Confidence Intervals for Two Proportions

Transcription

1 PASS Samle Size Software Chater 6 Cofidece Itervals for Two Proortios Itroductio This routie calculates the grou samle sizes ecessary to achieve a secified iterval width of the differece, ratio, or odds ratio of two ideedet roortios. Cautio: These rocedures assume that the roortios obtaied from future samles will be the same as the roortios that are secified. If the samle roortios are differet from those secified whe ruig these rocedures, the iterval width may be arrower or wider tha secified. Four Procedures Documeted Here There are four rocedures i the meus described i this chater. These rocedures are very similar excet for the tye of arameterizatio. The arameterizatio ca be i terms of roortios, differeces i roortios, ratios of roortios, ad odds ratios. Techical Details A bacgroud of the comariso of two roortios is give, followed by details of the cofidece iterval methods available i this rocedure. Comarig Two Proortios Suose you have two oulatios from which dichotomous (biary) resoses will be recorded. The robability (or ris) of obtaiig the evet of iterest i oulatio (the treatmet grou) is ad i oulatio (the cotrol grou) is. The corresodig failure roortios are give by q ad q. The assumtio is made that the resoses from each grou follow a biomial distributio. This meas that the evet robability i is the same for all subjects withi a oulatio ad that the resoses from oe subject to the ext are ideedet of oe aother. Radom samles of m ad idividuals are obtaied from these two oulatios. The data from these samles ca be dislayed i a -by- cotigecy table as follows Success Failure Total Poulatio a c m Poulatio b d Totals s f N 6-

2 PASS Samle Size Software Cofidece Itervals for Two Proortios The followig alterative otatio is sometimes used: Success Failure Total Poulatio x x Poulatio x x Totals m m N The biomial roortios ad are estimated from these data usig the formulae a x m ad b x Whe aalyzig studies such as these, you usually wat to comare the two biomial robabilities ad. The most direct methods of comarig these quatities are to calculate their differece or their ratio. If the biomial robability is exressed i terms of odds rather tha robability, aother measure is the odds ratio. Mathematically, these comariso arameters are Parameter Comutatio Differece δ Ris Ratio φ / Odds Ratio ψ / q / q q q The choice of which of these measures is used might at seem arbitrary, but it is imortat. Not oly is their iterretatio differet, but, for small samle sizes, the coverage robabilities may be differet. Differece The (ris) differece δ is erhas the most direct method of comariso betwee the two evet robabilities. This arameter is easy to iterret ad commuicate. It gives the absolute imact of the treatmet. However, there are subtle difficulties that ca arise with its iterretatio. Oe iterretatio difficulty occurs whe the evet of iterest is rare. If a differece of 0.00 were reorted for a evet with a baselie robability of 0.40, we would robability dismiss this as beig of little imortace. That is, there usually little iterest i a treatmet that decreases the robability from to However, if the baselie robably of a disease was 0.00 ad 0.00 was the decrease i the disease robability, this would rereset a reductio of 50%. Thus we see that iterretatio deeds o the baselie robability of the evet. A similar situatio occurs whe the amout of ossible differece is cosidered. Cosider two evets, oe with a baselie evet rate of 0.40 ad the other with a rate of 0.0. What is the maximum decrease that ca occur? Obviously, the first evet rate ca be decreased by a absolute amout of 0.40 which the secod ca oly be decreased by a maximum of 0.0. So, although creatig the simle differece is a useful method of comariso, care must be tae that it fits the situatio. Ratio The (ris) ratio φ / gives the relative chage i the disease ris due to the alicatio of the treatmet. This arameter is also direct ad easy to iterret. To comare this with the differece, cosider a treatmet that reduces the ris of disease for to Which sigle umber is most elighteig, the fact that the 6-

3 PASS Samle Size Software Cofidece Itervals for Two Proortios absolute ris of disease has bee decreased by , or the fact that ris of disease i the treatmet grou is oly 55.8% of that i the cotrol grou? I may cases, the ercetage (ris ratio) commuicates the imact of the treatmet better tha the absolute chage. Perhas the biggest drawbac to this arameter is that it caot be calculated i oe of the most commo exerimetal desigs: the case-cotrol study. Odds Ratio Chaces are usually commuicated as log-term roortios or robabilities. I bettig, chaces are ofte give as odds. For examle, the odds of a horse wiig a race might be set at 0-to- or 3-to-. How do you traslate from odds to robability? A odds of 3-to- meas that the evet will occur three out of five times. That is, a odds of 3-to- (.5) traslates to a robability of wiig of The odds of a evet are calculated by dividig the evet ris by the o-evet ris. Thus, i our case of two oulatios, the odds are o ad o For examle, if is 0.60, the odds are 0.60/ Rather tha rereset the odds as a decimal amout, it is re-scaled ito whole umbers. Thus, istead of sayig the odds are.5-to-, we say they are 3-to-. Aother way to comare roortios is to comute the ratio of their odds. The odds ratio of two evets is ψ o o Although the odds ratio is more comlicated to iterret tha the ris ratio, it is ofte the arameter of choice. Reasos for this iclude the fact that the odds ratio ca be accurately estimated from case-cotrol studies, while the ris ratio caot. Also, the odds ratio is the basis of logistic regressio (used to study the ifluece of ris factors). Furthermore, the odds ratio is the atural arameter i the coditioal lielihood of the two-grou, biomial-resose desig. Fially, whe the baselie evet-rates are rare, the odds ratio rovides a close aroximatio to the ris ratio sice, i this case,, so that ψ φ Cofidece Itervals for the Differece May methods have bee devised for comutig cofidece itervals for the differece betwee two roortios δ. Seve of these methods are available i the Cofidece Itervals for Two Proortios [Proortios] usig Proortios ad Cofidece Itervals for Two Proortios [Differeces] rocedures. The seve cofidece iterval methods are. Score (Farrigto ad Maig). Score (Miettie ad Nurmie) 6-3

