- Email: [email protected]
Comments on “Z. Govindarajulu (2005) robustness of sample size re-estimation with interim binary data for double-blind clinical trials”
Journal of Statistical Planning and Inference 139 (2009) 675 -- 676
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / j s p i
Letter to the Editor
Comments on “Z. Govindarajulu (2005) robustness of sample size reestimation with interim binary data for double-blind clinical trials” Xiaohui Luoa , Peng-Liang Zhaob,∗ a b
Clinical Biostatistics, Merck Research Laboratories, Rahway, NJ 07065-0900, USA Biostatistics and Programming, sanofi aventis, Bridgewater, NJ 08807-0890, USA
In Shih and Zhao (1997), a design was proposed for clinical trials with binary outcomes to reestimate the required sample size during the trial without unblinding the treatment codes. Their simulation study showed that the method has a small increase in the effective type I error rate (around 0.01 at = 0.05) and maintains an adequate effective power. Theoretical results were later derived by Govindarajulu (2005), which claimed that the increase in the effective type I error rate was much larger (about 0.03 at = 0.05) and the effective power was much lower. With the corrections in the errata in the current issue, the results from the corrected formulas of Govindarajulu (2005) are similar to the simulation results in Shih and Zhao (1997), but the differences, as presented in Tables 1 and 2, cannot be solely explained by a margin due to simulation. The following two facts further explain the differences between the theoretical results and the simulation results. First, in the design of Shih and Zhao (1997) the re-reestimated sample size is a random variable with 1 n∗ (e.g., 1 = 4.1) as the upper bound and 2 n∗ (e.g., 2 = 0.6) as the lower bound (where n∗ is the initial estimated sample size), while Govindarajulu (2005) treats the re-reestimated sample size as a fixed number, i.e., 1 n∗ (if an increase in sample size is needed) or 2 n∗ (if a decrease in sample size is needed) when deriving the formulas for the effective type I error rate and power. This discrepancy has minimal impact on the inflation of type I error (see Table 1), but tends to exaggerate the inflation in power (see Table 2). Second, 1 and 2 are two factors whose impacts on the type I error and power are largely affected by n∗ which depends on the working assumption of the effect size (assumed to be non-zero to avoid the n∗ = ∞ case). That is why Shih and Zhao (1997) provided a range for the inflation in type I error in Table 1. But the impact of n∗ is totally ignored when the formulas were derived in Govindarajulu (2005). By using the corrected formulas in Govindarajulu (2005), we can show that the inflation in power (or type I error under the null) can be decomposited into two parts, i.e., the inflation due to a possible increase in sample size and the inflation due to a possible decrease in sample size. It is quite intuitive that an increase in sample size will increase the power, but an inflation in power can also be triggered by a decrease in sample size. When the observed treatment difference in the interim analysis is greater than what is assumed in the initial power calculation, a decrease in sample size is required in order to maintain the targeted power, which magnifies the observed optimistic treatment difference in the final analysis and consequently might inflate the power. In conclusion, the method by Shih and Zhao (1997) can be used to re-estimate the sample size without unblinding the treatment codes, and the small inflation in type I error can be eliminated by using a nominal type I error rate slightly smaller than the targeted significance level. The corrected formulas of Govindarajulu (2005) can be utilized to calculate the approximate effective type I error and power to facilitate the planning process, but as it tends to overestimate the power, a simulation study is indispensable in order to adequately power the study. Table 1 Inflation in type I error (at = 0.05) in Examples 3.1 and 3.2 of Govindarajulu (2005) Example
Shih and Zhao (1997)
Corrected Govindarajulu (2005)
3.1 3.2
0.0048--0.0142 0.0076--0.0124
0.0136 0.0118
∗
Corresponding author. E-mail addresses: [email protected] (X. Luo), [email protected] (P.-L. Zhao).
