Simon’s two-stage designs are widely used in cancer phase II clinical trials for assessing the efficacy of a new treatment. control the type I error but they required a prespecified difference between the modified and planned sample sizes. Koyama and Chen [7] controlled the conditional type I error for the modified stage II sample size but the corresponding overall type I error could be highly deflated and the power would be lower than desired. Li obtains desirable frequentist properties under certain types of Cichoric Acid priors. In this paper we attempt to maximize the unconditional power while controlling for the type I error for the modified stage 2 sample size. Because enumerating all possible scenarios in the power calculation is computationally intensive we propose a normal approximation in the evaluation of the power and our numerical results show that the proposed approximation is very accurate even under small sample sizes. Finally we construct confidence intervals for the response rate by inverting the hypothesis test. The rest of this paper is organized as follows. In section 2 we describe the proposed method to account for sample size change in Simon’s two-stage design. Cichoric Acid Our method includes an explicit formula for the charged power calculation and an analytic derivation of the confidence intervals. Extensive simulations are conducted in section 3 to demonstrate the finite-sample performance of the proposed method. Some concluding remarks are given in the final section. 2 Method 2.1 Hypothesis Cichoric Acid testing in Simon’s two-stage design Suppose that Simon’s two-stage design is implemented to test the null hypothesis that the response rate (= = and type I error denote the Capn1 critical values for rejecting the null hypothesis in specific we precede to stage II if we observe ? while maximizing the charged power. Specifically we let denote the critical value Cichoric Acid for the modified stage II sample size when we observe to maximize the power of the test with the overall type I error controlled. This is equivalent to finding the such that by searching among all the possible combinations of in evaluating the corresponding power the computation is very intensive. Instead we consider the following approximation: the cumulative distribution function of the binomial random variable in the previous expression will be approximated by the cumulative distribution function of a normal random variable that is and is the Lagrange multiplier and from 0 to and to get a reasonable range of as to find the such that the type I error defined in equation (1) is as close to as possible. The corresponding { …versus then determine is chosen so that the conditional rejection probability for one example scenario (= 0.3 versus = 0.5 and we changed the stage II sample size are not necessarily the same for different decreases with in the second stage we report the corresponding type I error and power based on our method (AG) and the method of Koyama and Chen. Table II Calculation of type I error and power for extended stage 2 sample size. Table III Calculation of type I error and power for the reduced or same stage 2 sample size. Table II and III show that both AG and KC have protected type I error rate while in almost all scenarios AG has more power than KC. Bold numbers in the tables indicate the scenarios for which AG’s power is at least 0.03 higher than KC’s power. Because KC tries to control the conditional type I error rate for each stage I sample path the overall unconditional type I error rate could be possibly much less than = 0.3 versus = 0.5 stage II sample size doubles) where KC has larger protected type I error rate and a slightly larger power than AG. In calculating the critical value for the modified stage II sample size we adopt a normal approximation to simplify the computation. However for small sample sizes an exhaustive grid search could also be used to search exactly among all the possible combinations of { with true response rate is equal to the probability of observing the critical value = 0.05 1 ? = 0.8. = 17. If we observe and observe = 0.05 1 ? = 0.8and a modified stage II sample size. We consider three different possibilities for the true underlying is equal to the null response rate is equal to the alternative rate is larger than gets larger. In this case (A2) fails.