4 PASS Samle Size Software Cofidece Itervals for Two Proortios 3. Score with Correctio for Sewess (Gart ad Nam) 4. Score (Wilso) 5. Score with Cotiuity Correctio (Wilso) 6. Chi-Square with Cotiuity Correctio (Yates) 7. Chi-Square (Pearso) Newcombe (998b) coducted a comarative evaluatio of eleve cofidece iterval methods. He recommeded that the modified Wilso score method be used istead of the Pearso Chi-Square or the Yate s Corrected Chi- Square. Beal (987) foud that the Score methods erformed very well. The lower L ad uer U limits of these itervals are comuted as follows. Note that, uless otherwise stated, z z α / is the aroriate ercetile from the stadard ormal distributio. C.I. for Differece: Farrigto ad Maig s Score Farrigto ad Maig (990) roosed a test statistic for testig whether the differece is equal to a secified valueδ 0. The regular MLE s ad are used i the umerator of the score statistic while MLE s ad costraied so that δ 0 are used i the deomiator. The sigificace level of the test statistic is based o the asymtotic ormality of the score statistic. The test statistic formula is z FMD δ 0 q q where the estimates ad are comuted as i the corresodig test of Miettie ad Nurmie (985) give as δ 0 B cos ( A) A π cos 3 L 3L C B 3 3 B sig ( C) 3 L 9L L 3L L3 L L L L0 C 7 6 L L L3 ( ) [ N δ 0 N x] 0 ( N N ) 0 N 0 xδ 0 δ 0 L δ M L δ M L 3 N 3 6-4

5 PASS Samle Size Software Cofidece Itervals for Two Proortios Farrigto ad Maig (990) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z FMD z α / z FMD z α / C.I. for Differece: Miettie ad Nurmie s Score Miettie ad Nurmie (985) roosed a test statistic for testig whether the differece is equal to a secified valueδ 0. The regular MLE s ad are used i the umerator of the score statistic while MLE s ad costraied so that δ0 are used i the deomiator. A correctio factor of N/(N-) is alied to mae the variace estimate less biased. The sigificace level of the test statistic is based o the asymtotic ormality of the score statistic. The formula for comutig this test statistic is where δ 0 B cos ( A) A π cos 3 L 3L C B 3 3 z MND δ 0 q q N N B sig ( C) 3 L 9L L 3L L3 L L L L0 C 7 6 L L L3 ( ) [ N δ 0 N x] 0 ( N N ) 0 N 0 xδ 0 δ 0 L δ M L M L 3 δ N 3 Miettie ad Nurmie (985) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z z MND α / z MND z α / 6-5

6 PASS Samle Size Software Cofidece Itervals for Two Proortios C.I. for Differece: Gart ad Nam s Score Gart ad Nam (990) age 638 roosed a modificatio to the Farrigto ad Maig (990) differece test that δ stad for the Farrigto ad Maig differece test statistic described z FM corrected for sewess. Let ( ) above. The sewess corrected test statistic z GN is the aroriate solutio to the quadratic equatio where / V γ 6 ( δ ) q ( q ) q ( q ) 3 ( ) γ δ γ 0 ( ) zgnd ( ) zgnd zfmd( ) Gart ad Nam (988) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z GND z α / z GND z α / C.I. for Differece: Wilso s Score as Modified by Newcombe (with ad without Cotiuity Correctio) For details, see Newcombe (998b), age 876. where ( l ) u ( u ) l B z m ( u ) l ( l ) u C z m ad l ad u are the roots of ( ) z m ad l ad u are the roots of ( ) z 0 0 L B U C 6-6

7 PASS Samle Size Software Cofidece Itervals for Two Proortios 6-7 C.I. for Differece: Yate s Chi-Square with Cotiuity Correctio For details, see Newcombe (998b), age 875. m m z L ) ( ) ( m m z U ) ( ) ( C.I. for Differece: Pearso s Chi-Square For details, see Newcombe (998b), age 875. m z L ) ( ) ( m z U ) ( ) ( For each of the seve methods, oe-sided itervals may be obtaied by relacig α/ by α. For two-sided itervals, the distace from the differece i samle roortios to each of the limits may be differet. Thus, istead of secifyig the distace to the limits we secify the width of the iterval, W. The basic equatio for determiig samle size for a two-sided iterval whe W has bee secified is L U W For oe-sided itervals, the distace from the variace ratio to limit, D, is secified. The basic equatio for determiig samle size for a oe-sided uer limit whe D has bee secified is ( ) U D The basic equatio for determiig samle size for a oe-sided lower limit whe D has bee secified is ( ) L D Each of these equatios ca be solved for ay of the uow quatities i terms of the others. Cofidece Itervals for the Ratio (Relative Ris) May methods have bee devised for comutig cofidece itervals for the ratio (relative ris) of two roortios / φ. Six of these methods are available i the Cofidece Itervals for Two Proortios [Ratios] rocedure. The six cofidece iterval methods are. Score (Farrigto ad Maig). Score (Miettie ad Nurmie) 3. Score with Correctio for Sewess (Gart ad Nam) 4. Logarithm (Katz) 5. Logarithm / (Walter) 6. Fleiss

8 PASS Samle Size Software Cofidece Itervals for Two Proortios C.I. for Ratio: Farrigto ad Maig s Score Farrigto ad Maig (990) roosed a test statistic for testig whether the ratio is equal to a secified value φ 0. The regular MLE s ad are used i the umerator of the score statistic while MLE s ad costraied so that / φ0 are used i the deomiator. A correctio factor of N/(N-) is alied to icrease the variace estimate. The sigificace level of the test statistic is based o the asymtotic ormality of the score statistic. Here is the formula for comutig the test where φ 0 B A Nφ 0 B 4AC A z FMR [ N φ x N x ] B C M 0 φ0 as i the test of Miettie ad Nurmie (985). / φ0 q q φ0 Farrigto ad Maig (990) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z FMR z α / z FMR z α / C.I. for Ratio: Miettie ad Nurmie s Score Miettie ad Nurmie (985) roosed a test statistic for testig whether the ratio is equal to a secified value φ 0. The regular MLE s ad are used i the umerator of the score statistic while MLE s ad costraied so that / φ0 are used i the deomiator. A correctio factor of N/(N-) is alied to mae the variace estimate less biased. The sigificace level of the test statistic is based o the asymtotic ormality of the score statistic. Here is the formula for comutig the test z MNR / φ0 q q N φ0 N 6-8

9 PASS Samle Size Software where φ 0 Cofidece Itervals for Two Proortios B A Nφ 0 B 4AC A [ N φ x N x ] B C M 0 φ0 Miettie ad Nurmie (985) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z z MNR α / z MNR z α / C.I. for Ratio: Gart ad Nam s Score Gart ad Nam (988) age 39 roosed a modificatio to the Farrigto ad Maig (988) ratio test that corrected for sewess. Let z FM ( φ ) stad for the Farrigto ad Maig ratio test statistic described above. The sewess corrected test statistic z GN is the aroriate solutio to the quadratic equatio ϕ z z z φ ϕ 0 where ( q ) q ( q ) q ϕ 3 / 6 u q q u ( ) ( ) GNR ( ) GNR FMR( ) Gart ad Nam (988) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z z GNR α / zgnr z α / 6-9

