An Interactive SAS/AF System For Sample Size Calculation

Transcription

1 An Interactive SAS/AF System For Sample Size Calculation Deborah J. Matos CIBA-GEIGY Corporation, Drug Development Department, Summit, NJ ABSTRACT In the design of a clinical trial, the number of patients selected is typically. determined through use of statistical criteria. Sample sizes are cho.sen to ensure that a clinical effect of some prespecified magnitude can be detected with sufficiently high probability. Calculation of sample size with provision for control of Type I and II error rates is an essential part of the planning of an adequate and well-controlled clinical trial. This paper describes PCSIZE, a SAS/AF system, which makes the calculation of sample size estimates fast and easy. This PC-based application was developed under PC-SAS' Version 6.04 (for MS-DOS') using Base SAS', SAS/AF, and Screen Control language" (SCl). First, some basic background theory on sample size estimation is presented, and then focus is shifted to the interactive SASI AF system and its features. STATISTICAL BACKGROUND An understanding of the basic statistical conc'epts of hypothesis testing, significance level, and power isessential for discussion of sample size determination. A brief review of these concepts is provided in the following section. Selection Variable of the Primary Response The primary response variable used to judge the effectiveness of a treatment regimen or therapy must be identified before sample size can be calculated. In this paper only continuous response variables (e.g. vital signs, cholesterol levels, etc.) are considered. For continuous response variables, the true unknown mean of the test treatment (,lh) is to be compared with that of the control treatment (Pc). terms of the primary response variable, /it will be compared with /i6 however, the true values of the test and control means are unknown. Only estimates of these means, denoted by PT and Pc, respectively, are known. Hypothesis Testing In the clinical trial setting, one typically tests whether or not a true difference exists between the means of the two treatments. This is traditionally stated in terms of a null hypothesis, denoted by He' which states that no difference between the true means exists (He: /ic - /it ~ 0). The null hypothesis is assumed to be true unless sufficient evidence is obtained to prove it false. The null hypothesis is then tested and a decision of whether or not to reject it is made. However, since only estimates of the true means are available, it is quite possible that, even if the null hypothesis is true (/ic - /it ~ 0), the observed means might be different enough to exceed some specified criterion by chance. In this case, if the observed difference between the means is large enough by chance alone, one would incorrectly reject the null hypothesis. Significance level The error resulting from rejection of the null hypothesis when it is true is referred to as Type I error. The probability of committing a Type I error is called significance level, and is denoted by a. If the null hypothesis is not true, then another hypothesis, called the alternative or test hypothesis, and denoted by H.. must be true. That is, the true In 639

2 difference between the treatment means Pc and PT is some value 0, where 0;0<0. However, the observed difference, Pc - pp might be quite small by chance, even when the alternative hypothesis is true. Therefore, on the basis of small observed differences, one could fail to reject Ho when in fact Ho is false. This error is referred to as Type" error, and the probability of committing a Type II error is denoted by p. The value of P is dependent on 0, the true but unknown difference between the two group means, as well as on the sample size, and a. Power The probability of correctly rejecting He in favor of a particular alternative hypothesis is called power, and is equal to 1-p. Power quantifies the ability of a study to detect true differences of various values o. For a fixed sample size, an interesting relationship between Type I and Type II error rates is observed: as a increases, P decreases; and vice-versa. Thus, the only way to simultaneously control a and p at acceptably small values is to increase the amount of information available, i.e. to increase the sample size. For specified values of a (conventionally chosen as 0.05 or 0.01), and power (typically O.SO or larger), and 0 (the estimated minimum difference between treatments judged to be of clinical importance), sample size estimates can be calculated for a given hypothesis and its associated test statistic. SAMPLE SIZE DETERMINATION The objective is to determine the number of subjects required for a clinical trial, such that the treatment mean difference of specified magnitude 0 can be detected with power l-p, when the significance level of the test is controlled at level a. Setting and Assumptions The General Linear Model Given a general linear model with i.i.d. random error terms following a normal distribution, with a common mean 0 and unknown variance a/, we are interested in testing He: Pi - Pi = 0 vs. HA: Pi - Pi ;0< 0 (two-sided alternative), where Pi and Pi represent the unknown least squares means for the ith and jth treatments. The Test Statistic Let l1i and l1i denote the estimated least squares means for Pi and Pi' respectively. Then the test statistic for testing He against HA is: T " (,;. - ';)IS- - rot t'" ).1/ - "'I where S- - ~I- ).1/ is the standard deviation of l1i - l1 i The Decision Rule The decision rule is to reject Ho at the a level of significance if: I T I > t,. 12, df=v where t,. 12, df_v is from the t-distribution with v degrees of freedom (v is the degrees of freedom for MS,,,., in the analysis of variance). An equivalent decision rule is to use the F-distribution; i.e. reject He at the a level of significance if: T"!=[(il l - il)is., _.)2 > ',_.; df,", df,-v where f, _ a; df1 ='; df2 = v is from the F distribution with numerator degrees of freedom 1 and denominator degrees of freedom v. Power of the F- Test When the F-test is used in general linear hypothesis, the power of the test is a 640

