Confidence Intervals for the Difference Between Two Proportions

Transcription

1 PASS Samle Size Software Chater 6 Cofidece Itervals for the Differece Betwee Two Proortios Itroductio This routie calculates the grou samle sizes ecessary to achieve a secified iterval width of the differece betwee 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. Techical Details A backgroud 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 risk 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 the Differece Betwee 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 Risk 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. This rocedure focuses o the differece. Other rocedures are available i PASS for comutig cofidece itervals for the ratio ad odds ratio. Differece The (risk 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. were reorted for a evet with a baselie robability of.4, we would robability dismiss this as beig of little imortace. That is, there usually little iterest i a treatmet that decreases the robability from.4 to.99. However, if the baselie robably of a disease was. ad. was the decrease i the disease robability, this would rereset a reductio of 5%. 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.4 ad the other with a rate of.. What is the maximum decrease that ca occur? Obviously, the first evet rate ca be decreased by a absolute amout of.4 which the secod ca oly be decreased by a maximum of.. So, although creatig the simle differece is a useful method of comariso, care must be take that it fits the situatio. 6-

3 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 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. Score with Correctio for Skewess (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 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. Farrigto ad Maig s Score Farrigto ad Maig (99 roosed a test statistic for testig whether the differece is equal to a secified valueδ. The regular ME s ad are used i the umerator of the score statistic while ME s ~ ad ~ costraied so that ~ ~ δ 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 ~ q~ ~ ~ q where the estimates ~ ad ~ are comuted as i the corresodig test of Miettie ad Nurmie (985 give as ~ ~ δ ~ B cos ( A A π cos C B B sig ( C 9 C

4 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios ( x δ δ [ δ N x] δ m ( N N δ m N Farrigto ad Maig (99 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 / Miettie ad Nurmie s Score Miettie ad Nurmie (985 roosed a test statistic for testig whether the differece is equal to a secified valueδ. The regular ME s ad are used i the umerator of the score statistic while ME s ~ ad ~ costraied so that ~ ~ δ are used i the deomiator. A correctio factor of N/(N- is alied to make 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 ~ B cos ( A A π cos C B z MND δ ~ ~ ~ ~ q q N N B sig ( C 9 C 7 6 x ( [ δ N x] ( N N δ δ δ m δ m N 6-4

5 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Miettie ad Nurmie (985 roosed ivertig their score test to fid the cofidece iterval. The lower limit is foud by solvig z z MND α / ad the uer limit is the solutio of z MND z α / Gart ad Nam s Score Gart ad Nam (99 age 68 roosed a modificatio to the Farrigto ad Maig (99 differece test that corrected for skewess. et z FM ( δ stad for the Farrigto ad Maig differece test statistic described above. The skewess corrected test statistic z GN is the aroriate solutio to the quadratic equatio ~ γ z z z δ ~ γ where ~ / ~ V γ 6 ( δ ~ q~ ( q~ ~ ~ q~ ( q~ ~ ( ( GND ( GND FMD( 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 / 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 B U C 6-5

6 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 6-6 Yate s Chi-Square with Cotiuity Correctio For details, see Newcombe (998b, age 875. m m z ( ( m m z U ( ( Pearso s Chi-Square For details, see Newcombe (998b, age 875. m z ( ( 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 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 ( D Each of these equatios ca be solved for ay of the ukow quatities i terms of the others. Cofidece evel 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.

7 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Procedure Otios 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 The Desig tab cotais the arameters associated with this calculatio such as the roortios or differeces, samle sizes, cofidece level, ad iterval width. Solve For Solve For This otio secifies the arameter to be solved for from the other arameters. 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/ Skewess (Gart & Nam This formula is based o ivertig Gart ad Nam's score test, with a correctio for skewess. 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. 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. 6-7

8 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Cofidece Cofidece evel ( Alha 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.95 or.99 are used. You ca eter sigle values or a rage of values such as.9,.95 or.9 to.99 by.. 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. 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. 6-8

9 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 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. results i a Grou samle size that is double the samle size i Grou (e.g., N ad N, or N 5 ad N. R must be greater tha. 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 ad less tha. You ca eter a sigle or a series of values. 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. 6-9

10 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 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], For examle, settig R. results i a Grou samle size that is double the samle size i Grou. R must be greater tha. 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, 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 ad less tha. You ca eter a sigle value or a series of values. 6-