10 PASS Samle Size Software Cofidece Itervals for Two Proortios C.I. for Ratio: Logarithm (Katz) This was oe of the first methods roosed for comutig cofidece itervals for ris ratios. For details, see Gart ad Nam (988), age 34. L φ ex z q q U φ ex z q q where φ C.I. for Ratio: Logarithm (Walters) For details, see Gart ad Nam (988), age 34. where a φ ex l m b l u a m b q q V φ m φ q q q ( q ) ( m ) ( q ) ( ) 3/ q q µ 3 v v m q q L φ ex U φ ex ( z u ) ( z u ) 6-0

11 PASS Samle Size Software Cofidece Itervals for Two Proortios C.I. for Odds Ratio ad Relative Ris: Iterated Method of Fleiss Fleiss (98) resets a imroved cofidece iterval for the odds ratio ad relative ris. This method forms the cofidece iterval as all those value of the odds ratio which would ot be rejected by a chi-square hyothesis test. Fleiss gives the followig details about how to costruct this cofidece iterval. To comute the lower limit, do the followig.. For a trial value of ψ, comute the quatities X, Y, W, F, U, ad V usig the formulas X ψ ( m s) ( s) Y X 4msψ ψ X Y A ψ ( ) B s A C m A D f m A ( ) W A B C D F ( a A ) W ( ψ ) z α / ψ T Y ψ Y U B C A D [( a A ) ( )] U W a V T A [ X ( m s) ms( )] Fially, use the udatig equatio below to calculate a ew value for the odds ratio usig the udatig equatio ( ) ( ) F ψ V ψ. Cotiue iteratig util the value of F is arbitrarily close to zero. The uer limit is foud by substitutig for i the formulas for F ad V. Cofidece limits for the relative ris ca be calculated usig the exected couts A, B, C, ad D from the last iteratio of the above rocedure. The lower limit of the relative ris φ lower φ uer A B A B lower lower uer uer m m 6-

12 PASS Samle Size Software Cofidece Itervals for Two Proortios 6- Cofidece Itervals for the Odds Ratio May methods have bee devised for comutig cofidece itervals for the odds ratio of two roortios ψ Eight of these methods are available i the Cofidece Itervals for Two Proortios [Odds Ratios] rocedure. The eight cofidece iterval methods are. Exact (Coditioal). Score (Farrigto ad Maig) 3. Score (Miettie ad Nurmie) 4. Fleiss 5. Logarithm 6. Matel-Haeszel 7. Simle 8. Simle / C.I. for Odds Ratio: Coditioal Exact The coditioal exact cofidece iterval of the odds ratio is calculated usig the ocetral hyergeometric distributio as give i Sahai ad Khurshid (995). That is, a ( ) 00 α % cofidece iterval is foud by searchig for ψ L ad ψ U such that ( ) ( ) α ψ ψ L x L m m ad ( ) ( ) α ψ ψ U x U m m where ( ), 0 max m ad ( ), mi m

13 PASS Samle Size Software Cofidece Itervals for Two Proortios 6-3 Farrigto ad Maig s Test of the Odds Ratio Farrigto ad Maig (990) develoed a test statistic similar to that of Miettie ad Nurmie but with the factor N/(N-) removed. The formula for comutig this test statistic is ( ) ( ) q N q N q q z FMO where the estimates ad are comuted as i the corresodig test of Miettie ad Nurmie (985) as ( ) 0 0 ψ ψ A AC B B 4 ( ) 0 ψ N A ( ) 0 0 ψ ψ M N N B C M Farrigto ad Maig (990) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig z α / z FMO ad the uer limit is the solutio of z FMO z α / C.I. for Odds Ratio: Miettie ad Nurmie Miettie ad Nurmie (985) roosed a test statistic for testig whether the odds ratio is equal to a secified valueψ 0. Because the aroach they used with the differece ad ratio does ot easily exted to the odds ratio, they used a score statistic aroach for the odds ratio. The regular MLE s are ad. The costraied MLE s are ad, These estimates are costraied so that ψ ψ 0. A correctio factor of N/(N-) is alied to mae the variace estimate less biased. The sigificace level of the test statistic is based o the asymtotic ormality of the score statistic. The formula for comutig the test statistic is ( ) ( ) N N q N q N q q z MNO where ( ) 0 0 ψ ψ

14 PASS Samle Size Software Cofidece Itervals for Two Proortios B A N ψ ( ) 0 B 4AC A B N ψ C M ( ) ψ 0 N M 0 Miettie ad Nurmie (985) roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig ad the uer limit is the solutio of z MNO z α / z MNO z α / C.I. for Odds Ratio: Iterated Method of Fleiss Fleiss (98) resets a imrove cofidece iterval for the odds ratio. This method forms the cofidece iterval as all those value of the odds ratio which would ot be rejected by a chi-square hyothesis test. Fleiss gives the followig details about how to costruct this cofidece iterval. To comute the lower limit, do the followig.. For a trial value of ψ, comute the quatities X, Y, W, F, U, ad V usig the formulas X ψ ( m s) ( s) Y X 4msψ ψ X Y A ψ ( ) B s A C m A D f m A ( ) W A B C D F ( a A ) W ( ψ ) z α / ψ T Y.. ψ Y U B C A D [( a A ) ( )] U W a V T A [ X ( m s) ms( )] 6-4

15 PASS Samle Size Software Cofidece Itervals for Two Proortios Fially, use the udatig equatio below to calculate a ew value for the odds ratio usig the udatig equatio ( ) ( ) F ψ V ψ. Cotiue iteratig util the value of F is arbitrarily close to zero. The uer limit is foud by substitutig for i the formulas for F ad V. Cofidece limits for the relative ris ca be calculated usig the exected couts A, B, C, ad D from the last iteratio of the above rocedure. The lower limit of the relative ris φ lower φ uer A B A B lower lower uer uer m m C.I. for Odds Ratio: Matel-Haeszel The commo estimate of the logarithm of the odds ratio is used to create this estimator. That is ad bc ( ) l l ψ The stadard error of this estimator is estimated usig the Robis, Breslow, Greelad (986) estimator which erforms well i most situatios. The stadard error is give by where a d A N b c B N ad C N bc D N The cofidece limits are calculated as A AD BC se ( l( ψ )) C CD B D ( l( ψ ) z se( l( ψ ))) lower / ψ ex α ( ( ) α / ( ( ))) ψ ex l ψ l ψ uer z se 6-5