3 function of a quantity known as the noncentrality parameter, which is denoted as A. That is, the probability of rejecting Ho when /1; - /1; = 6 can be calculated through use of a noncentral F distribution. In mathematical notation, =p[t" > '1_.; 1 111/ - 1'/ =0,,,~ where the random variable T2 follows a noncentral F-distribution with non centrality parameter when /1; - /1; = 6 (6;0<01. For a given linear model and sample size, the power of the test can be computed when a, a/, and 6 are specified. The following value of A =,,2/,,2,. 1',- ~J is needed in evaluating the power. general, 02~ ~ = xu 2 ~, - P) for some K, where K depends on the specific model and sample size. Conversely, when a, a,2, and 6 are specified under a given model, the sample size required to achieve a predetermined power can be obtained through an iterative procedure by repeatedly adjusting the sample size until the desired power is reached (see flowchart in Figure 11. The power is calculated through use of the SAS PROBPfunction. THE SAS/AF SYSTEM: PCSIZE During the planning phase of clinical trials, statisticians are routinely requested to provide clinical research with sample size estimates. Therefore, it was determined that an interactive, user-friendly application which In automated this task would be quite useful. PCSIZE System Requirements PCSIZE was developed on a microcomputer under PC-SAS Version 6.04 for MS-DOS, using Base SAS, SAS/AF, and Screen Control language (SCl). SCl provided several key features which were then unavailable under our mainframe environment (SAS Release for MVS). Some of these features were: cross-field validation for userentered values control for communication between screens generation of customized error messages customized on-line help for all screens and input fields various styles of menu formats (e.g. selection list, pulldown, choice group, and pushbutton) PCSIZE's Features PCSIZE is installed on a shared PC in a central area which is accessible to all statisticians in the Biostatistics Group at Ciba. This machine is a model 486 CPU running at 33 MHz. The SAS/AF application has a menudriven user-friendly interface consisting of a series of menus created in SAS/AF via the BUilD procedure. Input data are collected through entry screens which were formatted and programmed in SCL. The main menu of PCSIZE is the model selection menu, and is displayed in Figure 2. The application provides the ability to calculate sample size estimates for six different types of models: one-way layout (single-center parallel group study), two-way layout (multi-center parallel group study), and three-way 641

4 layout (stratified, multi-center parallel group study), and 2x2 (two-treatment, two-period) crossover, 2x3 (twotreatment, three period) crossover, and 3x3 (three-treatment, three-period) crossover designs. Context-sensitive on-line help is available for all screens and input fields. PCSIZE's main menu help screen is displayed in Figure 3. Users can access the help facility by pressing the F1 function key at any screen. The F12 function key can be used to exit the application. Once a model has been selected from the primary menu, an input screen, which was created via a SAS/AF DISPLAY panel, prompts the statistician to input values for: Alpha la) (Type I error rate or significance level) Variance lu;) (Estimate of the variance for the error term for the primary variable) Desired Power (1 - p) (Power to be achieved) Delta (6) (Estimate of meaningful treatments) the minimum clinically difference between Number of Treatment Groups (For parallel group designs only) Number of Centers (For two-way and three-way layout only) Number of Strata (For three-way layout only) Once values for each of these input parameters have been given, and the fields have been validated, the user presses the F10 function key to run the program. Results are written to the OUTPUT window. design) is presented in Figure 4. The corresponding sample output is given in Figure 5. From this figure we see that for the multi-center parallel group study with three treatment groups and ten centers, the total number of subjects required is 120 (4 subjects per treatment group and center). CONCLUSION The SAS/AF application system presented in this paper provides a means for producing quick and accurate sample size estimates for a variety of different models. Using SAS software products, we were able to provide an effective user-friendly front-end to expedite the sample size calculation process. Statisticians have enjoyed using PCSIZE; they have learning quickly how to use the application since their input is minimal and is automatically validated. With this system in place, the time required to produce sample size estimates has been significantly reduced. ACKNOWLEDGEMENTS Special thanks to Michael Chen and Ching-Ming Yeh, statisticians in the Biostatistics Group in the Drug Development Department at Ciba, for their extensive contributions to the development of the methodology and to the design of the SASI AF system. An earlier version of this paper was presented at the PharmaSUG Fall meeting. TRADEMARKS SAS, PC-SAS, SAS/AF, and SCl are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. MS-DOS is a registered trademark of Microsoft Corp. indicates USA registration. A sample input screen for a two-way layout model (multi-center parallel group 642

