# Computes efficacy and futility boundaries

### Description

This computes efficacy and futility boundaries for interim analysis
and sequential designs. Two sided symmetric efficacy boundaries can
be computed by specifying half of the intended total type I error
probability in the argument, `Alpha.Efficacy`

. Otherwise,
especially in the case of efficacy and futility bounds only one sided
boundaries are currently computed. The computation allows for two
different time scales–one must be the variance ratio, and the second
can be a user chosen increasing scale beginning with 0 that takes the
value 1 at the conclusion of the trial.

### Usage

1 2 3 | ```
GrpSeqBnds(EfficacyBoundary = LanDemets(alpha = 0.05, spending = ObrienFleming),
FutilityBoundary = LanDemets(alpha = 0.1, spending = ObrienFleming),
NonBindingFutility = TRUE, frac, frac.ii = NULL, drift = NULL)
``` |

### Arguments

`EfficacyBoundary` |
This specifies the method used to construct the efficacy boundary. The available choices are: (i) (ii) (iii) (iv) User supplied boundary points in the form |

`FutilityBoundary` |
This specifies the method used to construct the futility boundary. The available choices are: (i) (ii) (iii) (iv) User supplied boundary points in the form |

`NonBindingFutility` |
When using a futility boundary and this is set to 'TRUE', the efficacy boundary will be constructed in the absence of the futility boundary, and then the futility boundary will be constructed given the resulting efficacy boundary. This results in a more conservative efficacy boundary with true type I error less than the nominal level. This is recommended due to the fact that futility crossings are viewed by DSMB's with much less gravity than an efficacy crossing and as such, the consensus is that efficacy bounds should not be discounted towards the null hypothesis because of paths which cross a futility boundary. Default value is 'TRUE'. |

`frac` |
The variance ratio. If the end of trial variance is unknown then normalize all previous variances by the current variance. In this case you must specify a second scale that is monotone increasing from 0 to 1 at the end of the trial. Required. |

`frac.ii` |
The second information scale that is used for type I and type II error probability spending. Optional (see above) |

`drift` |
The drift function of the underlying brownian motion, which is the expected value under the design alternative of the un-normalized weighted log-rank statistic, then normalized to have variance one when the variance ratio equals 1. See the examples below. |

### Value

An object of class `boundaries`

with components:
"table" "frac" "frac.ii" "drift" "call"

`call` |
The call that produced the returned results. |

`frac` |
The vector of variance ratios. |

`frac.ii` |
The vector of information ratios for type I and type II error probability spending, which differs from the
above if the user sets the argument |

`drift` |
The drift vector that is required as an argument when futility boundaries are calculated. |

`table` |
A matrix with components frac The information ratio for type I and type II error probability spending. b.f The calculated futility boundary (if requested). alpha.f The type II error probability spent at that analysis (if doing futility bounds). cum-alpha.f Cumulative sum of b.e The calculated efficacy boundary. alpha.e The type I error probability spent at that analysis. cum-alpha.e Cumulative sum of |

### Author(s)

Grant Izmirlian <izmirlian@nih.gov>

### References

Gu, M.-G. and Lai, T.-L. “Determination of power and sample size in the design of clinical trials with failure-time endpoints and interim analyses.” Controlled Clinical Trials 20 (5): 423-438. 1999

Izmirlian, G. “The PwrGSD package.” NCI Div. of Cancer Prevention Technical Report. 2004

Jennison, C. and Turnbull, B.W. (1999) Group Sequential Methods: Applications to Clinical Trials Chapman & Hall/Crc, Boca Raton FL

Proschan, M.A., Lan, K.K.G., Wittes, J.T. (2006), corr 2nd printing (2008) Statistical Monitoring of Clinical Trials A Unified Approach Springer Verlag, New York

### See Also

`PwrGSD`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | ```
## NOTE: In an unweighted analysis, the variance ratios and event ratios
## are the same, whereas in a weighted analysis, they are quite different.
##
## For example, in a trial with 7 or so years of accrual and maximum follow-up of 20 years
## using the stopped Fleming-Harrington weights, `WtFun' = "SFH", with paramaters
## `ppar' = c(0, 1, 10) we might get the following vector of variance ratios:
frac <- c(0.006995655, 0.01444565, 0.02682463, 0.04641363, 0.0585665,
0.07614902, 0.1135391, 0.168252, 0.2336901, 0.3186155, 0.4164776,
0.5352199, 0.670739, 0.8246061, 1)
## and the following vector of event ratios:
frac.ii <- c(0.1494354, 0.1972965, 0.2625075, 0.3274323, 0.3519184, 0.40231,
0.4673037, 0.5579035, 0.6080742, 0.6982293, 0.7671917, 0.8195019,
0.9045182, 0.9515884, 1)
## and the following drift under a given alternative hypothesis
drift <- c(0.06214444, 0.1061856, 0.1731267, 0.2641265, 0.3105231, 0.3836636,
0.5117394, 0.6918584, 0.8657705, 1.091984, 1.311094, 1.538582,
1.818346, 2.081775, 2.345386)
## JUST ONE SIDED EFFICACY BOUNDARY
## In this call, we calculate a one sided efficacy boundary at each of 15 analyses
## which will occur at the given (known) variance ratios, and we use the variance
## ratio for type I error probability spending, with a total type I error probabilty
## of 0.05, using the Lan-Demets method with Obrien-Fleming spending (the default).
gsb.all.just.eff <- GrpSeqBnds(frac=frac,
EfficacyBoundary=LanDemets(alpha=0.05, spending=ObrienFleming))
## ONE SIDED EFFICACY AND FUTILTY BOUNDARIES
## In this call, we calculate a one sided efficacy boundary at each of 15 analyses
## which will occur at the given (known) variance ratios, and we use the variance
## ratio for type I and type II error probability spending, with a total type I error
## probabilty of 0.05 and a total type II error probability of 0.10, using the Lan-Demets
## method with Obrien-Fleming spending (the default) for both efficacy and futilty.
gsb.all.eff.fut <- GrpSeqBnds(frac=frac,
EfficacyBoundary=LanDemets(alpha=0.05, spending=ObrienFleming),
FutilityBoundary=LanDemets(alpha=0.10, spending=ObrienFleming),
drift=drift)
## Now suppose that we are performing the 7th interim analysis. We don't know what the variance
## will be at the end of the trial, so we normalize variances of the current and previous
## statistics by the variance of the current statistic. This is equivalent to the following
## length 7 vector of variance ratios:
frac7 <- frac[1:7]/frac[7]
## To proceed under the "unknown variance at end of trial" case, we must use a second
## scale for spending type I and II error probabilty. Unlike the above scale
## which is renormalized at each analysis to have value 1 at the current analysis, the
## alpha spending scale must be monotone increasing and attain the value 1 only at the
## end of the trial. A natural choice is the event ratio, which is known in advance if
## the trial is run until a required number of events is obtained, a so called
## maximum information trial:
frac7.ii <- frac.ii[1:7]
## the first seven values of the drift function
drift7 <- drift[1:7]/frac[7]^0.5
gsb.1st7.eff.fut <- GrpSeqBnds(frac=frac7, frac.ii=frac7.ii,
EfficacyBoundary=LanDemets(alpha=0.05, spending=ObrienFleming),
FutilityBoundary=LanDemets(alpha=0.10, spending=ObrienFleming),
drift=drift7)
## Of course there are other options not covered in these examples but this should get you
## started
``` |

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