16 PASS Samle Size Software Cofidece Itervals for Two Proortios C.I. for Odds Ratio: Simle, Simle ½, ad Logarithm The simle estimate of the odds ratio uses the formula The stadard error of this estimator is estimated by q ψ q ad bc se( ψ ) ψ a b c d Problems occur if ay oe of the quatities a, b, c, or d are zero. To correct this roblem, may authors recommed addig oe-half to each cell cout so that a zero caot occur. Now, the formulas become ad se ( ψ ) ψ ψ ( a 0.5)( d 0.5) ( b 0.5)( c 0.5) a 0.5 b 0.5 c 0.5 d 0.5 The distributio of these direct estimates of the odds ratio do ot coverge to ormality as fast as does their logarithm, so the logarithm of the odds ratio is used to form cofidece itervals. The formula for the stadard error of the log odds ratio is ad se ( L ) a 0.5 ( ψ ) L l b 0.5 c 0.5 d 0.5 A 00( α )% cofidece iterval for the log odds ratio is formed usig the stadard ormal distributio as follows See Fleiss et al (003) for more details. ψ ψ ( L z α / se( L )) ( L z se( L )) lower ex uer ex α / Cofidece Level The cofidece level, α, has the followig iterretatio. If thousads of radom samles of size ad are draw from oulatios ad, resectively, ad a cofidece iterval for the true differece/ratio/odds ratio of roortios is calculated for each air of samles, the roortio of those itervals that will iclude the true differece/ratio/odds ratio of roortios is α. 6-6

17 PASS Samle Size Software Procedure Otios Cofidece Itervals for Two Proortios This sectio describes the otios that are secific to this rocedure. These are located o the Desig tab. For more iformatio about the otios of other tabs, go to the Procedure Widow chater. Desig Tab (Commo Otios) This chater covers four rocedures, each of which has differet otios. This sectio documets otios that are commo to all four rocedures. Followig this sectio, the uique otios for each rocedure (roortios, differeces, ratios, ad odds ratios) will be documeted. Solve For Solve For This otio secifies the arameter to be solved for from the other arameters. Oe-Sided or Two-Sided Iterval Iterval Tye Secify whether the iterval to be used will be a two-sided cofidece iterval, a iterval that has oly a uer limit, or a iterval that has oly a lower limit. Cofidece Cofidece Level The cofidece level, α, has the followig iterretatio. If thousads of radom samles of size ad are draw from oulatios ad, resectively, ad a cofidece iterval for the true differece/ratio/odds ratio of roortios is calculated for each air of samles, the roortio of those itervals that will iclude the true differece/ratio/odds ratio of roortios is α. Ofte, the values 0.95 or 0.99 are used. You ca eter sigle values or a rage of values such as 0.90, 0.95 or 0.90 to 0.99 by 0.0. Samle Size (Whe Solvig for Samle Size) Grou Allocatio Select the otio that describes the costraits o N or N or both. The otios are Equal (N N) This selectio is used whe you wish to have equal samle sizes i each grou. Sice you are solvig for both samle sizes at oce, o additioal samle size arameters eed to be etered. Eter N, solve for N Select this otio whe you wish to fix N at some value (or values), ad the solve oly for N. Please ote that for some values of N, there may ot be a value of N that is large eough to obtai the desired ower. Eter N, solve for N Select this otio whe you wish to fix N at some value (or values), ad the solve oly for N. Please ote that for some values of N, there may ot be a value of N that is large eough to obtai the desired ower. 6-7

18 PASS Samle Size Software Cofidece Itervals for Two Proortios Eter R N/N, solve for N ad N For this choice, you set a value for the ratio of N to N, ad the PASS determies the eeded N ad N, with this ratio, to obtai the desired ower. A equivalet reresetatio of the ratio, R, is N R * N. Eter ercetage i Grou, solve for N ad N For this choice, you set a value for the ercetage of the total samle size that is i Grou, ad the PASS determies the eeded N ad N with this ercetage to obtai the desired ower. N (Samle Size, Grou ) This otio is dislayed if Grou Allocatio Eter N, solve for N N is the umber of items or idividuals samled from the Grou oulatio. N must be. You ca eter a sigle value or a series of values. N (Samle Size, Grou ) This otio is dislayed if Grou Allocatio Eter N, solve for N N is the umber of items or idividuals samled from the Grou oulatio. N must be. You ca eter a sigle value or a series of values. R (Grou Samle Size Ratio) This otio is dislayed oly if Grou Allocatio Eter R N/N, solve for N ad N. R is the ratio of N to N. That is, R N / N. Use this value to fix the ratio of N to N while solvig for N ad N. Oly samle size combiatios with this ratio are cosidered. N is related to N by the formula: where the value [Y] is the ext iteger Y. N [R N], For examle, settig R.0 results i a Grou samle size that is double the samle size i Grou (e.g., N 0 ad N 0, or N 50 ad N 00). R must be greater tha 0. If R <, the N will be less tha N; if R >, the N will be greater tha N. You ca eter a sigle or a series of values. Percet i Grou This otio is dislayed oly if Grou Allocatio Eter ercetage i Grou, solve for N ad N. Use this value to fix the ercetage of the total samle size allocated to Grou while solvig for N ad N. Oly samle size combiatios with this Grou ercetage are cosidered. Small variatios from the secified ercetage may occur due to the discrete ature of samle sizes. The Percet i Grou must be greater tha 0 ad less tha 00. You ca eter a sigle or a series of values. 6-8

19 PASS Samle Size Software Cofidece Itervals for Two Proortios Samle Size (Whe Not Solvig for Samle Size) Grou Allocatio Select the otio that describes how idividuals i the study will be allocated to Grou ad to Grou. The otios are Equal (N N) This selectio is used whe you wish to have equal samle sizes i each grou. A sigle er grou samle size will be etered. Eter N ad N idividually This choice ermits you to eter differet values for N ad N. Eter N ad R, where N R * N Choose this otio to secify a value (or values) for N, ad obtai N as a ratio (multile) of N. Eter total samle size ad ercetage i Grou Choose this otio to secify a value (or values) for the total samle size (N), obtai N as a ercetage of N, ad the N as N - N. Samle Size Per Grou This otio is dislayed oly if Grou Allocatio Equal (N N). The Samle Size Per Grou is the umber of items or idividuals samled from each of the Grou ad Grou oulatios. Sice the samle sizes are the same i each grou, this value is the value for N, ad also the value for N. The Samle Size Per Grou must be. You ca eter a sigle value or a series of values. N (Samle Size, Grou ) This otio is dislayed if Grou Allocatio Eter N ad N idividually or Eter N ad R, where N R * N. N is the umber of items or idividuals samled from the Grou oulatio. N must be. You ca eter a sigle value or a series of values. N (Samle Size, Grou ) This otio is dislayed oly if Grou Allocatio Eter N ad N idividually. N is the umber of items or idividuals samled from the Grou oulatio. N must be. You ca eter a sigle value or a series of values. R (Grou Samle Size Ratio) This otio is dislayed oly if Grou Allocatio Eter N ad R, where N R * N. R is the ratio of N to N. That is, R N/N Use this value to obtai N as a multile (or roortio) of N. N is calculated from N usig the formula: where the value [Y] is the ext iteger Y. N[R x N], 6-9

20 PASS Samle Size Software Cofidece Itervals for Two Proortios For examle, settig R.0 results i a Grou samle size that is double the samle size i Grou. R must be greater tha 0. If R <, the N will be less tha N; if R >, the N will be greater tha N. You ca eter a sigle value or a series of values. Total Samle Size (N) This otio is dislayed oly if Grou Allocatio Eter total samle size ad ercetage i Grou. This is the total samle size, or the sum of the two grou samle sizes. This value, alog with the ercetage of the total samle size i Grou, imlicitly defies N ad N. The total samle size must be greater tha oe, but ractically, must be greater tha 3, sice each grou samle size eeds to be at least. You ca eter a sigle value or a series of values. Percet i Grou This otio is dislayed oly if Grou Allocatio Eter total samle size ad ercetage i Grou. This value fixes the ercetage of the total samle size allocated to Grou. Small variatios from the secified ercetage may occur due to the discrete ature of samle sizes. The Percet i Grou must be greater tha 0 ad less tha 00. You ca eter a sigle value or a series of values. Desig Tab (Proortios) This sectio documets otios that are used whe the arameterizatio is i terms of the values of the two samle roortios, P ad P. The corresodig rocedure is Cofidece Itervals for the Differece betwee Two Proortios usig Proortios. Cofidece Iterval Method Cofidece Iterval Formula Secify the formula to be i used i calculatio of cofidece itervals. Score (Farrigto & Maig) This formula is based o ivertig Farrigto ad Maig's score test. Score (Miettie & Nurmie) This formula is based o ivertig Miettie ad Nurmie's score test. Score w/ Sewess (Gart & Nam) This formula is based o ivertig Gart ad Nam's score test, with a correctio for sewess. Score (Wilso) This formula is based o the Wilso score method for a sigle roortio, without cotiuity correctio. Score (Wilso C.C.) This formula is based o the Wilso score method for a sigle roortio, with cotiuity correctio. Chi-Square C.C. (Yates) This is the commoly used simle asymtotic method, with cotiuity correctio. 6-0

21 PASS Samle Size Software Cofidece Itervals for Two Proortios Chi-Square (Pearso) This is the commoly used simle asymtotic method, without cotiuity correctio. Precisio Cofidece Iterval Width (Two-Sided) This is the distace from the lower cofidece limit to the uer cofidece limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Distace from Diff to Limit (Oe-Sided) This is the distace from the differece i samle roortios to the lower or uer limit of the cofidece iterval, deedig o whether the Iterval Tye is set to Lower Limit or Uer Limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Proortios (Differece P P) P (Proortio Grou ) Eter a estimate of the roortio for grou. The samle size ad width calculatios assume that the value etered here is the roortio estimate that is obtaied from the samle. If the samle roortio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee ad You ca eter a rage of values such as...3 or. to.5 by.. P (Proortio Grou ) Eter a estimate of the roortio for grou. The samle size ad width calculatios assume that the value etered here is the roortio estimate that is obtaied from the samle. If the samle roortio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee ad You ca eter a rage of values such as...3 or. to.5 by.. Desig Tab (Differeces) This sectio documets otios that are used whe the arameterizatio is i terms of the differece i samle roortios ad the value of the secod samle roortio, P. The corresodig rocedure is Cofidece Itervals for the Differece betwee Two Proortios usig Differeces. Cofidece Iterval Method Cofidece Iterval Formula Secify the formula to be i used i calculatio of cofidece itervals. Score (Farrigto & Maig) This formula is based o ivertig Farrigto ad Maig's score test. Score (Miettie & Nurmie) This formula is based o ivertig Miettie ad Nurmie's score test. 6-

22 PASS Samle Size Software Cofidece Itervals for Two Proortios Score w/ Sewess (Gart & Nam) This formula is based o ivertig Gart ad Nam's score test, with a correctio for sewess. Score (Wilso) This formula is based o the Wilso score method for a sigle roortio, without cotiuity correctio. Score (Wilso C.C.) This formula is based o the Wilso score method for a sigle roortio, with cotiuity correctio. Chi-Square C.C. (Yates) This is the commoly used simle asymtotic method, with cotiuity correctio. Chi-Square (Pearso) This is the commoly used simle asymtotic method, without cotiuity correctio. Precisio Cofidece Iterval Width (Two-Sided) This is the distace from the lower cofidece limit to the uer cofidece limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Distace from Diff to Limit (Oe-Sided) This is the distace from the differece i samle roortios to the lower or uer limit of the cofidece iterval, deedig o whether the Iterval Tye is set to Lower Limit or Uer Limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Proortios (Differece P P) Differece i Samle Proortios Eter a estimate of the differece betwee samle roortio ad samle roortio. The samle size ad width calculatios assume that the value etered here is the differece estimate that is obtaied from the samle. If the samle differece is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee - ad, ad such that P Differece P is betwee ad You ca eter a rage of values such as...3 or. to.5 by.. P (Proortio Grou ) Eter a estimate of the roortio for grou. The samle size ad width calculatios assume that the value etered here is the roortio estimate that is obtaied from the samle. If the samle roortio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee ad You ca eter a rage of values such as...3 or. to.5 by.. Desig Tab (Ratios) This sectio documets otios that are used whe the arameterizatio is i terms of the ratio of samle roortios ad the value of the secod samle roortio, P. The corresodig rocedure is Cofidece Itervals for the Differece betwee Two Proortios usig Ratios. 6-