5 CONTACT INFORMATION FIGURES For further information, feel free to contact the author at: CIBA GEIGY Corporation Drug Development Department Biostatistics Group 556 Morris Avenue Summit, NJ REFERENCES Lachin, J.M. "Introduction to Sample Size Determination and Power Analysis for Clinical Trials." Controlled Clinical Trials 2:93 113,1981. Mace, A.E. Sample Size Determination. New York: Reinhold Publishing Corporation, Meinert, C.L. Clinical Trials - Design, Conduct, and Analysis. New York: Oxford University Press, SASIAF Guide For Personal Computers, Version 6 Edition, SAS Institute, Inc. Cary, N.C. SASIAF Screen Control Language: Reference, Version 6, First Edition, SAS Institute, Inc. Cary, N.C. SASIAF Software Applications Using Screen Control Language Course Notes, Version 6 Edition, SAS Institute, Inc. Cary, N.C. SAS Language Guide For Personal Computers, Release 6.03 Edition, SAS Institute, Inc. Cary, N.C. SAS Procedures Guide,Release 6.03 Editi()IJ, SAS Institute, Inc. Cary, N.C. Snedecor, G.W., and Cochran, W.G. Statistical Methods. 6th Edition. Ames: Iowa State University Press, ModII I v 643

6 SAMPLE SIZE ESTIMATION PIRe an X neat to your choice and then Dr... centert SAMPLE SIZE ESTIMATION Pile. In X next to your cttolce and then ",... ENTE~ 2x2 Croaaovwr Tria' 2x3 Croaaover Tria' 3x3 erohcwer Tria' Slnv... cen... Parallel Group Study Multt-C.,..r Parallel Grauli' Study Stratified Multt-Center Parallel G_Study 2x2 ero owr Tria' 2x3 Crouowr Trial 3x3 erouovwr Trial Co-mmilnd~;;; MAIN MENu HEl P Type an X not to your chok:1 and thin p,... cente'" to run ttm program. Fl ENTER> F12 H... Run Program exit tho AoPIlcatlon To nit the ap~lcatlon PNaa: F12 TO Ixlt thla HELP len'" type: END at ttle HELP lel'mn command Jlne. Figure 2 - M Bin Menu Figure 3 - Main Menu Help PRSllNe MULTKENTEA MAAL I fl CiRQUP STYDY TYPE IN VAl.UES FOR THE FOLLOWING Pll.RAMETERS, ALPHA, (Ty.. I Error R... ) <OUTPUT) Command ) SAMPLE SIZE PER TREATMENT GROUP ANO CENTER FOR A MULTI-CENTER PARALLEL GROUP STUDY VARIANCe, (Exti... of tho Variance) SAMPLE SIZE TYPE I ESTIMATE OF ERROR VARIANCE DELTA (Power to bl DESIRED POWE'" ActIlewd) CEatlmate of ttl. Minimum Difference DELTA, to bl D end} " TREATMENTS " CENTERS 10 II TllEATMENT& 3 (In_.onI)'O NONCENTRALITY PARAMETER DESIRED POWER ACTUAL POWER i II CENTERs. 10 (In_r. onlyt) p,..., Fl H... FlO - Run Program Ft2 - Aeturn to Main Menu N ole: Type AF at the command 11,ne 10 relurn 10 the Main Menu. t f f Figure 4 - Input Screen For Two-Way Layout Modal 644 Figura 5 - Output For Two-Way Layout Model