Maximum inflation of the type 1 error rate when sample size and allocation rate are adapted in a pre-planned interim look

Abstract

We calculate the maximum type 1 error rate of the pre-planned conventional fixed sample size test for comparing the means of independent normal distributions (with common known variance) which can be yielded when sample size and allocation rate to the treatment arms can be modified in an interim analysis. Thereby it is assumed that the experimenter fully exploits knowledge of the unblinded interim estimates of the treatment effects in order to maximize the conditional type 1 error rate. The 'worst-case' strategies require knowledge of the unknown common treatment effect under the null hypothesis. Although this is a rather hypothetical scenario it may be approached in practice when using a standard control treatment for which precise estimates are available from historical data. The maximum inflation of the type 1 error rate is substantially larger than derived by Proschan and Hunsberger (Biometrics 1995; 51:1315-1324) for design modifications applying balanced samples before and after the interim analysis. Corresponding upper limits for the maximum type 1 error rate are calculated for a number of situations arising from practical considerations (e.g. restricting the maximum sample size, not allowing sample size to decrease, allowing only increase in the sample size in the experimental treatment). The application is discussed for a motivating example.

Figure 1

Worst case scenarios in partitions of the ($Z_0^{\left(1\right)}$, $Z_1^{\left(1\right)}$)-plane.

Figure 2

$E_{\alpha }^{\ast }(r^{(1)})∕E_{\alpha }^{\ast }(1)$ for different values of r⁽¹⁾. α was set to 0.01 (solid line), 0.025 (dashed line) and 0.05 (dotted line).

Figure 3

Maximum overall type 1 error rate ($E_{\alpha }^{\ast }$) for varying upper boundary $r_{1,\text{up}}^{\left(2\right)}$ for the sample size inflation factor $r_1^{\left(2\right)}$ in the treatment group. Solid lines: $r_{0,\mathit{lo}}^{\left(2\right)}=r_{1,\mathit{lo}}^{\left(2\right)}=0$ and $r_{0,\text{up}}^{\left(2\right)}=r_{1,\text{up}}^{\left(2\right)}$ (sample size decrease and early rejection possible); dashed lines: $r_{0,\mathit{lo}}^{\left(2\right)}=r_{1,\mathit{lo}}^{\left(2\right)}=1$ and $r_{0,\text{up}}^{\left(2\right)}=r_{1,\text{up}}^{\left(2\right)}$ (no sample size reduction in the second stage); dotted lines: $r_{0,\mathit{lo}}^{\left(2\right)}=r_{1,\mathit{lo}}^{\left(2\right)}=1$ and $r_{0,\text{up}}^{\left(2\right)}=1$ (sample size increase only in treatment group). α = 0.01 (black lines with squares), 0.025 (dark grey lines with triangles) and 0.05 (light grey lines with dots).