23 PASS Samle Size Software Cofidece Itervals for Two Proortios Cofidece Iterval Method Cofidece Iterval Formula Secify the formula to be i used i calculatio of cofidece itervals. Score (Farrigto & Maig) This formula is based o ivertig Farrigto ad Maig's score test. Score (Miettie & Nurmie) This formula is based o ivertig Miettie ad Nurmie's score test. Score w/ Sewess (Gart & Nam) This formula is based o ivertig Gart ad Nam's score test, with a correctio for sewess. Logarithm (Katz) This formula is based o the asymtotic ormality of log(p/p). Logarithm / (Walter) This formula is based o the asymtotic ormality of log(p/p), but / is used as a adjustmet. Fleiss This is a iterative method that was develoed for the odds ratio ad adated to the roortio ratio. Precisio Cofidece Iterval Width (Two-Sided) This is the distace from the lower cofidece limit to the uer cofidece limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Distace from Ratio to Limit (Oe-Sided) This is the distace from the ratio of samle roortios to the lower or uer limit of the cofidece iterval, deedig o whether the Iterval Tye is set to Lower Limit or Uer Limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Proortios (Ratio P/P) Ratio of Samle Proortios Eter a estimate of the ratio of samle roortio to samle roortio. The samle size ad width calculatios assume that the value etered here is the ratio estimate that is obtaied from the samles. If the samle ratio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be greater tha 0, ad such that P Ratio * P is betwee ad You ca eter a rage of values such as or.5 to.9 by.. P (Proortio Grou ) Eter a estimate of the roortio for grou. The samle size ad width calculatios assume that the value etered here is the roortio estimate that is obtaied from the samle. If the samle roortio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee ad

24 PASS Samle Size Software Cofidece Itervals for Two Proortios You ca eter a rage of values such as...3 or. to.5 by.. Desig Tab (Odds Ratios) This sectio documets otios that are used whe the arameterizatio is i terms of the odds ratio ad the value of the secod samle roortio, P. The corresodig rocedure is Cofidece Itervals for the Differece betwee Two Proortios usig Odds Ratios. Cofidece Iterval Method Cofidece Iterval Formula Secify the formula to be i used i calculatio of cofidece itervals. Exact (Coditioal) This coditioal exact cofidece iterval formula is calculated usig the o-cetral hyergeometric distributio. Score (Farrigto & Maig) This formula is based o ivertig Farrigto ad Maig's score test. Score (Miettie & Nurmie) This formula is based o ivertig Miettie ad Nurmie's score test. Fleiss This iterative method forms the cofidece iterval as all those value of the odds ratio which would ot be rejected by a chi-square hyothesis test. Logarithm This formula is similar to SIMPLE /, but with the logarithm of the odds ratio. Matel- Haeszel This formula is based o the Matel-Haeszel formula for the odds ratio. Simle This uses the simle odds ratio formula ad large samle stadard error estimate. Simle / This uses the simle odds ratio formula ad large samle stadard error estimate, but with / added to frequecies as a bias reductio device. Precisio Cofidece Iterval Width (Two-Sided) This is the distace from the lower cofidece limit to the uer cofidece limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Distace from OR to Limit (Oe-Sided) This is the distace from the odds ratio to the lower or uer limit of the cofidece iterval, deedig o whether the Iterval Tye is set to Lower Limit or Uer Limit. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. 6-4

25 PASS Samle Size Software Proortios (OR O/O) Cofidece Itervals for Two Proortios Odds Ratio Eter a estimate of the samle odds ratio (O/O). The samle size ad width calculatios assume that the value etered here is the odds ratio estimate that is obtaied from the samles. If the samle odds ratio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be greater tha 0. You ca eter a rage of values such as or.5 to.9 by.. P (Proortio Grou ) Eter a estimate of the roortio for grou. The samle size ad width calculatios assume that the value etered here is the roortio estimate that is obtaied from the samle. If the samle roortio is differet from the oe secified here, the width may be arrower or wider tha secified. The value(s) must be betwee ad You ca eter a rage of values such as...3 or. to.5 by.. Examle Calculatig Samle Size usig Proortios Suose a study is laed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the differece i roortios such that the width of the iterval is o wider tha 0.. The cofidece iterval method to be used is the Yates chi-square simle asymtotic method with cotiuity correctio. The cofidece level is set at 0.95, but 0.99 is icluded for comarative uroses. The roortio estimates to be used are 0.6 for Grou, ad 0.4 for Grou. Istead of examiig oly the iterval width of 0., a series of widths from 0.05 to 0.3 will also be cosidered. The goal is to determie the ecessary samle size. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Proortios. You may the mae the aroriate etries as listed below, or oe Examle by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Chi-Square C.C. (Yates) Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided) to 0.30 by 0.05 P P

26 PASS Samle Size Software Cofidece Itervals for Two Proortios Aotated Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Numeric Results for Two-Sided Cofidece Itervals for the Differece i Proortios Cofidece Iterval Method: Chi-Square - Simle Asymtotic with Cotiuity Correctio (Yates) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit Refereces Newcombe, R. G 'Iterval Estimatio for the Differece Betwee Ideedet Proortios: Comariso of Eleve Methods.' Statistics i Medicie, 7, Fleiss, J. L., Levi, B., Pai, M.C Statistical Methods for Rates ad Proortios. Third Editio. Joh Wiley & Sos. New Yor. Reort Defiitios Cofidece level is the roortio of cofidece itervals (costructed with this same cofidece level, samle size, etc.) that would cotai the true differece i oulatio roortios. N ad N are the umber of items samled from each oulatio. N is the total samle size, N N. Target Width is the value of the width that is etered ito the rocedure. Actual Width is the value of the width that is obtaied from the rocedure. P ad P are the assumed samle roortios for samle size calculatios. P - P is the differece betwee samle roortios at which samle size calculatios are made. Lower Limit ad Uer Limit are the lower ad uer limits of the cofidece iterval for the true differece i roortios (Poulatio Proortio - Poulatio Proortio ). Summary Statemets Grou samle sizes of 3030 ad 3030 roduce a two-sided 95% cofidece iterval for the differece i oulatio roortios with a width that is equal to whe the estimated samle roortio is 0.60 ad the estimated samle roortio is This reort shows the calculated samle sizes for each of the scearios. 6-6

27 PASS Samle Size Software Plots Sectio Cofidece Itervals for Two Proortios These lots show the grou samle size versus the cofidece iterval width for the two cofidece levels. 6-7

