This R Markdown document provides example code for the the definition of the most commonly used group sequential boundaries in rpact.

In rpact,
**sample size calculation for a group sequential trial proceeds by
following the same two steps regardless of whether the endpoint is a
continuous, binary, or a time-to-event endpoint**:

**Define the (abstract) group sequential boundaries**of the design using the function`getDesignGroupSequential()`

.**Calculate sample size for the endpoint of interest**by feeding the abstract boundaries from step 1. into specific functions for the endpoint of interest. This step uses functions such as`getSampleSizeMeans()`

(for continuous endpoints),`getSampleSizeRates()`

(for binary endpoints), and`getSampleSizeSurvival()`

(for survival endpoints).

The mathematical rationale for this two-step approach is that all group sequential trials, regardless of the chosen endpoint type, rely on the fact that the \(z\)-scores at different interim stages follow the same “canonical joint multivariate distribution” (at least asymptotically).

This document covers the more abstract first step, **Step 2 is
not covered in this document but it is covered in the separate
endpoint-specific R Markdown files for continuous, binary, and time to
event endpoints.** Of note, step 1 can be omitted for trials
without interim analyses.

These examples are not intended to replace the official rpact documentation and help pages but rather to supplement them.

In general, rpact supports both
one-sided and two-sided group sequential designs. If futility boundaries
are specified, however, only one-sided tests are permitted. **For
simplicity, it is often preferred to use one-sided tests for group
sequential designs** (typically, with \(\alpha = 0.025\)).

**First, load the rpact package**

```
library(rpact)
packageVersion("rpact") # version should be version 2.0.5 or later
```

`## [1] '3.3.2'`

**Example:**

- Interim analyses at information fractions 33%, 67%, and 100% (
`informationRates = c(0.33, 0.67, 1)`

). [Note: For equally spaced interim analyses, one can also specify the maximum number of stages (`kMax`

, including the final analysis) instead of the`informationRates`

.] - Lan & DeMets \(\alpha\)-spending approximation to the
O’Brien & Fleming boundaries
(
`typeOfDesign = "asOF"`

) - \(\alpha\)-spending approaches allow for flexible timing of interim analyses and corresponding adjustment of boundaries.

```
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1), typeOfDesign = "asOF"
)
```

The originally published O’Brien & Fleming boundaries are
obtained via `typeOfDesign = "OF"`

which is also the default
(therefore, if you do not specify `typeOfDesign`

, this type
is selected). Note that strict Type I error control is only guaranteed
for standard boundaries without \(\alpha\)-spending if the pre-defined
interim schedule (i.e., the information fractions at which interim
analyses are conducted) is exactly adhered to.

```
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1), typeOfDesign = "OF"
)
```

Pocock (`typeOfDesign = "P"`

for constant boundaries over
the stages, `typeOfDesign = "asP"`

for corresponding \(\alpha\)-spending version) or Haybittle
& Peto (`typeOfDesign = "HP"`

) boundaries (reject at
interim if \(z\)-value exceeds 3) is
obtained with, for example,

```
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1), typeOfDesign = "P"
)
```

- Kim & DeMets \(\alpha\)-spending
(
`typeOfDesign = "asKD`

) with parameter`gammaA`

(power function:`gammaA = 1`

is linear spending,`gammaA = 2`

quadratic) - Hwang, Shi & DeCani \(\alpha\)-spending
(
`typeOfDesign = "asHSD"`

) with parameter`gammaA`

(for details, see Wassmer & Brannath 2016, p. 76) - Standard Wang & Tsiatis Delta classes
(
`typeOfDesign = "WT"`

) and (`typeOfDesign = "WToptimum"`

)

```
# Quadratic Kim & DeMets alpha-spending
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1), typeOfDesign = "asKD", gammaA = 2
)
```

