This R Markdown document provides a brief introduction to group sequential designs with delayed responses as proposed by Hampson and Jennison (2013). It is shown how this is implemented in rpact. Examples for designing trials with delayed responses using the software are provided. We also describe an alternative approach that directly uses the \(\alpha\)-spending approach to derive the decision boundaries.

In rpact version 3.3, the
group sequential methodology from Hampson and Jennison
(2013) is implemented. As traditional group sequential designs are characterized
specifically by the underlying boundary sets, one main task was to write a
function returning the decision critical values according to the calculation
rules in Hampson and Jennison (2013). The function returning the respective
critical values has been validated, particularly via simulation towards Type I
error rate control in various settings (for an example see below). Subsequently,
functions characterizing a delayed response group sequential test in terms of
power, maximum sample size and expected sample size have been written.
These functions were integrated in the
rpact functions
`getDesignGroupSequential()`

, `getDesignCharacteristics()`

, and the
corresponding `getSampleSize...()`

and `getPower...()`

functions.

The classical group sequential methodology works on the assumptions of having no treatment response delay, i.e., it is assumed that enrolled subjects are observed upon recruitment or at least shortly after. In many practical situation, this assumption does not hold true. Instead, it might be that there is a latency between the timing of recruitment and the actual measurement of a primary endpoint. That is, at interim \(k\) there is some information \(I_{\Delta_t, k} > 0\) in pipeline.

One method to handle this pipeline information was proposed by Hampson &
Jennison (2013) and is called *delayed response group sequential design*. Assume
that, in a \(K\)-stage trial, given we will proceed to the trial end, we will
observe an information sequence \((I_1, \dots, I_K)\), and the corresponding
\(Z\)-statistics \((Z_1, \dots, Z_K)\). As we now have information in pipeline,
define \(\tilde{I}_k = I_k + I_{\Delta_t, k}\) as the information available after
awaiting the delay after having observed \(I_k\). Let
\((\tilde{Z}_1,\dots,\tilde{Z}_{K-1})\) be the vector of \(Z\)-statistics that are
calculated based upon the information levels
\((\tilde{I}_1,\dots,\tilde{I}_{K-1})\). Given boundary sets
\(\{u^0_1,\dots,u^0_{K-1}\}\), \(\{u_1,\dots,u_K\}\) and \(\{c_1,\dots,c_K\}\), a
\(K\)-stage delayed response group sequential design has the following structure: