SAMPLE SIZE RE-ESTIMATION FOR BINARY DATA VIA CONDITIONAL POWER

Transcription

1 SAPE SIZE RE-ESIAION FOR BINARY DAA VIA ONDIIONA POWER. Wang, D.S. Keller and K.K.G an. Wang and D.S. Keller, Pfizer Inc, Groton, K.K.G. an, Aventis Pharmaceuticals, Bridgewater, NJ Key Words: Sample size re-estimation, onditional power, Binary data, Noninferiority, ype I error rate Abstract: We extend an and rost s (997,999) conditional power approach for interim sample size re-estimation to binary data for clinical trials with a noninferiority objective. If conditional power is used to extend a trial, the α level will be inflated. On the contrary if it is used to stop a trial early to claim futility, the α level will be reduced. If inflation does not exceed deflation, then the α level will be maintained. Simulations were used to estimate probabilities of ype I error for a range of typical clinical trial situations commonly encountered in Veterinary edicine. he simulation results demonstrated that the procedure preserves ype I error rate. hroughout, hypothetical examples will be used to illustrate the procedure. his intuitive, simple and flexible procedure is recommended for use in clinical trials.. linical rial Setting/introduction onsider the following clinical trial situation. A sponsor was to test a novel compound for Veterinary edical use. A typical multi-center, double blind, randomized two -arm trial was envisioned. he clinical end point was binary success or failure of the treatment. Due to ethical concerns, a positive control was needed. Per IH Guideline and preliminary discussion with a regulatory agency, a noninferiority test of the test compound over the positive control was required. he sponsor and the regulatory agency mutually agreed ) that the minimum sample size per arm should be 00 patients, 2) that the noninferiority margin should be 5 percentage points, and 3) the significance level should be one-sided 5%. Also result of Freedom of Information, the treatment success rate of the positive control was known to be about 75%. Without sufficient early trial information, the sponsor was unsure about the efficacy of the test compound but thought it should be comparable to the positive control (75%). Under a fixed design, the trial statistician performed some sample size/power analyses for true effect sizes of the test compound up to 5% lower than the reference (able ). able. Sample Size and Power Percent Success (%) Positive Power SS** ontrol est (%)* * denotes power with 00 cases per arm, and ** denotes sample size (SS) per arm to achieve 80% of power What sample size should the sponsor use? If the efficacy was comparable to the positive control, the minimum sample size of 00 required by the agency should be sufficient to achieve a power of approximately 80%. On the other extreme, if the test compound performed 5 percentage points worse than the control, the sample size requirement increased substantially to 246. onservatively, to ensure success of the trial, a sample size of 246 would be needed. However, if the efficacy of the test was comparable to or better than the control, this would amount to a considerable waste of resources, since 00 subjects per arm would be sufficient. Fortunately, recent advances in flexible/adaptive trial design and interim analysis were used to resolve this dilemma. With concurrence from the regulatory agency, the sponsor proposed to perform an interim analysis to reassess the sample size requirement. Flexible/adaptive trial design and interim data analysis are active areas of research in recent years. Increased knowledge has contributed to greater use of these methods in clinical trials (O Neill, 994; IH E-9, 999). Gould (200) provided a comprehensive review of interim sample size re -estimation in methodology developments and their uses in practice. onditional power (P) (an et al., 982, 984; Halperin et al., 987; Proschan and Hunsberger, 995; Snapinn, 992) has been advanced as a useful tool to manage clinical trials. Its properties and uses have been elucidated in an and rost (997, 999), 362

