The SEQTEST Procedure

Analysis after a Sequential Test

Subsections:

At the end of a trial, the hypothesis is either rejected or accepted. But the p-value, median, and confidence limits depend on the ordering the sample space $(k, z)$, where k is the stage number and z is the standardized Z statistic.

Following the notations used in Jennison and Turnbull (2000, pp. 179–180), $(k’, z’) \succ (k, z)$ if $(k’, z’)$ has a higher order or more extreme than $(k, z)$. Then for a given ordering, the p-value, median, and confidence limits associated with the observed statistics $(k, z)$ can be derived.

p-value

With the observed pair of statistics $(k_{0}, z_{0})$ when the trial is stopped, a one-sided upper p-value is computed as

\[ \mr{Prob} \{ \, (k, z) \succeq (k_{0}, z_{0}) \, \} \]

A one-sided lower p-value is computed as

\[ \mr{Prob} \{ \, (k, z) \preceq (k_{0}, z_{0}) \, \} \]

A two-sided p-value is twice the smaller of the lower and upper p-values.

Median Unbiased Estimate

With the observed pair $(k_{0}, z_{0})$, a median unbiased estimate $\theta _{m}$ is computed from

\[ \mr{Prob} \{ \, (k, z) \succeq (k_{0}, z_{0}) \, |\, \theta _{m} \, \} = 0.50 \]

Confidence Limits

With the observed pair $(k_{0}, z_{0})$, a lower $(1-\alpha _{l})$ confidence limit for $\theta $,   $\theta _{l}$, is computed from

\[ \mr{Prob} \{ \, (k, z) \succeq (k_{0}, z_{0}) \, |\, \theta _{l} \, \} = \alpha _{l} \]

Similarly, an upper $(1-\alpha _{u})$ confidence limit for $\theta $,   $\theta _{u}$, is computed from

\[ \mr{Prob} \{ \, (k, z) \preceq (k_{0}, z_{0}) \, |\, \theta _{u} \, \} = \alpha _{u} \]