AN EXACT CONFIDENCE INTERVAL FOR THE RATIO OF MEANS USING NONLINEAR REGRESSION ThponRoy Boehringer Ingelheim PharmaceuticaIs, Inc.

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1 AN EXACT CONFIDENCE INTERVAL FOR THE RATIO OF MEANS USING NONLINEAR REGRESSION ThponRoy Boehringer Ingelheim PharmaceuticaIs, Inc. ABSTRACT In several statistical applications, the ratio of means is of primary interest. The construction of confidence intervals for such ratios can be done using various techniques, most of which are less than satisfactory. In this paper, a nonlinear regression model will be used to calculate an exact confidence interval for the ratio of means. This result will be compared to a commonly used approximate technique, and to the empirical~istnbution based bootstrap method. A pharmacokinetic example involving the bioequivalence of formulations will illustrate the methodology, and a SAS"" program to perform the calculations will be presented. THE REGRESSION MODEL AND CONFIDENCE INTERVAL The nonlinear regression model can be expressed as: Yi = j(x;, ~ ) + Ei where Yi are responses (i = 1,..., n) taken at design levels Xi (i = 1,...,11), and ~ is ap-element parameter vector. i (i = 1,...,11) are the errors, assumed to be independent, identically distributed normal random variables with mean zero and variance u'. The regression model for the two-sample problem in which the ratio of means is involved can be formulated as: Yi = 9,xi + 9,9,(1 - Xi), where X. is an indicator variable that takes the values o and 1 for means 1 and 2, with n, and n, as the group sample sizes, respectively. This is the Fieller-Creasy model (see Wallace, 1980; and Cook and Witmer, 1985 for a complete discussion). The ratio of means, 9" is of main interest. Following the derivation in Cook and Witmer (1985), the exact marginal confidence region for 9, is: E.(9,) = {9,1 8l (O,-9,t/(1 + Ii, ) ~ c,}, where c, = 202 X 2 (cr; 1)ln. The regression MSE is used to estimate (9,) will be an interval only if o I > c,; otherwise, 0.(9,) will be the complement of an interval or the entire real line. Thus, the exact interval for the estimate of the ratio, 0" is: [ 0, ± {r, (1- r,) +r, iii y'] 1(1- r,), where r,= c,/~ < 1. Aninterestingfeatureofthe Fieller-Creasy model is that though the intrinsic curvature, curvature, f"'/= 0, and the parameter effects r' = ujil,{(o; +n,/n, )U' + IO,IJ In, 10,1, is quite small, indicating low curvature, the linear approximation confidence regions are in poor agreement with the exact regions. This is in contrast to the Bates-Watts criterion guidelines (Bates and Watts, 1980). EXAMPLE An approximate confidence interval for the ratio of means can be represented as: y... ± t(cr/2;n-2)j2'mse / n Y... in the application of testing bioequivalence of pharmaceutical formulations, (std. and test). The MSE is taken from an ANOVA using subject, formulation, and other relevant variables as factors, using an n-subject crossover design. The formulation characteristic of interest could be area under the concentration-time curve (AVe) or a related measure. This type of confidence interval is widely used in bioequivalence testing despite its obvious limitations.' This will be referred to as the conventional method. Effons have been made to develop improved confidence intervals, (see Mandallaz and Mau, 1981; Steinijans and Diletti, 1983; and Locke, 1984). An illustrative example comparing different methods of confidence interval construction is provided in Thble 1 (from Roy, in press). (The data for this example is in the Appendix.) Included is the bootstrap confidence interval which reflects the empirical parameter distribution. Efron (1982) descnbes the bootstrap method and Roy and MacGregor (1991) have applied it to bioequivalence problems. Since in bioequivalence problems, the assumptions of independence and normality may be untenable, the bootstrap method is an attractive alternative to conventional methods. 859