2 ooper et al. (200), Siu and an (200), among others. In this paper, we extend this P approach (an and rost, 997, 999) to the aforementioned clinical trial situation. he proposed procedure does not require α spending or increase critical values to preserve overall α. It is well known that if a trial is extended, the α level will be inflated. On the contrary if a trial is terminated early to claim futility, the α level will be reduced. If inflation does not exceed deflation, then the α level will be maintained. he proposed procedure combines sample size reestimation with early trial termination for futility such that the nominal α level is preserved. Explicitly, the objectives of this paper are: ) to extend the an and rost s (997, 999) P approach to sample size re-estimation for binary data with a noninferiority study objective, 2) to validate the proposed procedure in terms of α level preservation using simulations under trial conditions applicable for Veterinary edicine, and 3) to illustrate the procedure using hypothetical examples. 2. Hypothesis et denote the test article and denote the positive control. Further let p and p be probabilities of treatment successes for test and control respectively. Noninferiority margin is denoted by δ (a positive number). he null hypothesis is that the test is inferior to the positive control by a noninferiority margin δ. Explicitly, the null and the alternative hypotheses are: H : p p δ H 0 A : p > p δ 3. est: One-sided Z test he Z test statistic is pˆ pˆ + δ λˆ + δ Z SE SE where pˆ pˆ n λˆ () pˆ is the estimated success rate of the test article, with n as the observed number of successful patients and as the total number of n successful patients in the test group, pˆ is the estimated success rate of the positive control, with n as the observed number of successful patients and as the total number of patients in the positive control group, and SE pˆ ( pˆ ) / + pˆ ( pˆ ) / /. Reject λ ˆ { } 2 H 0 (concluding noninferiority), if Z. Z α Equivalently, noninferiority will be concluded if the one-sided lower limit of the difference between the estimated success rates is greater than or equal to the negative of the noninferiority margin δ, i.e., if ( pˆ pˆ ) Z SE δ. 4. B-value 0.95 pˆ pˆ Without loss of generality, let be the total sample size per group, and r be the number of patients per group at the interim check. he information fraction is defined as r /. (2) he Z value at, denoted by Z can be calculated as in () using the available patients (total number of patients 2r). he B value calculated at the information fraction (an and Wittes, 988) is: B Z. (3) 5. onditional power he conditional power to declare noninferiority given the currently observed data, a total sample size of per treatment and the treatment effect is calculated as (an and Wittes, 988): P P[B Z, θ] B E(B Z P[ V(B α α V(B E(B, θ] Z α E(B P[N(0,) ] (4) where E(B B + ( θ is the mean and ), θ) V(B is the variance of the conditional normal distribution, with E (B ) Z (βype II error). θ α + Z β Assume the current treatment trend continues, then θ B / and E(B, θ ) B, so Eq. (4) becomes Z P P[ N(0,) α B / / ]. (5) 3622

3 6. Sample size required to achieve a certain level of conditional power From (4), we have Z α E(B, θ) ZP (6) with P Φ(Z ), and Φ Z ) the cumulative P ( P standard normal distribution evaluated at Z P. Assume the current treatment trend continues, Eq. (6) becomes Z α B / ZP (7) with defined in (2) and B defined in (3). Eq. (7) can be expressed as r ZP + Z Z α 0 (8) r he total sample size () required to achieve conditional power of P is obtained by solving Eq. (8). 7. An example calculation of P and sample size For illustration purposes, we provide example calculations for the following conditions: δ 0.5, r 50, 200, r / 50 / , pˆ 0.70, estimated at, and SE pˆ 0.75, estimated at. { pˆ ( pˆ ) / r + pˆ ( pˆ ) / r} / 2 λ ˆ he Z vale at is: pˆ ˆ pˆ + δ λ + δ Z SE SE pˆ pˆ he B vale at is: B Z For our case, α.05, Z λˆ he conditional power to declare noninferiority given the observed data, a total scheduled sample size of 200 patients per treatment, and the current treatment trend, is P P[B , θ ] B P[ ] P[N(0,) ] with the current treatment trend, θ B / / , and E(B B + ( ) θ ( 0.25) he total sample size required per group to achieve a certain level of conditional power is obtained by solving Eq. (8). For P80%, Z P , the equation becomes he solution is Interim sample size re-estimation procedure he procedure consists of the following steps and decision rules: With an interim sample size r, calculate success rates by treatment group. If the estimated success rate of the test is d (d>0) less than that of the positive control (i.e., pˆ pˆ ) d ), then stop ( the trial and fail to demonstrate noninferiority. If not, calculate Z as in Eq. (). Solve Eq. (8) for to achieve P c U (where c U is the desirable P for the trial) assuming the treatment trend estimated at the interim continues. If < min reset min, where min is the minimum sample size required; If > reset, where is the imum sample size affordable. alculate P as in Eq. (5) given assuming the treatment trend estimated at the interim continues. If P < c then stop the trial and fail to demonstrate noninferiority, where c is the minimu m acceptable P for continuing the trial. If P c then continue the trial with a sample size of min. Reject H 0 if Z Z α. 3623

