# 2.8: Bound Summary and Z-transformations

### Description

A tabular summary of a group sequential design's bounds and their properties are often useful.
The 'vintage' `print.gsDesign()`

function provides a complete but minimally formatted summary of a group sequential design derived by `gsDesign()`

.
A brief description of the overall design can also be useful (`summary.gsDesign()`

.
A tabular summary of boundary characteristics oriented only towards LaTeX output is produced by `xtable.gsSurv`

.
More flexibility is provided by `gsBoundSummary()`

which produces a tabular summary of a user-specifiable set of package-provided boundary properties in a data frame.
This can also be used to along with functions such as `print.data.frame()`

, `write.table()`

, `write.csv()`

, `write.csv2()`

or, from the RTF package, `addTable.RTF()`

(from the rtf package) to produce console or R Markdown output or output to a variety of file types.
`xprint()`

is provided for LaTeX output by setting default options for `print.xtable()`

when producing tables summarizing design bounds.

Individual transformation of z-value test statistics for interim and final analyses are obtained from
`gsBValue()`

, `gsDelta()`

, `gsHR()`

and `gsCPz()`

for B-values, approximate treatment effect (see details), approximate hazard ratio and conditional power, respectively.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
## S3 method for class 'gsDesign'
print(x,...)
## S3 method for class 'gsDesign'
summary(object, information=FALSE, timeunit="months",...)
gsBoundSummary(x, deltaname=NULL, logdelta=FALSE,
Nname=NULL, digits=4, ddigits=2, tdigits=0, timename="Month",
prior=normalGrid(mu=x$delta/2, sigma=10/sqrt(x$n.fix)), POS=FALSE, ratio=NULL,
exclude=c("B-value","Spending","CP","CP H1","PP"), r=18, ...)
## S3 method for class 'gsBoundSummary'
print(x,row.names=FALSE,digits=4,...)
xprint(x,include.rownames=FALSE,
hline.after=c(-1,which(x$Value==x[1,]$Value)-1,nrow(x)), ...)
gsBValue(z, i, x, ylab="B-value", ...)
gsDelta(z, i, x, ylab=NULL, ...)
gsHR(z, i, x, ratio=1, ylab="Estimated hazard ratio", ...)
gsRR(z, i, x, ratio=1, ylab="Estimated risk ratio",...)
gsCPz(z, i, x, theta=NULL, ylab=NULL, ...)
``` |

### Arguments

`x` |
An item of class |

`object` |
An item of class |

`information` |
indicator of whether |

`timeunit` |
Text string with time units used for time-to-event designs created with |

`deltaname` |
Natural parameter name. If default |

`logdelta` |
Indicates whether natural parameter is the natural logarithm of the actual parameter. For example, the relative risk or odds-ratio would be put on the logarithmic scale since the asymptotic behavior is 'more normal' than a non-transformed value. As with |

`Nname` |
This will normally be changed to |

`digits` |
Number of digits past the decimal to be printed in the body of the table. |

`ddigits` |
Number of digits past the decimal to be printed for the natural parameter delta. |

`tdigits` |
Number of digits past the decimal point to be shown for estimated timing of each analysis. |

`timename` |
Text string indicating time unit. |

`prior` |
A prior distribution for the standardized effect size. Must be of the format produced by |

`ratio` |
Sample size ratio assumed for experimental to control treatment group sample sizes. This only matters when |

`exclude` |
A list of test statistics to be excluded from design boundary summary produced; see details or examples for a list of all possible output values. A value of |

`POS` |
This is an indicator of whether or not probability of success (POS) should be estimated at baseline or at each interim based on the prior distribution input in |

`r` |
See |

`row.names` |
indicator of whether or not to print row names |

`include.rownames` |
indicator of whether or not to include row names in output. |

`hline.after` |