28 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle Validatio (Proortios ad Differeces) usig Newcombe Newcombe (998b) age 877 gives a examle of a calculatio for a cofidece iterval for the differece i roortios whe the cofidece level is 95%, the samle roortios are 0.9 ad 0.3, ad the iterval width is for the Chi-Square (Pearso) method, for the Chi-Square C.C. (Yates) method, for the Score (Miettie ad Nurmie) method, for the Score (Wilso) method, ad for the Score C.C. (Wilso) method. The ecessary samle size i each case is 0 er grou. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Proortios. You may the mae the aroriate etries as listed below, or oe Examle by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Varies [Chi-Square (Pearso), Chi-Square C.C. (Yates), Score (Miettie & Nurmie), Score (Wilso), Score C.C. (Wilso)] Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided).. Varies (0.6790, , , , ) P P Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Chi-Square (Pearso) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 0 er grou. Chi-Square C.C. (Yates) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 0 er grou. 6-8

29 PASS Samle Size Software Cofidece Itervals for Two Proortios Score (Miettie & Nurmie) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 0 er grou. Score (Wilso) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 0 er grou. Score C.C. (Wilso) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 0 er grou. 6-9

30 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 3 Validatio (Proortios ad Differeces) usig Gart ad Nam Gart ad Nam (990) age 640 give a examle of a calculatio for a cofidece iterval for the differece i roortios whe the cofidece level is 95%, the samle roortios are 0.8 ad 0.08, ad the iterval width is 0.48 for the Score (Gart ad Nam) method. The ecessary samle size i each case is 5 er grou. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Proortios. You may the mae the aroriate etries as listed below, or oe Examle 3 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Score w/sewess (Gart & Nam) Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided) P P Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit PASS also calculates the ecessary samle size to be 5 er grou. 6-30

31 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 4 Calculatig Samle Size usig Differeces Suose a study is laed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the differece i roortios such that the width of the iterval is o wider tha 0.. The cofidece iterval method to be used is the Yates chi-square simle asymtotic method with cotiuity correctio. The cofidece level is set at 0.95, but 0.99 is icluded for comarative uroses. The differece estimate to be used is 0.05, ad the estimate for roortio is 0.3. Istead of examiig oly the iterval width of 0., a series of widths from 0.05 to 0.3 will also be cosidered. The goal is to determie the ecessary samle size. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Differeces rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Differeces. You may the mae the aroriate etries as listed below, or oe Examle 4 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Chi-Square C.C. (Yates) Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided) to 0.30 by 0.05 Differece i Samle Proortios P Aotated Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Numeric Results for Two-Sided Cofidece Itervals for the Differece i Proortios Cofidece Iterval Method: Chi-Square - Simle Asymtotic with Cotiuity Correctio (Yates) Cofidece Target Actual Lower Uer Level N N N Width Width P P P - P Limit Limit

32 PASS Samle Size Software Cofidece Itervals for Two Proortios Refereces Newcombe, R. G 'Iterval Estimatio for the Differece Betwee Ideedet Proortios: Comariso of Eleve Methods.' Statistics i Medicie, 7, Fleiss, J. L., Levi, B., Pai, M.C Statistical Methods for Rates ad Proortios. Third Editio. Joh Wiley & Sos. New Yor. Reort Defiitios Cofidece level is the roortio of cofidece itervals (costructed with this same cofidece level, samle size, etc.) that would cotai the true differece i oulatio roortios. N ad N are the umber of items samled from each oulatio. N is the total samle size, N N. Target Width is the value of the width that is etered ito the rocedure. Actual Width is the value of the width that is obtaied from the rocedure. P ad P are the assumed samle roortios for samle size calculatios. P - P is the differece betwee samle roortios at which samle size calculatios are made. Lower Limit ad Uer Limit are the lower ad uer limits of the cofidece iterval for the true differece i roortios (Poulatio Proortio - Poulatio Proortio ). Summary Statemets Grou samle sizes of 769 ad 769 roduce a two-sided 95% cofidece iterval for the differece i oulatio roortios with a width that is equal to whe the estimated samle roortio is 0.35, the estimated samle roortio is 0.30, ad the differece i samle roortios is This reort shows the calculated samle sizes for each of the scearios. Plots Sectio These lots show the grou samle size versus the cofidece iterval width for the two cofidece levels. Validatio usig Differeces The validatio for the rocedure Cofidece Itervals for the Differece betwee Two Proortios usig Differeces is show i Examles ad 3, which is the validatio for the roortio secificatio. 6-3

33 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 5 Calculatig Samle Size usig Ratios Suose a study is laed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the ratio of roortios such that the width of the iterval is o wider tha 0.. The cofidece iterval method to be used is the Logarithm (Katz) method. The cofidece level is set at 0.95, but 0.99 is icluded for comarative uroses. The ratio estimate to be used is., ad the estimate for roortio is 0.6. Istead of examiig oly the iterval width of 0., a series of widths from 0. to 0.3 will also be cosidered. The goal is to determie the ecessary samle size. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Ratios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Ratios. You may the mae the aroriate etries as listed below, or oe Examle 5 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Logarithm (Katz) Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided) to 0.30 by 0.05 Ratio of Samle Proortios.... P Aotated Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Numeric Results for Two-Sided Cofidece Itervals for the Ratio of Proortios Cofidece Iterval Method: Logarithm (Katz) Cofidece Target Actual Lower Uer Level N N N Width Width P P P/P Limit Limit

34 PASS Samle Size Software Cofidece Itervals for Two Proortios Refereces Gart, Joh J. ad Nam, Ju-mo 'Aroximate Iterval Estimatio of the Ratio of Biomial Parameters: A Review ad Correctios for Sewess.' Biometrics, Volume 44, Kooma, P. A. R 'Cofidece Itervals for the Ratio of Two Biomial Proortios.' Biometrics, Volume 40, Issue, Katz, D., Batista, J., Aze, S. P., ad Pie, M. C 'Obtaiig Cofidece Itervals for the Ris Ratio i Cohort Studies.' Biometrics, Volume 34, Reort Defiitios Cofidece level is the roortio of cofidece itervals (costructed with this same cofidece level, samle size, etc.) that would cotai the true ratio of oulatio roortios. N ad N are the umber of items samled from each oulatio. N is the total samle size, N N. Target Width is the value of the width that is etered ito the rocedure. Actual Width is the value of the width that is obtaied from the rocedure. P ad P are the assumed samle roortios for samle size calculatios. P/P is the ratio of samle roortios at which samle size calculatios are made. Lower Limit ad Uer Limit are the lower ad uer limits of the cofidece iterval for the true ratio of roortios (Poulatio Proortio / Poulatio Proortio ). Summary Statemets Grou samle sizes of 337 ad 337 roduce a two-sided 95% cofidece iterval for the ratio of oulatio roortios with a width that is equal to 0.00 whe the estimated samle roortio is 0.7, the estimated samle roortio is 0.60, ad the ratio of the samle roortios is.0. This reort shows the calculated samle sizes for each of the scearios. Plots Sectio These lots show the grou samle size versus the cofidece iterval width for the two cofidece levels. 6-34