0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.02.016
676
X. Luo, P.-L. Zhao / Journal of Statistical Planning and Inference 139 (2009) 675 -- 676
Table 2 Power (at targeted power of 0.90) in Examples 5.1 and 5.2 of Govindarajulu (2005) Example
Shih and Zhao (1997)
Corrected Govindarajulu (2005)
5.1 5.2
0.9430 0.9400
1.00 0.9994
References Govindarajulu, Z., 2005. Robustness of sample size re-estimation with interim binary data for double-blind clinical trials. J. Statist. Plann. Inference 129, 125--144. Shih, W.J., Zhao, P.L., 1997. Design for sample size re-estimation with interim data for double-blind clinical trials with binary outcomes. Statist. Med. 16, 1913--1923.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / j s p i
Letter to the Editor
Comments on “Z. Govindarajulu (2005) robustness of sample size reestimation with interim binary data for double-blind clinical trials” Xiaohui Luoa , Peng-Liang Zhaob,∗ a b
Clinical Biostatistics, Merck Research Laboratories, Rahway, NJ 07065-0900, USA Biostatistics and Programming, sanofi aventis, Bridgewater, NJ 08807-0890, USA
In Shih and Zhao (1997), a design was proposed for clinical trials with binary outcomes to reestimate the required sample size during the trial without unblinding the treatment codes. Their simulation study showed that the method has a small increase in the effective type I error rate (around 0.01 at = 0.05) and maintains an adequate effective power. Theoretical results were later derived by Govindarajulu (2005), which claimed that the increase in the effective type I error rate was much larger (about 0.03 at = 0.05) and the effective power was much lower. With the corrections in the errata in the current issue, the results from the corrected formulas of Govindarajulu (2005) are similar to the simulation results in Shih and Zhao (1997), but the differences, as presented in Tables 1 and 2, cannot be solely explained by a margin due to simulation. The following two facts further explain the differences between the theoretical results and the simulation results. First, in the design of Shih and Zhao (1997) the re-reestimated sample size is a random variable with 1 n∗ (e.g., 1 = 4.1) as the upper bound and 2 n∗ (e.g., 2 = 0.6) as the lower bound (where n∗ is the initial estimated sample size), while Govindarajulu (2005) treats the re-reestimated sample size as a fixed number, i.e., 1 n∗ (if an increase in sample size is needed) or 2 n∗ (if a decrease in sample size is needed) when deriving the formulas for the effective type I error rate and power. This discrepancy has minimal impact on the inflation of type I error (see Table 1), but tends to exaggerate the inflation in power (see Table 2). Second, 1 and 2 are two factors whose impacts on the type I error and power are largely affected by n∗ which depends on the working assumption of the effect size (assumed to be non-zero to avoid the n∗ = ∞ case). That is why Shih and Zhao (1997) provided a range for the inflation in type I error in Table 1. But the impact of n∗ is totally ignored when the formulas were derived in Govindarajulu (2005). By using the corrected formulas in Govindarajulu (2005), we can show that the inflation in power (or type I error under the null) can be decomposited into two parts, i.e., the inflation due to a possible increase in sample size and the inflation due to a possible decrease in sample size. It is quite intuitive that an increase in sample size will increase the power, but an inflation in power can also be triggered by a decrease in sample size. When the observed treatment difference in the interim analysis is greater than what is assumed in the initial power calculation, a decrease in sample size is required in order to maintain the targeted power, which magnifies the observed optimistic treatment difference in the final analysis and consequently might inflate the power. In conclusion, the method by Shih and Zhao (1997) can be used to re-estimate the sample size without unblinding the treatment codes, and the small inflation in type I error can be eliminated by using a nominal type I error rate slightly smaller than the targeted significance level. The corrected formulas of Govindarajulu (2005) can be utilized to calculate the approximate effective type I error and power to facilitate the planning process, but as it tends to overestimate the power, a simulation study is indispensable in order to adequately power the study. Table 1 Inflation in type I error (at = 0.05) in Examples 3.1 and 3.2 of Govindarajulu (2005) Example
Shih and Zhao (1997)
Corrected Govindarajulu (2005)
3.1 3.2
0.0048--0.0142 0.0076--0.0124
0.0136 0.0118
∗
Corresponding author. E-mail addresses: [email protected] (X. Luo), [email protected] (P.-L. Zhao).
0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.02.016
676
X. Luo, P.-L. Zhao / Journal of Statistical Planning and Inference 139 (2009) 675 -- 676
Table 2 Power (at targeted power of 0.90) in Examples 5.1 and 5.2 of Govindarajulu (2005) Example
Shih and Zhao (1997)
Corrected Govindarajulu (2005)
5.1 5.2
0.9430 0.9400
1.00 0.9994
References Govindarajulu, Z., 2005. Robustness of sample size re-estimation with interim binary data for double-blind clinical trials. J. Statist. Plann. Inference 129, 125--144. Shih, W.J., Zhao, P.L., 1997. Design for sample size re-estimation with interim data for double-blind clinical trials with binary outcomes. Statist. Med. 16, 1913--1923.