table lines after which horizontal separation lines should be set; default is to put lines between each analysis as well as at the top and bottom of the table. |

`z` |
A vector of z-statistics |

`i` |
A vector containing the analysis for each element in |

`theta` |
A scalar value representing the standardized effect size used for conditional power calculations; see |

`ylab` |
Used when functions are passed to |

`...` |
This allows many optional arguments that are standard when calling |

### Details

The `print.gsDesign`

function is intended to provide an easier output to review than is available from a simple list of all the output components.
The `gsBoundSummary`

function is intended to provide a summary of boundary characteristics that is often useful for evaluating boundary selection; this outputs an extension of the `data.frame`

class that sets up default printing without row names using `print.gsBoundSummary`

.
`summary.gsDesign`

, on the other hand, provides a summary of the overall design at a higher level; this provides characteristics not included in the `gsBoundSummary`

summary and no detail concerning interim analysis bounds.

In brief, the computed descriptions of group sequential design bounds are as follows:
`Z:`

Standardized normal test statistic at design bound.

`p (1-sided):`

1-sided p-value for `Z`

. This will be computed as the probability of a greater EXCEPT for lower bound when a 2-sided design is being summarized.

`delta at bound:`

Approximate value of the natural parameter at the bound. The approximate standardized effect size at the bound is generally computed as `Z/sqrt(n)`

. Calling this `theta`

, this is translated to the `delta`

using the values `delta0`

and
`delta1`

from the input `x`

by the formula `delta0 + (delta1-delta0)/theta1*theta`

where `theta1`

is the alternate hypothesis value of the standardized parameter. Note that this value will be exponentiated in the case of relative risks, hazard ratios or when the user specifies `logdelta=TRUE`

. In the case of hazard ratios, the value is computed instead by `gsHR()`

to be consistent with `plot.gsDesign()`

. Similarly, the value is computed by `gsRR()`

when the relative risk is the natural parameter.

`Spending: `

Incremental error spending at each given analysis. For asymmetric designs, futility bound will have beta-spending summarized. Efficacy bound always has alpha-spending summarized.

`B-value: `

`sqrt(t)*Z`

where `t`

is the proportion of information at the analysis divided by the final analysis planned information. The expected value for B-values is directly proportional to `t`

.

`CP: `

Conditional power under the estimated treatment difference assuming the interim Z-statistic is at the study bound

`CP H1: `

Conditional power under the alternate hypothesis treatment effect assuming the interim test statistic is at the study bound.

`PP: `

Predictive power assuming the interim test statistic is at the study bound and the input prior distribution for the standardized effect size. This is the conditional power averaged across the posterior distribution for the treatment effect given the interim test statistic value.
`P{Cross if delta=xx}: `

For each of the parameter values in `x`

, the probability of crossing either bound given that treatment effect is computed. This value is cumulative for each bound. For example, the probability of crossing the efficacy bound at or before the analysis of interest.

### Value

`gsBValue()`

, `gsDelta()`

, `gsHR()`

and `gsCPz()`

each returns a vector containing the B-values, approximate treatment effect (see details), approximate hazard ratio and conditional power, respectively, for each value specified by the interim test statistics in `z`

at interim analyses specified in `i`

.

`summary`

returns a text string summarizing the design at a high level. This may be used with `gsBoundSummary`

