The SEQDESIGN Procedure

Type I and Type II Errors

Subsections:

The Type I error is the error of rejecting the null hypothesis when the null hypothesis is correct, and the Type II error is the error of not rejecting the null hypothesis when the null hypothesis is incorrect. The level of significance $\alpha $ is the probability of making a Type I error. The Type II error depends on the hypothetical reference of the alternative hypothesis, and the Type II error probability $\beta $ is defined as the probability of not rejecting the null hypothesis when a specific alternative reference is true. The power $1-\beta $ is then defined as the probability of rejecting the null hypothesis at the alternative reference.

In a sequential design, if the maximum information and alternative reference are not both specified, the critical values are created such that both the specified Type I and the specified Type II error probability levels are maintained in the design. Otherwise, the critical values are created such that either the specified Type I error probability or the specified Type II error probability is maintained.

One-Sided Tests

For a K-stage group sequential design with an upper alternative hypothesis $H_{1}: \theta =\theta _1$ and early stopping to reject or accept the null hypothesis $H_{0}: \theta =0$, the boundaries contain the upper $\alpha $ critical values $a_{k}$ and upper $\beta $ critical values $b_{k}$, $k=1, 2, \ldots , K$. At each interim stage, $b_{k} < a_{k}$, the null hypothesis $H_{0}$ is rejected if the observed statistic $z_{k} \geq a_{k}$, $H_{0}$ is accepted if $z_{k} < b_{k}$, or the process is continued to the next stage if $b_{k} \leq z_{k} < a_{k}$. At the final stage $b_{K} = a_{K}$, the hypothesis is either rejected or accepted.

The overall Type I error probability $\alpha $ is given by

\[ \alpha = \sum _{k=1}^{K} {\alpha _{k}} \]

where $\alpha _{k}$ is the $\alpha $ spending at stage k. That is, at stage 1,

\[ {\alpha _{1}} = P_{\theta =0} ( \, z_{1} \geq a_{1} \, ) \]

At a subsequent stage k,

\[ {\alpha _{k}} = P_{\theta =0} ( \, b_{j} \leq z_{j} < a_{j}, j=1, 2, \ldots , k-1, \, z_{k} \geq a_{k}) \]

Similarly, the Type II error probability

\[ \beta = \sum _{k=1}^{K} {\beta _{k}} \]

where $\beta _{k}$ is the $\beta $ spending at stage k. That is, at stage 1,

\[ {\beta _{1}} = P_{\theta =\theta _1} ( \, z_{1} < b_{1} \, ) \]

At a subsequent stage k,

\[ {\beta _{k}} = P_{\theta =\theta _1} ( \, b_{j} \leq z_{j} < a_{j}, j=1, 2, \ldots , k-1, \, z_{k} < b_{k} \, ) \]

With an upper alternative hypothesis $H_{1}: \theta =\theta _1 > 0$, the power $1-\beta $ is the probability of rejecting the null hypothesis for the upper alternative.

\[ 1-\beta = 1 - \sum _{k=1}^{K} {\beta _{k}} = \sum _{k=1}^{K} P_{\theta =\theta _1} ( \, \, b_{j} \leq z_{j} < a_{j}, j=1, 2, \ldots , k-1, \, z_{k} \geq a_{k}) \]

For a design with early stopping to reject $H_{0}$ only, the interim upper $\beta $ critical values are set to $-\infty $, $b_{k} = -\infty , k=1, 2, \ldots , K-1$, and $\beta = \beta _{K}$. For a design with early stopping to accept $H_{0}$ only, the interim upper $\alpha $ critical values are set to $\infty $, $a_{k} = \infty , k=1, 2, \ldots , K-1$, and $\alpha = \alpha _{K}$.

Similarly, the Type I and Type II error probabilities for a K-stage design with a lower alternative hypothesis $H_{0}: \theta =-\theta _1$ can also be derived.