User-defined \(\alpha\)-spending
functions (`typeOfDesign = "asUser"`

) can be obtained via the
argument `userAlphaSpending`

which must contain a numeric
vector with elements \(0< \alpha_1 <
\ldots < \alpha_{kMax} = \alpha\) that define the values of
the cumulative alpha-spending function at each interim analysis.

```
# Example: User-defined alpha-spending function which is very conservative at
# first interim (spend alpha = 0.001), conservative at second (spend an additional
# alpha = 0.01, i.e., total cumulative alpha spent is 0.011 up to second interim),
# and spends the remaining alpha at the final analysis (i.e., cumulative
# alpha = 0.025)
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1),
typeOfDesign = "asUser",
userAlphaSpending = c(0.001, 0.01 + 0.001, 0.025)
)# $stageLevels below extract local significance levels across interim analyses.
# Note that the local significance level is exactly 0.001 at the first
# interim, but slightly >0.01 at the second interim because the design
# exploits correlations between interim analyses.
$stageLevels design
```

`## [1] 0.00100000 0.01052883 0.02004781`

- The argument
`futilityBounds`

contains a vector of futility bounds (on the \(z\)-value scale) for each interim (but not the final analysis). - A futility bound of \(z = 0\) corresponds to an estimated treatment effect of zero or “null”, i.e., in this case futility stopping is recommended if the treatment effect estimate at the interim analysis is zero or “goes in the wrong direction”. Futility bounds of \(z = -\infty\) (which are numerically equivalent to \(z = -6\)) correspond to no futility stopping at an interim.
- Due to the design of rpact, it is not
possible to directly define futility boundaries on the treatment effect
scale. If this is desired, one would need to manually convert the
treatment effect scale to the \(z\)-scale or, alternatively, experiment by
varying the boundaries on the \(z\)-scale until this implies the targeted
critical values on the treatment effect scale. (Critical values on
treatment effect scales are routinely provided by sample size functions
for different endpoint types such as
`getSampleSizeMeans()`

(for continuous endpoints),`getSampleSizeRates()`

(for binary endpoints), and`getSampleSizeSurvival()`

(for survival endpoints). Please see the R Markdown files for these endpoint types for further details.) - By default, all futility boundaries are non-binding
(
`bindingFutility = FALSE`

). Binding futility boundaries (`bindingFutility = TRUE`

) are not recommended although they are provided for the sake of completeness.

```
# Example: non-binding futility boundary at each interim in case
# estimated treatment effect is null or goes in "the wrong direction"
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025,
informationRates = c(0.33, 0.67, 1), typeOfDesign = "asOF",
futilityBounds = c(0, 0), bindingFutility = FALSE
)
```

Formal \(\beta\)-spending functions
are defined in the same way as for \(\alpha\)-spending functions, e.g., a Pocock
type \(\beta\)-spending can be
specified as `typeBetaSpending = "bsP"`

and `beta`

needs to be specified, the default is `beta = 0.20`

.

```
# Example: beta-spending function approach with O'Brien & Fleming alpha-spending
# function and Pocock beta-spending function
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025, beta = 0.1,
typeOfDesign = "asOF",
typeBetaSpending = "bsP"
)
```

Another way to formally derive futility bounds is through the
Pampallona and Tsiatis approach. This is through defining
`typeBetaSpending = "PT"`

, and the specification of two
parameters, `deltaPT1`

(shape of decision regions for
rejecting the null) and `deltaPT0`

(shape of shifted decision
regions for rejecting the alternative), for example

```
# Example: beta-spending function approach with O'Brien & Fleming boundaries for
# rejecting the null and Pocock boundaries for rejecting H1
<- getDesignGroupSequential(
design sided = 1, alpha = 0.025, beta = 0.1,
typeOfDesign = "PT",
deltaPT1 = 0, deltaPT0 = 0.5
)
```

Note that both the \(\beta\)-spending as well as the Pampallona
& Tsiatis approach can be selected to be one-sided or two-sided, the
bounds for rejecting the alternative to be binding
(`bindingFutility = TRUE`

) or non-binding
(`bindingFutility = FALSE`

).

Such designs can be implemented by using a user-defined \(\alpha\)-spending function which spends all of the Type I error at the final analysis. Note that such designs do not allow stopping for efficacy regardless how persuasive the effect is.

```
# Example: non-binding futility boundary using an O'Brien & Fleming type
# beta spending function. No early stopping for efficacy (i.e., all alpha</
```