the drug to 500 people with high blood pressure, ad ot give the drug to aother 500 people with high blood pressure, ad see how may i both groups experiece lower blood pressure levels. Which is the better way to test this drug? Results from the GSS Sigle populatio proportio The GSS asks the same questio, below is the distributio of resposes from the 2010 survey: All 1000 get the drug get the drug 500 do t 571 Total 670 (a) All 1000 get the drug (b) 500 get the drug, 500 do t Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sigle populatio proportio Parameter ad poit estimate We would like to estimate the proportio of all Americas who have good ituitio about experimetal desig, i.e. would aswer 500 get the drug 500 do t. What are the parameter of iterest ad the poit estimate? Parameter of iterest: Proportio of all Americas who have good ituitio about experimetal desig. p (a populatio proportio) Poit estimate: Proportio of sampled Americas who have good ituitio about experimetal desig. Sigle populatio proportio Iferece o a proportio What percet of all Americas have a good ituitio about experimetal desig, i.e. would aswer 500 get the drug 500 do t? We ca aswer this research questio usig a cofidece iterval, which we kow is always of the form poit estimate ± ME Ad we also kow that ME = critical value stadard error of the poit estimate. SEˆp =? CV =? ˆp (a sample proportio) Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
3 Sigle populatio proportio Sigle populatio proportio Idetifyig whe a sample proportio is early ormal Proportios ad the CLT What kid of probability model ca we use for ˆp? It may be useful to istead thik about ˆp, what distributio will that have? ˆp Biom(, p) ˆp X N(µ = p, σ 2 = p(1 p)) We ca the fid the distributio of ˆp by dividig by, ˆp X / N(µ = p, σ 2 = p(1 p)/) Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Cetral limit theorem for proportios Sample proportios will be early ormally distributed with mea equal to p (1 p) the populatio proportio, p, ad stadard error equal to. ( ) p (1 p) ˆp N mea = p, But of course this is true oly uder certai coditios... ay guesses? Assumptios/coditios: 1. Idepedece: Radom sample 10% coditio: If samplig without replacemet, < 10% of the populatio. 2. Normality: At least 10 successes (p 10) ad 10 failures ((1 p) 10). Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sigle populatio proportio Cofidece itervals for a proportio Sigle populatio proportio Cofidece itervals for a proportio Back to experimetal desig... The GSS foud that 571 out of 670 (85%) of Americas aswered the questio o experimetal desig correctly. Estimate (usig a 95% cofidece iterval) the proportio of all Americas who have a good ituitio about experimetal desig? Give: = 670, ˆp = = Are CLT coditios met? 1. Idepedece: The sample is radom, ad 670 < 10% of all Americas, therefore we ca assume that oe respodet s respose is idepedet of aother. 2. Success-failure: 571 people aswered correctly (successes) ad 99 aswered icorrectly (failures), both are greater tha 10. Calculatig the Cofidece Iterval We are give that = 670, ˆp = 0.85, we also just leared that the p(1 p) stadard error of the sample proportio is. What is the 95% cofidece iterval for this proportio? CI = poit estimate ± margi of error = poit estimate ± critical value SE = ˆp ± z SE = 0.85 ± 1.96 = (0.82, 0.88) 670 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
4 Sigle populatio proportio Choosig a sample size whe estimatig a proportio Sigle populatio proportio Choosig a sample size whe estimatig a proportio Choosig a sample size How may people should you sample i order to reduce the margi of error of a 95% cofidece iterval dow to 1%. ME = z SE What if there is t a previous study?... use ˆp = 0.5. Why? if you do t kow ay better, is a good guess ˆp = 0.5 gives the most coservative estimate largest stadard error ad thus the largest possible sample size. p (1 p) Usig ˆp from previous study should be at least 4,899 p * (1 - p) p Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sigle populatio proportio Hypothesis testig for a proportio Sigle populatio proportio Hypothesis testig for a proportio CI vs. HT for proportios Back to the GSS Success-failure coditio: CI: At least 10 observed successes ad failures HT: At least 10 expected successes ad failures, calculated usig the ull value, p 0 Stadard error: CI: calculate usig observed sample proportio: p(1 p) ˆp(1 ˆp) HT: calculate usig the ull value: p 0 (1 p 0 ) With meas SE oly depeded o s or σ ad, so it was ot determied i ay way by H 0. With proportios the mea ad SE both deped o p, therefore H 0 affects the SE. Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 The GSS foud that 571 out of 670 (85%) of Americas aswered the questio o experimetal desig correctly. Do these data provide covicig evidece that more tha 80% of Americas have a good ituitio about experimetal desig? H 0 : p = 0.80 H A : p > = Z = = p value = = sample proportios Sice p-value is small we reject H 0. The data provide covicig evidece that more tha 80% of Americas have a good ituitio o experimetal desig. Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
5 Sigle populatio proportio Commo Misiterpretatios Hypothesis testig for a proportio Differece of two proportios Example - Meltig ice cap survey 11% of 1,001 Americas respodig to a 2006 Gallup survey stated that they have objectios to celebratig Hallowee o religious grouds. At 95% cofidece level, the margi of error for this survey a is ±3%. A ews piece o this study s fidigs states: More tha 10% of all Americas have objectios o religious grouds to celebratig Hallowee. Is this statemet justified? Scietists predict that global warmig may have big effects o the polar regios withi the ext 100 years. Oe of the possible effects is that the orther ice cap may completely melt. Would this bother you a great deal, some, a little, or ot at all if it actually happeed? (a) A great deal (b) Some (c) A little (d) Not at all Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Differece of two proportios Results from the GSS & Duke The GSS asks this questio, below is the distributio of resposes from the 2010 survey: A great deal 454 Some 124 A little 52 Not at all 50 Total 680 The same questio was asked of 88 Duke studets, of which 56 said it would bother them a great deal. We will collapse the data such that we cosider oly the resposes of a great deal or ot a great deal. Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Collapsed Results Differece of two proportios US Duke Total A great deal Not a great deal Total This is a example of a 2 x 2 cotigecy table. We are iterested i comparig proportio of Duke studets who say it would both them a gread deal (P(GD Duke) = 56/88) to the proportio of all Americas who say it would both them a gread deal (P(GD US) = 454/680). What does it mea if P(GD Duke) = P(GD US)? Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
6 Differece of two proportios Parameter ad poit estimate Differece of two proportios Iferece for comparig proportios Parameter of iterest: Differece betwee the proportios of all Duke studets ad all Americas who would be bothered a great deal by the orther ice cap completely meltig. p Duke p US Poit estimate: Differece betwee the proportios of sampled Duke studets ad sampled Americas who would be bothered a great deal by the orther ice cap completely meltig. ˆp Duke ˆp US The details are the same as before... CI: poit estimate ± margi of error HT: Use Z = poit estimate ull value SE to fid appropriate p-value. We just eed the appropriate stadard error of the poit estimate (SE pduke p US ), SE (p1 p 2 ) = p 1 (1 p 1 ) 1 + p 2(1 p 2 ) 2 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Differece of two proportios Cofidece itervals for differece of proportios Differece of two proportios Cofidece itervals for differece of proportios Coditios for iferece o the differece of proportios 1 Idepedece withi groups: The US group is sampled radomly ad we re assumig that the Duke group represets a radom sample as well. Duke < 10% of all Duke studets ad 680 < 10% of all Americas. We ca assume that the attitudes of Duke studets i the sample are idepedet of each other, ad attitudes of US residets i the sample are idepedet of each other as well. 2 Idepedece betwee groups: The sampled Duke studets ad the US residets are idepedet of each other. 3 Success-failure: At least 10 observed successes ad 10 observed failures i both groups. CI for differece of proportios Costruct a 95% cofidece iterval for the differece betwee the proportios of Duke studets ad Americas who would be bothered a great deal by the meltig of the orther ice cap (p Duke p US ). Duke US A great deal Not a great deal Total Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