Two-Sided Tests

For a K-stage group sequential design with two-sided alternative hypotheses $H_{1u}: \theta =\theta _{1u}$ and $H_{1l}: \theta =\theta _{1l}$, and early stopping to reject or accept the null hypothesis $H_{0}: \theta =0$, the boundaries contain the upper $\alpha $ critical values $a_{k}$, upper $\beta $ critical values $b_{k}$, lower $\beta $ critical values $\_ b_{k}$, and lower $\alpha $ critical values $\_ a_{k}$, $k=1, 2, \ldots , K$. At each interim stage, $\_ a_{k} < \_ b_{k} \leq b_{k} < a_{k}$, the null hypothesis $H_{0}$ is rejected if the observed statistic $z_{k} \geq a_{k}$ or $z_{k} \leq \_ a_{k}$, $H_{0}$ is accepted if $\_ b_{k} < z_{k} < b_{k}$, or the process is continued to the next stage if $b_{k} \leq z_{k} < a_{k}$ or $\_ a_{k} < z_{k} \leq \_ b_{k}$. At the final stage $b_{K} = a_{K}$ and $\_ b_{K} = \_ a_{K}$, the hypothesis is either rejected or accepted.

The overall upper Type I error probability $\alpha _{u}$ is given by

\[ \alpha _{u} = \sum _{k=1}^{K} {\alpha _{uk}} \]

where $\alpha _{uk}$ is the $\alpha $ spending at stage k for the upper alternative. That is, at stage 1,

\[ {\alpha _{u1}} = P_{\theta =0} ( \, z_{1} \geq a_{1} \, ) \]

At a subsequent stage k,

\[ {\alpha _{uk}} = P_{\theta =0} ( \, \_ a_{j} < z_{j} \leq \_ b_{j} \, \, \mr{or} \, \, b_{j} \leq z_{j} < a_{j}, j=1, 2, \ldots , k-1, \, z_{k} \geq a_{k} \, ) \]

Similarly, the overall lower Type I error probability $\alpha _{l}$ can also be derived, and the overall Type I error probability $\alpha = \alpha _{l} + \alpha _{u}$.

The overall upper Type II error probability $\beta _ u$ is given by

\[ \beta _ u = \sum _{k=1}^{K} {\beta _{uk}} \]

where $\beta _{uk}$ is the upper $\beta $ spending at stage k. That is, at stage 1,

\[ {\beta _{u1}} = P_{\theta =\theta _{1u}} ( \, z_{1} < \_ a_{1} \, \, \mr{or} \, \, \_ b_{1} < z_{1} < b_{1} \, ) \]

At a subsequent stage k,

\begin{eqnarray*} \beta _{uk} = P_{\theta =\theta _{1u}} ( \, \_ a_{j} < z_{j} \leq \_ b_{j} \, \, \textrm{or} \, \, b_{j} \leq z_{j} < a_{j}, \, j=1, 2, \ldots , k-1, \, z_{k} < \_ a_{k} \, \, \textrm{or} \, \, \_ b_{k} < z_{k} < b_{k} \, ) \end{eqnarray*}

With an upper alternative hypothesis $H_{1}: \theta =\theta _{1u} > 0$, the power $1-\beta _ u$ is the probability of rejecting the null hypothesis for the upper alternative:

\[ 1-\beta _ u = 1 - \sum _{k=1}^{K} {\beta _{uk}} \]

which is

\[ P_{\theta =\theta _{1u}} ( \, \_ a_{j} < z_{j} \leq \_ b_{j} \, \, \mr{or} \, \, b_{j} \leq z_{j} < a_{j}, \, j=1, 2, \ldots , k-1, \, z_{k} \geq a_{k} \, ) \]

The overall lower Type II error probability $\beta _ l$ and power $1-\beta _ l$ can be similarly derived.

For a design with early stopping only to reject $H_{0}$, both the interim lower and upper $\beta $ critical values are set to missing, $k=1, 2, \ldots , K-1$, and $\beta _{lK} = \beta _{l}$, $\beta _{uK} = \beta _{u}$. For a design with early stopping only to accept $H_{0}$, the interim upper $\alpha $ critical values are set to $\infty $, $a_{uk} = \infty $, and the interim lower $\alpha $ critical values are set to $-\infty $, $a_{lk} = -\infty $, $k=1, 2, \ldots , K-1$, and $\alpha _{uK} = \alpha _{u}$, $\alpha _{lK} = \alpha _{l}$.