The package `mtdesign`

provides implementations of both
Simon (1989) and Mander & Thompson (2010). Other implementations of
Simon’s methods are available - for example, the `ph2simon`

function in the `clinfun`

package (Seshan 2018), but these do
not provide easy access to non-optimal solutions in the way that
`mtdesign`

does. I am not aware of any other R-based
implementations of Mander & Thompson’s extension to Simon.

Once available on CRAN, you can install `mtdesign`

in the
usual way:

`install.packages("mtdesign")`

You can install the development version of `mtdesign`

from
GitHub with:

`devtools::install_github("openpharma/mtdesign")`

```
# By policy, on CRAN, use only two cores, no matter how many are available.
if (requireNamespace("parallel", quietly = TRUE)) {
<- parallel::detectCores()
maxCores <- ifelse(identical(Sys.getenv("NOT_CRAN"), "true"), maxCores, min(maxCores, 2))
maxCores else {
} <- 1
maxCores }
```

Suppose that treatments with a response rate of less than 5% are of no interest but those with a response rate of at least 25% are worthy of further development. A Simon’s 2-stage design to seek an efficacy signal with a significance level of 5% and a power of 80% is required.

```
library(mtdesign)
library(knitr)
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
<- obtainDesign(p0 = 0.05, p1 = 0.25, alpha = 0.05, beta = 0.2, mander = FALSE, parallel = FALSE)
simonDesign
%>%
simonDesign select(-Alpha, -Beta, -p0, -p1, -PETAlt, -AveSizeAlt) %>%
kable(digits = c(0, 0, 0, 0, 3, 3, 2, 1, NA))
```

nTotal | nStage1 | rTotal | rFutility | Type1 | Type2 | PETNull | AveSizeNull | Criterion |
---|---|---|---|---|---|---|---|---|

17 | 9 | 2 | 0 | 0.047 | 0.188 | 0.63 | 12.0 | optimal |

16 | 12 | 2 | 0 | 0.043 | 0.199 | 0.54 | 13.8 | minimax |

The table shows that the optimal design for these requirements is 0/9 2/17. The expected sample size is 12.0 and the probability of early termination is 63%. The significance level actually achieved is 4.7% and the power level achieved is 100% - 18.8% = 81.2%.

The power curves for both designs are easily plotted.

`powerPlot(simonDesign)`

Obtaining the equivalent Mander & Thompson designs requires only a small change to the calls.

```
<- obtainDesign(
manderDesign p0 = 0.05,
p1 = 0.25,
alpha = 0.05,
beta = 0.2,
cores = maxCores
)
%>%
manderDesign select(-Alpha, -Beta, -p0, -p1) %>%
kable(digits = c(0, 0, 0, 0, 3, 3, 2, 2, 2, 1, NA))
```

nTotal | nStage1 | rTotal | rFutility | rSuccess | Type1 | Type2 | PETNull | PETAlt | AveSizeNull | AveSizeAlt | Criterion |
---|---|---|---|---|---|---|---|---|---|---|---|

17 | 9 | 2 | 0 | 2 | 0.047 | 0.19 | 0.64 | 0.47 | 11.9 | NA | optimalNull |

16 | 12 | 2 | 0 | 2 | 0.043 | 0.20 | 0.56 | 0.64 | 13.8 | NA | minimaxNull |

17 | 9 | 2 | 0 | 2 | 0.047 | 0.19 | 0.64 | 0.47 | 11.9 | NA | optimalAlt |

16 | 12 | 2 | 0 | 2 | 0.043 | 0.20 | 0.56 | 0.64 | 13.8 | NA | minimaxAlt |

`powerPlot(manderDesign)`

Suppose a trial, for whatever reason, is restricted to using 8
participants in each stage. As shown above, the optimal Simon’s two
stage design is 0/9 2/17. That’s close to n_{1} = 8, n = 16. Is
there a (slightly) sub-optimal design that has n_{1} = 8, n =
16?

```
<- createGrid(p0 = 0.05, p1 = 0.25, alpha = 0.05, beta = 0.2, mander = FALSE)
x
<- x %>% filter(nStage1 == 8, nTotal == 16)
y <- y %>% obtainDesign(cores = maxCores)
z #> Warning: No acceptable designs were found.
if (nrow(z) == 0) {
print("No acceptable designs were found.")
else {
} select(-Alpha, -Beta, -p0, -p1, -PETAlt, -AveSizeAlt) %>%
z() %>%
select(-Alpha, -Beta, -p0, -p1, -PETAlt, -AveSizeAlt) %>%
kable(digits = c(0, 0, 0, 0, 3, 3, 2, 1, NA))
}#> [1] "No acceptable designs were found."
```

No, there isn’t. How close can we get?

```
<- y %>% augmentGrid()
z1
<- z1 %>%
bestSize filter(Type1 < Alpha) %>%
slice_min(Type2)
%>%
bestSize select(-Alpha, -Beta, -p0, -p1, -PETAlt, -AveSizeAlt) %>%
kable(
caption = "Best sub-optimal design with required significance level",
digits = c(0, 0, 0, 0, 3, 3, 2, 1, NA)
)
```

nTotal | nStage1 | rTotal | rFutility | Type1 | Type2 | PETNull | AveSizeNull |
---|---|---|---|---|---|---|---|

