Using SAS to Evaluate Integrals and Reverse Functions in Power and Sample Size Calculations

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1 Usig SAS to Evaluate Itegrals ad everse Fuctios i Power ad Sample Size Calculatios Xigshu Zhu, Merck &Co., Ic. Blue Bell, PA 94 Shupig Zhag, Merck &Co., Ic. Blue Bell, PA 94 ABSTACT We have recetly created SAS codes for calculatig itegratio ad reverse fuctio. The codes were coceived durig the developmet of macro %SSEQ_BE, which is used to calculate power, sample size, ad other parameters, icludig expected mea differece ad equivalece limits. These SAS codes have bee successfully executed i both SAS V6. ad SAS V8. withi the Widows 98 operatig system. I this paper, we use the calculatio of power ad sample size as a example to demostrate how SAS tools ca be used to evaluate itegrals ad reverse fuctios. KEY WOS: Itegrals, everse Fuctios, Power Calculatios, Sample Size Calculatios INTOUCTION Performig power aalysis ad sample size estimatio is a importat aspect of experimetal desig. Without such calculatios, the selected sample size could be either too large or too small, resultig i a study that would be uecessarily costly or uable to produce eough data to make the results statistically sigificat. Power aalysis ad sample size estimatio allow cliical researchers to develop the most sesitive ad productive study desig for their available resources. The macro that we have developed, %SSEQ_BE, completes the power ad sample size calculatios for testig the equivalece betwee populatio meas. (See Item of the Appedix.) The macro ca be used i three differet types of studies: the parallel study, the cross-over study, ad the paired-observatio/oe-sample study. For each study type, there are eight parameters ivolved i the calculatios. Whe seve of the eight parameters are give, macro %SSEQ_BE ca be used to calculate the remaiig ukow parameter (except stadard deviatio). Sice all three study types ca be treated similarly, the example preseted i this paper ivolves a parallel study oly. The parallel study discussed here highlights the eed for SAS codes of itegratio ad reverse fuctio. I this study, which compared the ifluece of two drugs o diastolic blood pressure, the software PASS had bee used to calculate power ad sample size. The results the served as stadards agaist which macro %SSEQ_BE could be measured i order to validate the macro s effectiveess. The aim of the study was to show that the diastolic blood pressure would fall withi 0% of the pressure o the referece drug. The diastolic blood pressure was 96 mmhg with the referece drug ad was assumed to be 9 mmhg with the experimetal drug. The power was calculated with differet sample sizes, ragig from 3 to 60, while the stadard deviatio was set at 8, ad the sigificace level was Table presets the results geerated by PASS. Table Existig esults Geerated by PASS Power µ s N α

2 urig the developmet of macro %SSEQ_BE, we geerated our ow results for this data by first startig with a approximate formula which used the Schuirma s Two-Oe-Sided-Tests Procedure (TOST). (See Item of the Appedix.) However, the approximate formula did ot work as expected, ad the macro geerated the output displayed below i Table. Table esults Geerated by Macro %SSEQ_BE Usig the Approximate Formula Power µ s N α -? ? The first two rows of Table demostrate that the power was ot calculated correctly whe the sample size was small. Whe the sample size was below 0, a error was usually itroduced ito the calculatio that was performed with the approximate formula. As the sample size icreased, the approximatio improved, ad the results were more comparable to the output i Table. I order to provide the correct power for smaller sample sizes, we could have used a itegral equatio, which is defied i Item 3 of the Appedix. The questio was how to icorporate this equatio ito SAS codes sice there were o such fuctios or procedures available i the SAS library. We aswered this questio by developig SAS codes of itegratio. We also developed SAS codes of reverse fuctio i order to calculate sample size whe power was provided. Although these codes were coceived durig the developmet of macro %SSEQ_BE, the purpose of this paper is ot to describe our macro, or eve to modify existig statistical methods. ather, it is to demostrate how to utilize SAS tools to calculate itegratio ad reverse fuctio whe the variable is cotaied i a complicated mathematical itegral fuctio, like the oe give i the macro. This issue was critical to power ad sample size calculatio durig the developmet of the %SSEQ_BE macro. I additio, the ability to use itegratio ad reverse fuctio i SAS could have applicatios for statistical aalysis that exted well beyod the example preseted i this paper. METHOS FO CALCULATING POWE AN SAMPLE SIZE Our calculatio of power ad sample size begis with the itegral equatio that is displayed i full i Item 3 of the Appedix. If the complex formula betwee the symbols ad dx is replaced with the fuctio f(,,,,, N, x), the equatio is the simplified as follows: power = With this equatio as the basis of our aalysis, we ca accurately calculate power ad sample size i SAS with macro %SSEQ_BE. Power Calculatios f ( α, µ,,, σ, N, x) Oe of our objectives for developig macro %SSEQ_BE was to be able to calculate power whe the other parameters are kow. I terms of the itegral equatio, this objective is defied as calculatig power whe the parameters,,,,, ad N are give. As log as these give parameters are costat, we ca simplify the itegral equatio eve further so that we eed oly solve the followig calculatio: f ( x) dx I order to icorporate this itegral calculatio ito SAS, it is importat to cosider the defiitio of a itegral. A itegral is a mathematical object that ca be displayed dx