4 For the hypothetical trial introduced in Section, we set d 0.3, c 0.3, c 0. 8, r50, min 00 and 250. U 9. ype I error rate evaluation for the proposed interim sample size re-estimation procedure using simulation he question is: does the procedure proposed in Section 8 preserve ype I error rate? Due to flexibility of the procedure, no attempt was made to evaluate this property analytically. Instead, simulations were conducted to evaluate the ype I error rates of the procedure for the following conditions under the null hypothesis that test is inferior to the positive control by an noninferiority margin of 0.5 in terms of success rate: One-sided significance level: 5% inimum sample size required per group: 00 aximum sample size affordable per group: 250 Noninferiority margin: 0.5 Sample size per group at the interim check: 50 inimum estimated success rate difference at the interim check to stop the trial early to claim futility: d 0., 0.2, and 0.3. inimum conditional power at the interim check to continue the trial: c 0.2, 0.3, and 0. 4 Desirable conditional power at the interim check: c U 0.8 rue success rates for the positive control ( p ): 0.75, 0.8, 0.85, 0.9, and Under null hypothesis, the corresponding true success rates for the test article ( p ): 0.60, 0.65, 0.70, 0.75, and he above parameter specifications resulted in a total of 45 combinations. Each simulation combination was replicated,000,000 times. Probabilities of ype I error were estimated. For example, the estimated probability of ype I error was for d 0.3, c 0. 3 and p able 2 summarizes estimated probabilities of ype I error for all the selected simulated combinations. able 2. Estimated probabilities of ype I error for selected combinations d rue Success Rate for Positive ontrol (est) c.75 (.60).80 (.65).85 (.70).90 (.75).95 (.80) he results in able 2 demonstrated that under a wide range of use conditions, the proposed interim sample size re-estimation procedure preserves the overall α level. Given d 0.3 and c 0. 3 with the true success rates for the positive control between.75 and.90, the estimated probabilities of ype I error were less than or equal to the nominal level of 5%. 0. Example uses able 3 provides 8 hypothetical trials, and their interim results and impact on decision making under the following conditions specified a priori: α5%, δ0.5, d 0.3, c 0. 3, c U 0. 8, r50, min 00 and 250. For trials and 2, a total of 00 subjects per arm will provide a P of 97% and 87%, respectively, assuming the current treatment trend continues. So the interim decisions are to continue the trials. For trials 3 and 4, extending the trials to 9 (trial 3) and 227 (rial 4) subjects per arm will reach the desired P of 80%, so the trials will be continued. With 250 subjects per arm, rial 5 will yield a P of 65%. Should the trial be continued? It is up to the sponsor to decide to continue the trial or not with P65%. However, it is not proper to extend the trial beyond 250 subjects per arm to reach a P of 80%, because 3624

5 the a priori imum sample size was set at 250. For rial 6, according to the a priori specifications, the sponsor has an option to continue the trial to 250 subjects per arm, realizing the P is only 32%. It is more likely that the sponsor would terminate the trial early for futility owing to such a low P. his will have an effect of further deflating the α. In realistic trial situations, c can be raised further to a higher level, say 0.5. How likely would a sponsor continue a trial given a 50% of P? learly, rials 7 and 8 will be terminated early for futility due to low Ps (<c 0.3) for both trials and low efficacy of the test group ( pˆ pˆ ) 0.3 d ) for trial 8. ( able 3. Examples of interim sample size reestimation and decision making rial pˆ pˆ P (%) Interim Decision ontinue ontinue ontinue ontinue Pending Pending erminate erminate. Discussion We have proposed an interim sample size reestimation procedure via conditional power for binary data in clinical trials with a noninferiority objective. he procedure is an extension of an and rost (997, 999). he essence of the procedure is that if a trial is extended, the α level will be inflated and that on the contrary if a trial is stopped early to claim futility, the α level will be reduced. If inflation does not exceed deflation, then the α level will be maintained. he simulations demonstrated that the procedure preserves the nominal α under a wide range of trial conditions commonly encountered in Veterinary edicine clinical trials. Other trial situations can be simulated accordingly. As such, the procedure does not require α spending or increase critical values of the statistical test. he procedure is flexible to handle realistic trial conditions. In our settings, the procedure incorporated minimum, imum sample size requirements, imum efficacy deficiency of the test compound to the control for continuing the trial, minimum and desirable P to extend the trial. Any other trial conditions can be accommodated. Due to its flexibility, no analytical proof of α preservation was attempted. Rather, it is recommended to conduct simulations for each trial situation to demonstrate α preservation. onditions under which α will be maintained are easy to find. For example, increasing the lower limit of P (c ) will have a dramatic effect on α deflation. he procedure has been investigated to compare two means and two survival curves (an and rost, 997, 999; Siu and an, 200) for superiority testings. We extended the procedure to binary data for sample size re -estimation with a noninferiority objective. With a modification to the test statistics, the procedure is applicable to superiority and equivalence test situations as well. he procedure can be easily extended to multi-stages of interim sample size re-estimation. he P approach does require unblinding the data at interim. his raises the issue of interim data integrity, scope of information sharing, potential for introduction of operational bias and trial management. With proper planning and management, these concerns can be addressed satisfactorily. It is anticipated that with more understanding of the procedure and better trial and data management, this intuitive, simple and flexible procedure will find more use in clinical trials. 2. References ooper, J., K. Anderson and. akshminarayanan (200) Evaluation of sample size re -estimation using group sequential design and conditional power. American Statistical Association, Proceedings of the Biopharmaceutical Section, 200. Gould, A.. (200) Sample size re-estimation: recent developments and practical considerations. Statistics in edicine 20: Halperin,., K.K.G. an, E.. Wright and.a. Foulkes (987) Stochastic curtailing for comparison of slopes in longitudinal studies. ontrolled linical rials 8: IH E-9 Expert Working Group (999) Statistical principles for clinical trials (IH Harmonized ripartite Guideline E-9). Statistics in edicine 8: an, K.K.G., R. Simon and. Halperin (982) Stochastically curtailed tests in long term 3625

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