16 | 8 | 2 | 0 | 0.039 | 0.229 | 0.66 | 10.7 |

Best sub-optimal design with required significance level

```
<- z1 %>%
bestPower filter(Type2 < Beta) %>%
slice_min(Type1)
%>%
bestPower select(-Alpha, -Beta, -p0, -p1, -PETAlt, -AveSizeAlt) %>%
kable(
caption = "Best sub-optimal design with required power",
digits = c(0, 0, 0, 0, 3, 3, 2, 1, NA)
)
```

nTotal | nStage1 | rTotal | rFutility | Type1 | Type2 | PETNull | AveSizeNull |
---|---|---|---|---|---|---|---|

16 | 8 | 1 | 0 | 0.151 | 0.127 | 0.66 | 10.7 |

Best sub-optimal design with required power

So the choice lies between a design which achieves the required significance level but has a power of only 77.1% or one which has the required power but which has a significance level of 15.1%. Both designs accept the null hypothesis when no responders are seen in the first group of eight participants. They differ in the critical value at the end of stage 2: 1 to maintain the power, 2 to maintain the significance level.

The power curve for each of these designs can be compared with that for the globally optimal design.

```
<- simonDesign %>%
plotData1 filter(Criterion == "optimal") %>%
bind_rows(list(bestSize, bestPower))
powerPlot(plotData1)
```

The `mtdesign`

package consists of three main
functions:

`createGrid`

creates the grid (of nStage1, rFutility, nTotal and rTotal for Simon’s design or nStage1, rFutility, rSuccess, nTotal and rTotal for a Mander & Thompson design) over which the brute force search for the required design(s) is conducted`augmentGrid`

takes a grid created by`createGrid`

and adds columns for probability of early termination, Type 1 error, Type 2 error and expected sample size to it.`obtainDesign`

takes an augmented grid and identifies the optimal and minimax designs

The `mtdesign`

package supports logging via the
`logger`

package (Daróczi 2021). Most functions simply report
Entry and Exit at the `DEBUG`

level.

The `augmentGrid`

function reports steps of the
parallelisation process at the `TRACE`

level.

There is no known closed form solution to obtaining solutions to
either Simon’s original equations nor Mander & Thompson’s
extensions. The `mtdesign`

package uses a brute force
approach to evaluate the operating characteristics of all reasonable
potential designs. The grids can be quickly become large, particularly
for Mander & Thompson designs. For example,
`createGrid(0.2, 0.4, alpha=0.1, beta=0.1)`

creates a grid of
almost 11 million candidate designs. `mtdesign`

uses
paralellisation to attempt to speed up the evaluation of candidate
designs.

The `augmentGrid`

function allows users some control over
the parallelisation process:

- The
`parallel`

parameter defaults to`TRUE`

and defines whether or not paralellisation is to be used. - The
`cores`

parameter specifies how many cores are to be used. The default value,`NA`

tells`mtdesign`

to use all available (as defined by`parallel::detectCores()`

), cores. - The
`minChunkSize`

determines the smallest grid of candidate designs that will trigger paralellisation. The default value is`100000`

.

The `parallel`

package is required for parallelisation. If
parallelisation is both needed (ie the grid size exceeds
`minChunkSize`

) and requested but the `parallel`

package has not been installed, an error message is thrown and
augmentation of the grid stops. If paralellisation is not requested and
the grid contains one million or more rows, a warning is produced.

If, when installing or using the `mtdesign`

package, you
get an error regarding a syntax error in an`.hpp`

file,
similar to the following

`/BH/include/boost/math/tools/fraction.hpp:84:48: error: ‘long double’ is not a class, struct, or union type using value_type = typename T::value_type; ...`

the issue is most likely a mismatch between the g++ compiler being
used and the headers supplied by the `BH`

package. There are
only two solutions that I know of:

- Upgrade g++
- Downgrade the version of the
`BH`

package you are using. The appropriate package version depends on the version of the g++ compiler you are using.

Daróczi, Gergely. 2021. *Logger: A Lightweight, Modern and Flexible
Logging Utility*. https://daroczig.github.io/logger/.

Mander, AP, and Thompson, SG. 2010. “Two-Stage Designs Optimal Under the
Alternative Hypothesis for Phase II Cancer Clinical Trials.”
*Contemporary Clinical Trials* 31 (6): 572–78. https://doi.org/https://www.doi.org/10.1016/j.cct.2010.07.008.

Seshan, VE. 2018. “Clinfun: Clinical Trial Design and Data Analysis
Functions.” Software. https://CRAN.R-project.org/package=clinfun.

Simon, R. 1989. “Optimal Two-Stage Designs for Phase II Clinical
Trials.” *Controlled Clinical Trials* 10 (1): 1–10.
https://doi.org/https://www.doi.org/10.1016/0197-2456(89)90015-9.