TWO-STAGE GROUP SEQUENTIAL DESIGNS FOR COMPARATIVE CLINICAL TRIALS

Group sequential designs are widely used in Phase II clinical trials, which are usually undertaken to evaluate the response probability of specific treatment regimen. In most randomized clinical trials with sequential patient entry, fixed sample size design is unjustified on ethical grounds and sequential designs are often impractical. However group sequential designs are generally more practical and they provide much of the saving possible from sequential designs. Optimal restricted two-stage design is the simplest form of a group sequential design. In this study, group sequential design obtained by ( ) t ∗ α functions characterized using the type-I error probability and optimal restricted two stage design has been compared for the cases that the group sizes are equal. Furthermore, their efficiency regarding fixed sample size design has been calculated and the results have been discussed. KeywordsGroup sequential design, Clinical trials, Two-stage design, Efficiency


INTRODUCTION
For ethical, scientific and economic reasons, clinical trials are often repeatedly monitored for evidence of treatment benefit or harm.To achieve this, statisticians conduct interim analyses periodically on accumulating data [1].Repeated testing at conventional critical values can substantially inflate the overall type I error rate, and various group sequential testing procedures have been proposed to achieve the desired levels of type I error [2,3].
Pocock [2], O'Brien&Fleming [3] were among the first workers to develop group sequential design by modifying the initial work of Armitage [4].DeMets&Ware [5,6] considered asymmetric group sequential boundaries adapted from Pocock and O'Brien&Fleming designs and from Wald's sequential probability ratio test.These designs are based on equally spaced analyses.Jennison [7] and Eales&Jennison [8] explored the extent of possible reductions in expected sample size by searching for optimal symmetric group sequential one-sided tests.Barber&Jennison [9] extend the optimal symmetric group sequential tests of Eales&Jennison [8] to the broader class of asymmetric designs.Jennison&Turnbull [10] describe the parametric family of tests proposed by Emerson&Fleming [11] and extended by Pampallona&Tsiatis [12] to include asymmetric tests with unequal Type I and Type II error probabilities.We refer to the recent book by Jennison&Turnbull [10] for more details about group sequential test.
Lan&DeMets [13], Kim&DeMets [14] proposed the group sequential design obtained by ( ) t * α , use function, which characterizes spending the type I error probability (α).This design is more useful because it is not necessary that each group has equal observation.So that in this test group size is equal or unequal.Two-stage group sequential design is the simplest form of a group sequential design.Owen [15] described two-stage tests for one-sided hypotheses about a normal mean with known variance.Hald [16] derived optimal designs for this same problem using minimax and Bayes weighted average optimality criteria.Calton&McPherson [17] considered hypothesis tests for normal and binomial responses and presented optimal two-stage designs, which did not allow acceptance of the null hypothesis at the first stage.Dewith [18] extended the work of Calton and McPherson for binomial responses by developing optimal designs that allowed acceptance or rejection at the first stage none of these designs used the fixed sample critical value at the final stage.Case et.all [19] developed optimal two-stage designs that have the restriction of using the fixed sample critical value at the final stage.
In this paper, we compared the statistical properties of the group sequential design based on ( ) t * α function, and optimal restricted two-stage design in the setting of onesided comparative clinical trials with normal response.

GROUP SEQUENTIAL DESIGN BASED ON THE TYPE-1 ERROR SPENDING RATE FUNCTION
Lan&DeMets [13] and Kim&DeMets [14] have suggested a flexible group sequential design based on the use function approach.The design, needs only the specification in advance an α spending rate function, ( ) where τ is the first time that a standard Brownian process W t .In this paper the following ( ) ( ) ( ) When the group sizes are equal ( ) , it generates ( ) .A group sequential test is specified by the maximum number of analysis, K, the total number of observations at each analysis n 1, …., nk.Then at the k th analysis, the standardized test statistic after each group of observations is defined as, ( ) , representing the clinically meaningful mean treatment difference to be detected at given α, 1-β and K.The test statistics Z k , is compared with group sequential boundaries ( k C ) as below following; 1.We stop to reject H 0 if Z k ≥C k k=1,…, K-1, otherwise, we continue to the next stage.2. We stop to reject H 0 if Z k ≥C k k=K, Otherwise we stop to accept H 0 .
Therefore, the sample size for each treatment in a clinical trial is determined by and maximum sample size is given by K n 2 n k = . When 1 K = , that is for a fixed sample size design, this formula becomes the familiar sample size formula for normal data [10,13]

OPTIMAL RESTRICTED TWO-STAGE DESIGNS
The two-stage design is the simplest form of a group sequential design.In this design, the null hypothesis where θ ˆ is computed from data on the first 1 n patients.where θ ˆ is computed from data on all n patients. i One may want to test the mean of normal distribution; in this case, Z 1 and Z statistics are defined as following, Z 1 and Z are distributed standard normal distribution and their joint distribution is bivariate normal with zero means, unit variances, and correlation (n 1 /n) 1/2 .The maximum sample size for the two-stage design is n and is realized whenever a second stage is necessary.The expected sample size (ESS) of the two-stage design is given, ( ) [ ] where ( ) θ S P denote the probability that the trial will be stopped at the first stage, and p is the rate of the number of patients at the first stage to the number of total patients at the second stage n n p 1