3 graphically as the area eclosed by the boudaries represeted by the mathematical fuctio ad the abscissa, as displayed i Figure. The evaluatio of the itegral ca thus be coverted to the calculatio of the area eclosed by f(x) ad the x-axis betwee the two limits x= ad x=. Figure Geometric epresetatio of a Itegral ca be used to achieve the evaluatio of the itegral as follows: x=0.; blocks=it(( - )/x); x i = -x/; itegral=0; do to blocks; x i =x i +x; itegral=itegral+f(x i )*x; ed; The SAS fuctio MEAN ca also be used to evaluate a itegral because we ca rewrite the approximate represetatio of the itegral i the followig form: f ( xi) x = x xi f ( ) = x f ( xi) = To use SAS tools to solve this itegral calculatio, a programmer must first make a approximatio of the geometric represetatio of the itegral by dividig the shaded area ito small rectagles ad the addig the areas of these rectagles as show i the figure below. Figure Approximatio of the Itegral f(x i ) x Therefore, f ( x) dx ca be approximately represeted as f ( xi) x, ad the SAS O loop ( - ) MEANf(x ), f(x ), f(x ) Although the idea of usig area to represet a itegral is simple i cocept, there are may complicatios ad subtleties whe implemetig this idea ito a SAS program. I order to accurately evaluate a itegral i a program, two caveats must be heeded. First, if the fuctio f(x) is a liear fuctio of x, the the area calculatio is exact. Otherwise, the sum of the areas of the rectagles may oly be a approximatio of the area to be evaluated. I the itegral evaluatio above, we assumed that f(x) was at least a cotiuous fuctio. The secod caveat ivolves the selectio of the width (x). For a properly behaved fuctio f(x), the itegral evaluatio is more accurate, the smaller the x. However, more computatio time is required as x decreases because the umber of rectagles icreases. epedig o the specific fuctio ad its properties, differet values of x may be selected. For the power aalysis performed durig the developmet of macro %SSEQ_BE, we used a value of 0. for x i the calculatio.

4 Sample Size Calculatios A secod objective for developig macro %SSEQ_BE was to be able to estimate sample size (N) if power ad the parameters,,,, ad are give. I this case, we eed to fid the reverse fuctio N=f (power,,,,,, x) from the itegral equatio i Item 3 of the Appedix. Ufortuately, it is difficult to fid a explicit reverse fuctio for N from this itegral equatio. We therefore eed a algorithm to solve the followig math statemet: Give the fuctios y = f(x) ad y = Y 0, fid a value X 0 for the variable x, such that f(x 0 ) = Y 0. This statemet is represeted graphically i Figure 3 below. As show i figure 4, we take the middle poit X m of the regio [lower, upper] = [( ), ( )], ad compare f(x m ) with the give value for Y 0. Because f(x) is a icreasig fuctio ad f(x m )<Y 0, so that f(x)<y 0 for all x [lower, X m ], we move the lower limit to X m, arrowig the searchig regio to [X m, upper]. Figure 5 Cotiuatio of the Biary Search Whe f(x m )<Y 0 Figure 3 Graphical epresetatio of the everse Fuctio Next, as idicated i Figure 5, we take the middle poit X m of the ew regio [lower, upper] = [(X m ), ( )], ad compare f(x m ) with Y 0. Agai, because f(x) is a icreasig fuctio ad f(x m )>Y 0, so that f(x) > Y 0 for all x [X m, upper], we move the upper limit to X m, arrowig the searchig regio to [lower, X m ]. There are o geeral algorithms for evaluatig this statemet. However, we ca make the calculatio it describes by takig advatage of the fact that, i our sample size ad power aalysis, most of the fuctios are mootoic, either icreasig or decreasig. For this calculatio we ca use the biary search algorithm as demostrated below for a ascedig mootoic fuctio. Figure 4 Start of the Biary Search for a Ascedig Mootoic Fuctio Figure 6 Cotiuatio of the Biary Search Whe f(x m )>Y 0 These steps are repeated i Figure 6. That is, if f(x m )<Y 0, move lower to X m, ad if f(x m )>Y 0, move upper to X m. " If f(x m )=Y 0, or (upper-lower) < tolerace, the the search is stopped. We ca ow fid either the exact value of X 0, or a acceptable approximate value of X 0 withi the rage of [lower, upper].

5 A SAS WHILE-loop is used to achieve the biary search algorithm described above: lower = ; upper = ; accuracy = required_precisio; do while (upper -lower > accuracy); X m = (lower + upper)/; if f(x m ) = Y 0 the leave; else if f(x m )<Y 0 the lower= X m ; else if f(x m )>Y 0 the upper= X m ; ed; X 0 = X m ; Although the biary search algorithm for solvig the reverse fuctio was oly preseted i the case of a mootoic fuctio, this method ca also be applied to other types of fuctios, such as a o-mootoic fuctio. I this case, we eed to first idetify the mootoic regios of the fuctio, ad the apply the biary search algorithm i these specific regios to fid the correspodig values of the reverse fuctio. APPLICATION I order to use the methods described above to calculate power ad sample size with the macro %SSEQ_BE that we developed, it is ecessary to replace the fuctio f(x i ) ad the itegral limits ad with the terms listed i Item 4 of the Appedix. To test the macro s effectiveess with the itegral approach, we repeated the power calculatios for the parallel study o diastolic blood pressure described i the itroductio to this paper. Macro %SSEQ_BE geerated the output preseted below i Table 3. Table 3 esults Geerated by Macro %SSEQ_BE Usig the Itegral Method Power µ s N α The ext step is to compare the results i Table, geerated by macro %SSEQ_BE with the approximate formula, to the results i Table 3, geerated by macro %SSEQ_BE with the itegral method. This compariso demostrates that the power aalysis with the itegral method produces more accurate results tha the power aalysis with the approximate formula. The mai differece betwee the results i Tables ad 3 lies i the smaller sample sizes i the first two lies. The accurate results for these small sample sizes i Table 3 could be geerated by the macro with the itegral method, but ot without it. I fact, the overall results i Table 3 are almost idetical to those geerated by PASS i Table. CONCLUSION Power ad sample size aalysis ca be crucial to the desig of a study i order for it to be costeffective ad scietifically reliable. Therefore, proper calculatio of the power ad sample size for a specific desig is a very importat issue i cliical trials. We have developed SAS codes that make it possible to use the itegral method to calculate power, ad the biary search algorithm for reverse fuctio to calculate sample size. These methods produce accurate results for all values of sample size, while the existig approximate equatios used i SAS work oly for larger sample sizes. The methods worked correctly ad effectively whe applied to a example of a diastolic blood pressure data set i a parallel-group desig study comparig the ifluece of two drugs. The evaluatio of itegrals ad reverse fuctios has ever bee performed i SAS before, ad we believe that the SAS codes for these calculatios will fid may more applicatios i various aspects of data aalysis. APPENIX. %SSEQ_BE is a macro used to calculate power, sample size, equivalece, boud, ad

6 other parameters for the problem of testig the equivalece betwee populatio meas usig Schuirma s Two-Oe-Sided-Tests Procedure (TOST), which is idetical to Westlake s Cofidece Iterval Approach.. Approximate formula for power calculatio: Power = F( t α, µ + ) F ( t α, σ f ( N ) µ + ) σ f ( N ) where F = Cumulative istributio Fuctio of a cetral t distributio with degrees of freedom; f ( N ) = N t + N c t, = upper 00( - ) th percetile of a cetral t distributio with degrees of freedom; = N t + N c for a parallel study, where N t, N c are sample sizes for the groups; = the equivalece lower limit; = the equivalece upper limit; = the actual mea differece t - c betwee the two groups; ad σ = the estimated stadard deviatio. Ghosh, K., Cha, I.S.F. ad Biswas, N. (000), Equivalece Trials: Statistical Issues, Oe-ay Course at Joit Statistical Meetigs, Idiaapolis. 3. for a parallel study; x x [ Φ t + µ Φ t + µ ( α, ) ( α, ] f ( x)dx Power= 0 ) σ f ( N ) σ f ( N ) where Φ = Cumulative istributio Fuctio of a stadard ormal distributio; f = probability desity fuctio of a chisquare distributio with degrees of freedom; t, = upper 00( - ) th percetile of a cetral t distributio with degrees of freedom. 4. I the power ad sample size calculatios performed by macro %SSEQ_BE, the fuctio f(x i ) ad the itegral limits ad are replaced with the followig terms: = 0 = ( ) σ f ( N ) t, α xi xi µ µ f( xi) = Φ ( tα, + ) Φ( tα, + ) f σ f( N) σ f ( N ) TAEMAKS SAS ad all other SAS Istitute Ic. product or service ames are registered trademarks or trademarks of SAS Istitute Ic. i the USA ad other coutries. idicates USA registratio. Other brad ad product ames are trademarks of their respective compaies. ACKNOWLEGEMENTS The authors would like to thak Alla Glaser, Amy Gillespie, oa Usavage ad Jodi Bejami for their valuable suggestios ad commets. CONTACT INFOMATION Your commets ad questios are valued ad ecouraged. Cotact the authors at: Author Name: Xigshu Zhu Compay Merck &Co., Ic. Address : UNA-0, 785 Jolly road, Blue Bell, PA 94 Work phoe: Fax: xigshu_zhu@merck.com Author Name: Shupig Zhag Compay Merck &Co., Ic. Address : 0 Setry Parkway, Blue Bell, PA 94 Work phoe: Fax: Shupig_zhag@merck.com ( xi)