35 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 6 Validatio (Ratios) usig Gart ad Nam Gart ad Nam (988) age 33 give a examle (Examle ) of a calculatio for a cofidece iterval for the ratio of roortios whe the cofidece level is 95%, the samle roortio ratio is ad the samle roortio is 0.3, the samle size for grou is 0, ad the iterval width is for the Logarithm / (Walter) method, 3.75 for the Score (Farrigto ad Maig) method, ad 4.33 for the Score w/sewess (Gart ad Nam) method. The ecessary samle size for grou i each case is 0. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Ratios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Ratios. You may the mae the aroriate etries as listed below, or oe Examle 6 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Varies [Logarithm / (Walter), Score (Farrigto ad Maig), Score w/sewess (Gart ad Nam)] Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Eter N, solve for N N... 0 Cofidece Iterval Width (Two-Sided).. Varies (3.437, 3.75, 4.33) Ratio of Samle Proortios... P Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Logarithm / (Walter) Cofidece Target Actual Lower Uer Level N N N Width Width P P P/P Limit Limit PASS also calculates the ecessary samle size for Grou to be 0. Score (Farrigto ad Maig) Cofidece Target Actual Lower Uer Level N N N Width Width P P P/P Limit Limit PASS also calculates the ecessary samle size for Grou to be

36 PASS Samle Size Software Cofidece Itervals for Two Proortios Score w/sewess (Gart ad Nam) Cofidece Target Actual Lower Uer Level N N N Width Width P P P/P Limit Limit PASS also calculates the ecessary samle size for Grou to be 0. Examle 7 Validatio (Ratios) usig Katz et al Katz et al (978) ages give a examle of a calculatio for a lower limit cofidece iterval for the ratio of roortios whe the cofidece level is 97.5%, the samle roortio ratio is ad the samle roortio is , the samle size for grou is, ad the distace from the ratio to the limit is 0.63 for the Logarithm (Katz) method. The ecessary samle size for grou is 5. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Ratios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Ratios. You may the mae the aroriate etries as listed below, or oe Examle 7 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Logarithm (Katz) Iterval Tye... Lower Limit Cofidece Level Grou Allocatio... Eter N, solve for N N... Distace to from Ratio to Limit Ratio of Samle Proortios P Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Logarithm (Katz) Target Actual Dist from Dist from Cofidece Ratio Ratio Lower Uer Level N N N to Limit to Limit P P P/P Limit Limit If PASS also calculates the ecessary samle size for grou to be

37 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 8 Calculatig Samle Size usig Odds Ratios Suose a study is laed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the odds ratio such that the width of the iterval is o wider tha 0.5. The cofidece iterval method to be used is the Logarithm method. The cofidece level is set at 0.95, but 0.99 is icluded for comarative uroses. The odds ratio estimate to be used is.5, ad the estimate for roortio is 0.4. Istead of examiig oly the iterval width of 0.5, a series of widths from 0. to.0 will also be cosidered. The goal is to determie the ecessary samle size. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Odds Ratios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Odds Ratios. You may the mae the aroriate etries as listed below, or oe Examle 8 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Logarithm Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Equal (N N) Cofidece Iterval Width (Two-Sided).. 0. to.0 by 0. Odds Ratio....5 P Aotated Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Numeric Results for Two-Sided Cofidece Itervals for the Odds Ratio Cofidece Iterval Method: Logarithm Odds Cofidece Target Actual Ratio Lower Uer Level N N N Width Width P P O/O Limit Limit

38 PASS Samle Size Software Cofidece Itervals for Two Proortios Refereces Fleiss, J. L., Levi, B., Pai, M.C Statistical Methods for Rates ad Proortios. Third Editio. Joh Wiley & Sos. New Yor. Reort Defiitios Cofidece level is the roortio of cofidece itervals (costructed with this same cofidece level, samle size, etc.) that would cotai the true odds ratio of oulatio roortios. N ad N are the umber of items samled from each oulatio. N is the total samle size, N N. Target Width is the value of the width that is etered ito the rocedure. Actual Width is the value of the width that is obtaied from the rocedure. P ad P are the assumed samle roortios for samle size calculatios. Odds Ratio O/O is the samle odds ratio at which samle size calculatios are made. Lower Limit ad Uer Limit are the lower ad uer limits of the cofidece iterval for the true odds ratio of roortios (Poulatio Odds / Poulatio Odds ). Summary Statemets Grou samle sizes of 844 ad 844 roduce a two-sided 95% cofidece iterval for the oulatio odds ratio with a width that is equal to 0.00 whe the estimated samle roortio is 0.50, the estimated samle roortio is 0.40, ad the samle odds ratio is.50. This reort shows the calculated samle sizes for each of the scearios. Plots Sectio These lots show the grou samle size versus the cofidece iterval width for the two cofidece levels. 6-38

39 PASS Samle Size Software Cofidece Itervals for Two Proortios Examle 9 Validatio (Odds Ratios) usig Fleiss et al Fleiss et al (003) ages 7, 9 give a examle of a calculatio for a cofidece iterval for the odds ratio whe the cofidece level is 95%, the samle odds ratio is.5 ad the samle roortio is 0., the samle size for grou is 50, ad the iterval width is for the Logarithm method, ad for the Fleiss method. The ecessary samle size for grou i each case is 50. Setu This sectio resets the values of each of the arameters eeded to ru this examle. First, from the PASS Home widow, load the Cofidece Itervals for Two Proortios usig Odds Ratios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clicig o Cofidece Iterval, ad the clicig o Cofidece Itervals for Two Proortios usig Odds Ratios. You may the mae the aroriate etries as listed below, or oe Examle 9 by goig to the File meu ad choosig Oe Examle Temlate. Otio Value Desig Tab Solve For... Samle Size Cofidece Iterval Formula... Varies [Logarithm, Fleiss] Iterval Tye... Two-Sided Cofidece Level Grou Allocatio... Eter N, solve for N N Cofidece Iterval Width (Two-Sided).. Varies (4.387, 4.980) Odds Ratio....5 P Outut Clic the Calculate butto to erform the calculatios ad geerate the followig outut. Logarithm Odds Cofidece Target Actual Ratio Lower Uer Level N N N Width Width P P O/O Limit Limit PASS also calculates the ecessary samle size for Grou to be 50. Fleiss Odds Cofidece Target Actual Ratio Lower Uer Level N N N Width Width P P O/O Limit Limit PASS also calculates the ecessary samle size for Grou to be