=
. θ value can be computed for 1 0 , θ θ or max θ , where 0 θ is the θ value when 0 H is true; 1 θ is the θ value when 1 H is true; and max θ is the maximum value of θ.
There are five unknown parameters in the two-stage design, namely: C and 3 C .The critical value at the second stage, 3 C , will be set to equal that of the fixed sample test (11) where ( ) denotes the standard normal distribution function.The other four parameters of interest are chosen to satisfy the two equations: where, ( ) So that the probability of rejecting 0 H at the first stage plus the probability of continuing the trial and rejecting 0 H at the second stage is equal to α, when assuming 0 H is true.The desired power of the trial β − 1 is the same probability under the alternative hypothesis.Eq. ( 12) and Eq. ( 13) are solved iteratively by numerical integration of the bivariate normal distribution using a double precision function [19,21,22].
With five parameters and only three constraints given by Eq. ( 11), ( 12), ( 13) optimality criteria are used to determine the parameter values.So, this test is called optimal restricted two-stage design.In this study, we have examined Minimax and Bayes criteria.
Minimax Criterion: It is described as minimizing the maximum expected sample size with respect to the five design parameters . For the two-stage design max θ is; ( ) function, the function which will be minimized for one-sided test is: (15) Bayes Criterion: Minimize a weighted average of the ESS under 0 H and the 1 H , minimize ( ) ( ) ( ) ( ) Using a weight of 0 for this criterion gives the most efficient designs if the null hypothesis is true while a weight of 1 gives the most efficient designs if the specified alternative is true [19,21].
The design parameters ( 1 C , 2 C , 3 C , p), the probabilities p s (θ) and maximum (n) and expected sample sizes (ESS) obtained using the different criteria are given in Table 2 . The choice of p is sometimes determined by factors unrelated to optimal designs.For some studies it might be practical to choose equal samples at each stage.Therefore, if 50 .0 p = , each stage has equal sizes.Table 2 gives the optimal design parameter for 50 .0 p = .In tables, f n denotes the sample size required using a fixed sample design.Example: In this section, we described a clinical trial to evaluate the potential chemoprevention agent tamoxifen in women with early stage breast cancer and showed how to determine the maximum sample size for group sequential and optimal restricted two-stage design.
The drug tamoxifen was evaluated at the University of Wisconsin-Madison as a potential chemoprevention agent in women with early stage breast cancer and compared with a placebo in a double blind clinical trial [23].In addition to evaluating the recurrence of cancer, this study measured the effect the drug might have on the lipid cholesterol level.If the level were decreased, a reduction in cardiovascular risk would be induced.If the cholesterol level were increased, the drug might be increasing the risk of heart disease and thus diminish the potential benefit.For women eligible for this study, the mean cholesterol level on the placebo was expected to be 220 mg/dl with a standard deviation of 30 mg/dl.A clinically significant change induced by the drug was thought to be a 20 mg/dl decrease or increase in mean cholesterol level.
Suppose we wish to design this trial, using a 5 per cent significance level for a one-sided test of hypothesis with 90 per cent power to distinguish between mean cholesterol levels of 220 mg/dl and 200mg/dl.Assuming that the measurement of cholesterol level is normally distributed, the maximum sample size determination for group sequential design based on ( )

COMPARISON OF THE TEST DESIGNS AND RESULTS
In this section, efficiency of the designs regarding fixed sample size design has been examined.Therefore, the obtained expected sample sizes have been used for the cases in which the group sizes are equal for 1 , 50 .0 t = in group sequential design (Table 1) and 50 .0 p = in optimal restricted two-stage design (Table 2).
The expected sample sizes of the designs relative to the fixed sample size are called the efficiency.
The efficiencies of the tests are computed as; • The efficiency of the designs for equal group sizes have been presented in Bottom number is R 0 (when θ = θ 0 ) When Table 3 is examined, we obtain the following results: • Since Group Sequential designs mentioned in this study do not allow acceptance of H 0 until the final stage, their performance under the null hypothesis is poor.They perform much better under the alternative hypothesis.• However, in each case, the ESS is the smallest for the optimal restricted twostage designs under the null hypothesis.• Moreover, the ESS under the alternative hypothesis for the optimal restricted two-stage designs is usually smaller than that of the Group Sequential designs.• ( ) α and Pocock's designs performance is better than other designs under the alternative hypothesis for large powers.
• If we examined the tables in second and third sections; the expected sample sizes are approximately the same in two-stage designs and group sequential designs under the alternative hypothesis.However, the expected sample sizes are rather small in two-stage designs under the null hypothesis.So, two-stage designs result approximately in an expected % 40 savings in the expected sample size under the null hypothesis.• So, if we compare the optimal restricted two-stage designs with the group sequential designs in case of 2 N = , we can say two-stage design is preferable in terms of sample sizes and performance.
• However, the test design to be used in a clinical trial is generally predetermined.
The existence of parameters in an optimal two-stage design has some restrictions.Therefore, it is more advisable to use group sequential design in clinical trials where there is a probability of continuation.
two-stage design for one-sided test as follows:

Table 2 .
Optimal Restricted Two-Stage One-Sided Designs for Minimax and Bayes a Multiply each value by (σ/δ)2

Table 3
a Top number is R 1