for a nicely formatted, concise group sequential design description.

`gsBoundSummary`

returns a table in a data frame providing a variety of boundary characteristics. The tabular format makes formatting particularly amenable to place in documents either through direct creation of readable by Word (see the `rtf`

package) or to a csv format readable by spreadsheet software using `write.csv`

.

`print.gsDesign`

prints an overall summary a group sequentia design. While the design description is complete, the format is not as ‘document friendly’ as `gsBoundSummary`

.

`print.gsBoundSummary`

is a simple extension of `print.data.frame`

intended for objects created with `gsBoundSummary`

. The only extension is to make the default to not print row names. This is probably ‘not good R style’ but may be helpful for many lazy R programmers like the author.

### Note

The manual is not linked to this help file, but is available in library/gsdesign/doc/gsDesignManual.pdf in the directory where R is installed.

### Author(s)

Keaven Anderson keaven\_anderson@merck.

### References

Jennison C and Turnbull BW (2000), *Group Sequential Methods with Applications to Clinical Trials*.
Boca Raton: Chapman and Hall.

### See Also

gsDesign, Plots for group sequential designs, `gsProbability`

, `xtable.gsSurv`

### 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 | ```
# survival endpoint using gsSurv
# generally preferred over nSurv since time computations are shown
xgs <- gsSurv(lambdaC=.2, hr=.5, eta=.1, T=2, minfup=1.5)
gsBoundSummary(xgs,timename="Year",tdigits=1)
summary(xgs)
# survival endpoint using nSurvival
# NOTE: generally recommend gsSurv above for this!
ss <- nSurvival(lambda1=.2 , lambda2=.1, eta = .1, Ts = 2, Tr = .5,
sided=1, alpha=.025, ratio=2)
xs <- gsDesign(nFixSurv=ss$n,n.fix=ss$nEvents, delta1=log(ss$lambda2/ss$lambda1))
gsBoundSummary(xs,logdelta=TRUE, ratio=ss$ratio)
# generate some of the above summary statistics for the upper bound
z <- xs$upper$bound
# B-values
gsBValue(z=z, i=1:3, x=xs)
# hazard ratio
gsHR(z=z, i=1:3, x=xs)
# conditional power at observed treatment effect
gsCPz(z=z[1:2], i=1:2, x=xs)
# conditional power at H1 treatment effect
gsCPz(z=z[1:2], i=1:2, x=xs, theta=xs$delta)
# information-based design
xinfo <- gsDesign(delta=.3,delta1=.3)
gsBoundSummary(xinfo, Nname="Information")
# show all available boundary descriptions
gsBoundSummary(xinfo, Nname="Information",exclude=NULL)
# add intermediate parameter value
xinfo <- gsProbability(d=xinfo, theta=c(0,.15,.3))
class(xinfo) # note this is still as gsDesign class object
gsBoundSummary(xinfo, Nname="Information")
# now look at a binomial endpoint; specify H0 treatment difference as p1-p2=.05
# now treatment effect at bound (say, thetahat) is transformed to
# xp$delta0 + xp$delta1*(thetahat-xp$delta0)/xp$delta
np <- nBinomial(p1=.15, p2=.10)
xp <- gsDesign(n.fix=np, endpoint="Binomial", delta1=.05)
summary(xp)
gsBoundSummary(xp,deltaname="p[C]-p[E]")
# estimate treatment effect at lower bound
# by setting delta0=0 (default) and delta1 above in gsDesign
# treatment effect at bounds is scaled to these differences
# in this case, this is the difference in event rates
gsDelta(z=xp$lower$bound, i=1:3, xp)
# binomial endpoint with risk ratio estimates
n.fix<-nBinomial(p1=.3, p2=.15, scale="RR")
xrr <- gsDesign(k=2,n.fix=n.fix,delta1=log(.15/.3),endpoint="Binomial")
gsBoundSummary(xrr,deltaname="RR",logdelta=TRUE)
gsRR(z=xp$lower$bound, i=1:3, xrr)
plot(xrr,plottype="RR")
# delta is odds-ratio: sample size slightly smaller than for relative risk or risk difference
n.fix<-nBinomial(p1=.3, p2=.15, scale="OR")
xOR <- gsDesign(k=2,n.fix=n.fix,delta1=log(.15/.3/.85*.7),endpoint="Binomial")
gsBoundSummary(xOR,deltaname="OR",logdelta=TRUE)
# for nice LaTeX table output, use xprint
xprint(xtable(gsBoundSummary(xOR, deltaname="OR", logdelta=TRUE), caption="Table caption."))
``` |