7 Differece of two proportios HT for comparig proportios Differece of two proportios HT for comparig proportios Hypotheses for testig the differece of two proportios Which of the followig is the correct set of hypotheses for testig if the proportio of all Duke studets who would be bothered a great deal by the meltig of the orther ice cap differs from the proportio of all Americas who do? H 0 : p Duke = p US H 0 : p Duke p US = 0 H A : p Duke p US H A : p Duke p US 0 Flashback to workig with oe proportio Whe costructig a cofidece iterval for a populatio proportio, we check if the observed umber of successes ad failures are at least 10. ˆp 10 (1 ˆp) 10 Whe coductig a hypothesis test for a populatio proportio, we check if the expected umber of successes ad failures are at least 10. p 0 10 (1 p 0 ) 10 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Differece of two proportios HT for comparig proportios Differece of two proportios HT for comparig proportios A slight wrikle... I the case of comparig two proportios where H 0 : p 1 = p 2, there is t a ull value we ca use to calculated the expected umber of successes ad failures i each sample or the SE. Therefore, we eed to first fid a commo (pooled) proportio for the two groups, ad use that i our aalysis. This ivolves fidig the proportio of total successes amog all observatios. Pooled estimate of a proportio Calculate the estimated pooled proportio of Duke studets ad Americas who would be bothered a great deal by the meltig of the orther ice cap. Duke US A great deal Not a great deal Total ˆp pooled = # of successes i 1 + # of successes i = 1 ˆp ˆp ˆp pooled = = Which sample proportio (ˆp Duke or ˆp US ) is closer to the pooled estimate? Why? Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
8 Differece of two proportios HT for comparig proportios Differece of two proportios HT for comparig proportios Implicatios for the SE HT for comparig proportios Uder the ull hypothesis we are statig that p 1 = p 2 which does ot uiquely idetify either p 1 or p 2. Therefore we are usig the pooled proportio (ˆp) as our best guess for p 1 ad p 2 uder the ull hypothesis. For a cofidece iterval we have see that p1 (1 p 1 ) + p 2(1 p 2 ) ˆp1 (1 ˆp 1 ) + ˆp 2(1 ˆp 2 ) Do these data suggest that the proportio of all Duke studets who would be bothered a great deal by the meltig of the orther ice cap differs from the proportio of all Americas who do? ˆp pooled = 0.664, 1 = 88, 2 = 680 Therefore, for a hypothesis test we will use ˆp as our approximatio for p 1 ad p 2 p1 (1 p 1 ) + p 2(1 p 2 ) ˆp(1 ˆp) ˆp(1 ˆp) ( 1 ˆp(1 ˆp) + 1 ), where ˆp = 1 ˆp ˆp Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Recap Recap - iferece for oe proportio Recap Recap - comparig two proportios Populatio parameter: p, poit estimate: ˆp Coditios: idepedece - radom sample ad 10% coditio at least 10 successes ad failures - observed for CI - expected for HT Stadard error: for CI: use ˆp for HT: use p 0 p(1 p) Populatio parameter: (p 1 p 2 ), poit estimate: (ˆp 1 ˆp 2 ) Coditios: idepedece withi groups - radom sample ad 10% coditio met for both groups idepedece betwee groups at least 10 successes ad failures i each group - observed for CI - expected for HT SE (ˆp1 ˆp 2 ) = p1 (1 p 1 ) 1 + p 2(1 p 2 ) 2 for CI: use ˆp 1 ad ˆp 2 for HT: whe H 0 : p 1 = p 2 : use ˆp pool = #suc 1+#suc whe H 0 : p 1 p 2 = (some value other tha 0): use ˆp 1 ad ˆp 2 - this is pretty rare Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33 Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
9 Recap Referece - stadard error calculatios oe sample two samples mea σ σ σ2 2 2 proportio p(1 p) p1 (1 p 1 ) 1 + p 2(1 p 2 ) 2 Whe workig with meas, it s very rare that σ is kow, so we usually use s as a approximatio. Whe workig with proportios, we will ot kow p therefore if doig a hypothesis test, p comes from the ull hypothesis if costructig a cofidece iterval, use ˆp istead Sta102 / BME102 (Coli Rudel) Lec 13 October 15, / 33
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