2 The bootstrap technique incorporates distnbutional features, correlation structure, and factor effects when applied appropriately. Since a crossover design was used, the (std., test) pairs of response values were resampled. This ensures that both formulation balance and correlation features are accounted for in the analysis. The distnbution of individual ratios (or their means or medians) can also be characterized using this procedure. For parallel group designs, the resampling should be stratified by formulation. Stratified resampling can be used for any design with multiple factors to guarantee proportionally representative samples. From Thble I it is evident that the Fieller-Creasy and bootstrap intervals are inclose agreement, and the conventional procedure interval is quite different. Since bioequivalence is generally concluded when the 90% confidence interval for the ratio is entirely within the (0.80, 1.20) range «0.80, 1.25) for 10g~tranSformed data), the conventional procedure would indicate bioequivalence while the Fieller~reasy and bootstrap methods do not. 10 examine the validity and accuracy of the bootstrap, bioequivalence study (AUC) data was simulated following the normal, lognormal, and uniform distnbutions. For each distnbution, a two-period crossover (n = 24) design to compare a test formulation with a standard was simulated, incorporating the necessary correlation structure. TABLE I COMPARISON OF CONFIDENCE INTERVALS 90% Confidence Intervals RatiO of Conven- Fieller- Bootstrap, AUC Means tional Creasy B=4, (0.89, (0.85, (0.86, 1.14) 1.21) 1.21) This procedure was repeated 200 times for each distnbution. B = 500 bootstrap replications were performed for each simulated sample (for the lognormal simulations, the 95% confidence intervals are also presented, as this is customary in bioequivalence studies). The conventional procedure for the lognormal simulation data was the usual confidence interval on the difference of the log-transformed means, exponentiating the endpoints to return to the original scale. This construction would have given a correct reference interval if the means were uncorrelated. The results, from Roy and MacGregor (1991), are summarized in Thbles II-Iv. From Thbles II and III, it is evident that the bootstrap intervals are considerably narrower than the conventional intervals. From Thble rv; it is clear that coverage probabilities (the proportion of confidence intervals containing the true values) for the bootstrap analyses are far closer to the nominal levels than those for the conventional intervals, especially for the. lognormal and uniform cases. Theoretically, the assumptions of normality and independence must be tenable for the validity of the exact Fieller-Creasy interval. In Thble V, the stability of the bootstrap intervals is demonstrated for various values of B, the number of bootstrap replications used for normally distnbuted, simulated data. These results indicate that the bootstrap method can be used when the situation defies conventional analysis. In fact, if confidence intervals on the ratio of medians or inferences on individual ratios are desired, it