11 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 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 imit (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 ower imit or Uer imit. You ca eter a sigle value or a list of values. The value(s must be greater tha zero. Proortios (Differece P P Iut Tye Idicate what tye of values to eter to secify the differece. Regardless of the etry tye chose, the calculatios are the same. This otio is simly give for coveiece i secifyig the differece. P P (Differece i Samle Proortios This otio is dislayed oly if Iut Tye Differeces 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... or. to.5 by.. P (Proortio Grou This otio is dislayed oly if Iut Tye Proortios 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... 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... or. to.5 by.. 6-

12 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Examle 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.. The cofidece iterval method to be used is the Yates chi-square simle asymtotic method with cotiuity correctio. The cofidece level is set at.95, but.99 is icluded for comarative uroses. The differece estimate to be used is.5, ad the estimate for roortio is.. Istead of examiig oly the iterval width of., a series of widths from.5 to. 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 the Differece Betwee Two Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for the Differece Betwee Two Proortios. You may the make 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 evel Grou Allocatio... Equal (N N Cofidece Iterval Width (Two-Sided....5 to. by.5 Iut Tye... Differeces P P (Differece i Samle Proortios....5 P.... Aotated Outut Click 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 ower Uer evel N N N Width Width P P P - P imit imit

13 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Refereces Newcombe, R. G 'Iterval Estimatio for the Differece Betwee Ideedet Proortios: Comariso of Eleve Methods.' Statistics i Medicie, 7, Fleiss, J.., evi, B., Paik, M.C.. Statistical Methods for Rates ad Proortios. Third Editio. Joh Wiley & Sos. New York. 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. ower imit ad Uer imit 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.5 whe the estimated samle roortio is.5, the estimated samle roortio is., ad the differece i samle roortios is.5. 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-

14 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios 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.. The cofidece iterval method to be used is the Yates chi-square simle asymtotic method with cotiuity correctio. The cofidece level is set at.95, but.99 is icluded for comarative uroses. The roortio estimates to be used are.6 for Grou, ad.4 for Grou. Istead of examiig oly the iterval width of., a series of widths from.5 to. 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 the Differece Betwee Two Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for the Differece Betwee Two Proortios. You may the make 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 evel Grou Allocatio... Equal (N N Cofidece Iterval Width (Two-Sided....5 to. by.5 Iut Tye... Proortios P....6 P....4 Outut Click 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 ower Uer evel N N N Width Width P P P - P imit imit This reort shows the calculated samle sizes for each of the scearios. 6-4

15 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Examle Validatio usig Newcombe (998b 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.9 ad., ad the iterval width is.679 for the Chi-Square (Pearso method,.895 for the Chi-Square C.C. (Yates method,.6764 for the Score (Miettie ad Nurmie method,.685 for the Score (Wilso method, ad.774 for the Score C.C. (Wilso method. The ecessary samle size i each case is 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 the Differece Betwee Two Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for the Differece Betwee Two Proortios. You may the make the aroriate etries as listed below, or oe Examle (a-e 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 evel Grou Allocatio... Equal (N N Cofidece Iterval Width (Two-Sided... Varies (.679,.895,.6764,.685,.774 Iut Tye... Proortios P....9 P.... Outut Click the Calculate butto to erform the calculatios ad geerate the followig outut. Chi-Square (Pearso Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be er grou. Chi-Square C.C. (Yates Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be er grou. 6-5

16 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Score (Miettie & Nurmie Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be er grou. Score (Wilso Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be er grou. Score C.C. (Wilso Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be er grou. 6-6

17 PASS Samle Size Software Cofidece Itervals for the Differece Betwee Two Proortios Examle 4 Validatio usig Gart ad Nam (99 Gart ad Nam (99 age 64 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.8 ad.8, ad the iterval width is.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 the Differece Betwee Two Proortios rocedure widow by exadig Proortios, the Two Ideedet Proortios, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for the Differece Betwee Two Proortios. You may the make 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... Score w/skewess (Gart & Nam Iterval Tye... Two-Sided Cofidece evel Grou Allocatio... Equal (N N Cofidece Iterval Width (Two-Sided Iut Tye... Proortios P....8 P....8 Outut Click the Calculate butto to erform the calculatios ad geerate the followig outut. Numeric Results Cofidece Target Actual ower Uer evel N N N Width Width P P P - P imit imit PASS also calculates the ecessary samle size to be 5 er grou. 6-7