may be the only applicable method. The Fieller-Creasy method is recommended if departures from normality and independence are not excessive. Factor effects can easily be incorporated into the Fieller-Creasy model by the addition of indicator variables, if desired. COMPUTER PROGRAM In the APPendix, a SAS program is given that will calculate the exact Fieller-Creasy confidence intervals. It is written to produce 90% intervals, though this can be changed to any desired level by changing the argument to CINV in the program. The example shown in Thble r is used to demonstrate the input and output for the program. This Version 6.07 program was written for the CMS environment on an IBM 9121 time-sharing system. It should be easily adaptable to any operating system and platform. DISCUSSION In this paper, the computation of an exact confidence interval for the ratio of means has been presented. Comparison of the exact method to a conventional procedure and to the bootstrap technique was performed, and the conditions under which the exact method and bootstrap. technique are valid were examined. Under standard assumptions, the exact method is valid from theoretical considerations. When the usual assumptions do not hold or are difficult to verify, additional factors are present, or other unusual conditions exist, the bootstrap can be used to obtain accurate and reliable confidence intervals. The conventional procedure appears less than satisfactory and should be used with caution. 860

3 TABLE II SIMULATION STUDY RESULTS, 200 TRIALS COMPARISON OF CONFIDENCE INTERVALS FOR RATIOS OF MEANS Conventional Bootstrap, B = % Confidence 95% Confidence 90% Confidence 95% Confidence Distribution Intervals Intervals Intervals Intervals Lower Upper Lower Upper Lower Upper Lower Upper Umlt Umlt Umlt Umlt Umlt Umlt Umlt Umlt Mean Mean Mean Mean Mean Mean Mean Mean (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) Normal ( ) ( ) ( ) ( ) Lognormal ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Uniform ( ) (0.01S01) ( ) ( ) TABLE III SIMULATION STUDY RESULTS, 200 TRIALS COMPARISON OF CONFIDENCE INTERVAL LENGTHS FOR RATIOS OF MEANS.'.. Conventional Bootstrap, B = % Coi.fldence 95% Confidence 90% Confidence 95% Confidence Distribution Interval Interval Interval Interval Mean Length (S.D.) Mean Length (S.D.) Mean Length (S.D.) Mean Length (S.D.). Normal ( ) ( ) Lognormal , ( ) ( ) ( ) ( ) Uniform (0.0148) ( ) TABLE IV SIMULATION STUDY RESULTS, 200 TRIALS CONFIDENCE INTERVAL COVERAGIO PROBABILITIES FOR RATIOS OF MEANS Conventional Bootstrap, B Distribution Nominal = Normal Lognormal Uniform

4 TABLE V COMPARISON OF RESULTS AT DIFFERENT NUMBERS OF BOOTSTRAP REPLICATIONS: POINT ESTIMATES AND 90% BOOTSTRAP CONADENCE INTERVALS Normal DIStribution B=5OO B=1,000 B=2,000 B=3,000 B=4,000 Ratio of Means (0.8258, ) (0.8254,0.9211) (0.8241, ) (0.8241,0.9199) (0.8241, ) Ratio of Medians (0.7829, ) (0.7829, ) (0.7829, ) (0.7829, ) (0.7829, ) Entries are estimate (5th percentile, 95th percentile) where estimate is the median of the bootstrap distribution. ACknowledgment: The author wishes to thank the BIPI Document Processing Group for their expert word processing. REFERENCES Bates, D.M. and Watts, D.G. (1980). Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society, Ser. B 42, Cook, R.D. and Witmer, J.A. (1985). A note on parameter-effects curvature. Journal of the American Statistical Association 80, Efron, B. (1982). The Jackknife. the Bootstrap, and Other Resampling Plans. Philadelphia: Society for Industrial and Applied Mathematics. Locke, C.S. (1984). An exact confidence interval from untransformed data for the ratio of two formulation means. Journal of Phannacokinetics and Biophannaceutics 12, Mandallaz, D. and Mall, J. (1981). Comparison of different methods for decision-making in bioequivalence assessment. Biometrics 37, Roy, T. and MacGregor, T.R. (1991). Bootstrap confidence intervals for bioequivalence ratios. Paper presented at the 1991 American Association of Pharmaceutical Scientists meeting. Pharmaceutical Research 8, S-63. Roy, T. An exact confidence interval for the ratio of means usfug regression methods: Journal of Statistical Computation and Simulation, in press. Steinijans, v.w. and Diletti, E. (1983). Statistical analysis of bioavaiiability studies:. parametric and nonparametric confidence intervals. European Journal of Clinical Phannacology 24, Wallace, D.L. (1980). The Behrens-Fisher and FieBer-CreasY problems. In RA. FISher: An Appreciation (S.E. Fienberg and D.V. Hinkley, eds.) New York: Springer-Verlag, pp ; SAS is a registered trademark or trademark of SAS Institute, Inc. in the U.S.A. and other countries. indicates U.S.A. registration. Other brand and product names are registered trademarks \>r trademarks of their respective companies. 862

5 APPENDIX: PROGRAM AND OUTPUT.. ", EXACT CONFIDENCE INTERVAL CALCULATION FOR THE RATIO. OF MEANS USING THE FIELLER-CREASY MODEL..:..,, SEE COOK AND WITMER (1985), JABA, 80, AUTHOR: TAPON ROY DATE: 9/10/ ~....~... ': :.:, ; OPTIONS NODATE NONUMBER: READ INPUT DATA IN ; Y VALUES ARE FOR COMPUTATION OF THE TWO ~ANS '00: '00 THE VALUE OF X IS 0 FOR MEAN 1 AND 1 FOR MEAN 2 0": DATA A: SET A.EQVCONF: CALL SYlIPUT( 'N', _N_): PROC PRINT: TITLE 'SAMPLE DATA': PRoe SORT: BY X: PROC MEANs: VAR Y: BY X' TITrE 'GROUP MEANS': '00 IN THE PROCEDURE BELOW, SET THE INITIAL VALUE OF B1 "': TO BE - THE VALUE OF MEAN 2, AND THE INITIAL VALUE OF B2 TO BE - THE RATIO OF MEANl TO MBAH2 ; PROC NLIN BEST=Z5 PLOT METHOD-MARQUARDT OUTEST=POUT: PARMS B1-1Z B2= 1: MODEL Y- (B1 X) + (B1'BZ"(1-X»: DER.BI- X + (BZO(l-X»: DER.B2= B1 (1-X): OUTPUT OUT=D1 P=PD1 R=RD1: TITLE 'NONLINEAR REGRESSION MODEL': TITLE 'PREDICTED VALUES AND RESIDUALS FOR METHOD-MARQUARDT': DATA POUT: SET POUT: IF _fype_='final'; DATA AA1: SET POUT: KEEP _TYPE SSE_ B1 B2: DATA AA1: SET AA1: N=&N: DFE- N-2: C1= (2 (_SSE IDFE) CINV(0.90,l»/N: FOR 90 PCT C.I.: CUTOFF= (Bl T2): R1= Cl/(B1"2): B218= (B2 - SQRT(R1'(1-Rl) (Rl'(82 '2»»/(1-Rl): B2UE= (B2 SQRT(Rl (l-rl) (R1'(82002»»/(1-R1): PROC PRINT: TITLE1 'PARAMETER ESTIMATES AND THE EXACT CONFIDENCE INTERVAL': UTLE2 'FOR THE RATIO OF MEANS':. TITLE3 '(B2UE AND B2LE ARE ONLY VALID IF CUTOFF) C1)':. SAMPLE DATA abs y X

6 VARIABL.E... STANDARD GROUP JiEANS MINIMUM MUIMtJM DEVIATION VALUE VALUE -._. XaO _ STD ERROl OF MEAN.'"' VA.IAHc:E C.V Y S5405UI ; $ X-I y ' & & NOTE: CONVERGENCE CRITERION MET. ITERATION o NDNLINEAR REGRESSION MODEL NON-LINEAR LEAST SQUARES ITERATIVE PHASE DEPENDENT VARIABLE: Y METHOD: MARQUARDT B RESIDUAL SS NONLINEAR REGRESSION MODEL NON-LINEAR LEAST SQUARES SUMMARY STATISTICS DEPENDENT VARIABLE Y SOURCE REGRESSION RESIDUAL OF SUM OF SQUARES MEAN SQUARE UNCORRECTED (CORRECTED TOTP,L TOTAL] PARAMETER Bl B2 ESTIMATE ASYMPTOTIC ASYMPTOTIC 95 % STD. ERROR CONFIDENCE INTERVAL LOWER UPPER USE EXACT C.I. BELOW --- ASYMPTOTIC CORRELATION MATRIX OF THE PARAMETERS CORR B1 B2 B1 B ! l,,, f OBS., PARAMETER ESfIMATES AND THE EXACT CONFIDENCE INTERVAL FOR THE RATIO Dr MEANS (B2UE AND 82LE ARE ONLY VALID IF CUTOFF) ell.2 N DFE C1 CUTOFF R1 12, o EXACT CONFIDENCE INTERVAL B2L.E B2UE