ldBands {Hmisc} | R Documentation

## Group Sequential Boundaries using the Lan-DeMets Approach

### Description

This function computes and plots group sequential stopping boundaries from the Lan-DeMets method with a variety of α-spending functions using the `ld98` program from the Department of Biostatistics, University of Wisconsin written by DM Reboussin, DL DeMets, KM Kim, and KKG Lan. Such stopping boundaries are useful for early termination of clinical trials for safety problems or for efficacy. Simple `plot` and `print` methods are implemented. Simple sample size and minimally detectable effect sizes given sample sizes may be obtained with a `summary` method if `power` was specified to `ldBands`. Alternatively, `summary` computes, for each look, the difference in means that must be achieved to cross a boundary if `n` and `sd` are specified, or the minimum difference in proportions and the odds ratios that must be achieved to cross a boundary if `n` and `pbar` are specified.

### Usage

    ldBands(n = length(times), times = NULL, alpha = 0.05, sided = 2,
            alphaLower=alpha/2, alphaUpper=alpha/2,
            information = NULL,
            spending=c('OBrien-Fleming','Pocock','alpha*t^phi',
                       'Hwang-Shih-DeCani'),
            phi=1,
            spending2=c('OBrien-Fleming','Pocock','alpha*t^phi',
                        'Hwang-Shih-DeCani'),
            phi2=phi,
            truncate = Inf, power = NULL, pr = TRUE)
    ## S3 method for class 'ldBands':
    print(x, ...)
    ## S3 method for class 'ldBands':
    plot(x, xlab='Time', ylab='Z', actual=NULL,
            type='b', labels=NULL, ...)

    ## S3 method for class 'ldBands':
    summary(object, stdiff=NULL, n=NULL,
            p1=NULL, p2=NULL,  hr=NULL, events=NULL,
            pbar=NULL, sd=NULL, ...)
    ## S3 method for class 'summary.ldBands':
    print(x, ...)

### Arguments

`n` | number of interim analyses. If `times` is given, is automatically taken as the length of `times`. For `summary.ldBands`, `n` is the sample size, to obtain detectable standardized difference.
`times` | times at which interim analyses are done
`alpha` | overall α level for the multiple tests. Default is `0.05`. If `sided=3` is the α-level for the lower bounds, otherwise is the total α.
`sided` | set to `1` to use a one-sided test, `3` for asymmetric two-sided bounds
`alphaLower` | α-level for lower bound if `sided=3`. Defaults to α/2. When `sided=3` `alpha` is recalculated from `alphaLower+alphaUpper`.
`alphaUpper` | α-level for upper bound if `sided=3`. Defaults to α/2.
`information` | a vector of information times if different from `times`. Used for computing covariances of test statistics.
`spending` | an α spending function. Default is O'Brien-Fleming function. If `sided=3` is the spending function for the lower bounds.
`phi` | parameter for the third and fourth types of spending functions (exponent of time for the third, default is `1`)
`spending2` | spending function for the upper bounds if `sided=3`. Defaults to same spending function for lower bounds.
`phi2` | parameter for third and fourth spending functions if `sided=3`. Default is `phi`.
`truncate` | a value at which to truncate Z critical values so that early looks will have some power without really affecting the overall α level. Default is no truncation.
`power` | A power for which to compute a drift parameter; useful in sample size calculations
`pr` | set to `FALSE` to supress the actual output of the `ld98` program
`x` | an object created by `ldBands` or `summary.ldBands`
`xlab` | x-axis label
`ylab` | y-axis label
`actual` | an optional list with two components: times and z values to add as points to the first plot
`type` | default is `"b"` causing both points and lines to be drawn for boundaries. Use `type="l"` for example to suppress drawing circles at points.
`labels` | an optional character vector to be used to label time points corresponding to those generated by `ldBands`
`object` | an object created by `ldBands`
`stdiff` | standardized difference to detect
`p1` | probability of event in group 1
`p2` | probability of event in group 2, to compare with `p1` in order to obtain sample size for two-sample binomial
`hr` | hazard ratio to detect, to obtain needed number of events at end of study for either treatment arm using the logrank test
`events` | number of events per treatment arm at end of study, to obtain detectable hazard ratio
`pbar` | mean of actual probabilities of events in two treatment arms
`sd` | standard deviation of an observation
`...` | unused

### Details

This function assumes that you have stored the `ld98` executable in a subdirectory that is in your system path. Obtain `ld98` program from the URL given in the reference below.

The `plot` method makes two plots if `power` is specified, the second containing exit and cumulative exit probabilities as a function of time. If `par(mfrow=c())` is not already set to something besides `c(1,1)`, `par(mfrow=c(2,1))` is set temporarily.

### Value

A list of class `"ldBands"` with the following components. When `summary` is used and `n` and either `pbar` or `sd` are given, `summary` returns the same object returned by `ldBands` but with possible components `diff.lower,diff.lower` (needed difference means or proportions) and `or.lower,or.upper` (needed odds ratios).

`data` | a data frame containing the main calculations
`power` | power specified, if any
`drift` | drift calculated, if `power` specified
`type` | `"boundaries"` if `power` not specified, `"power"` otherwise
`n` | number of interim analyses, for `ldBands`. For `summary` is the number of observations per arm, and it must be a vector with length equal to the number of looks if `pbar` or `sd` are given.
`alpha` | input value of `alpha`
`sided` | `1-3`
`alphaLower` | lower α
`alphaUpper` | upper α
`spending` | name of α spending function used
`phi` | parameter for spending function
`spending2` | name of spending function for upper boundary. Defaults to `spending`
`phi2` | parameter for second spending function. Defaults to `phi` but is ignored if `spending2` is the first or second type of spending function.
`truncate` | value at which Z statistics truncated (default is `Inf`)

### Author(s)

Frank E Harrell Jr

### References

Reboussin DM, DeMets DL, Kim K-M, Lan KKG (1996): Programs for computing group sequential boundaries using the Lan-DeMets method. http://www.medsch.wisc.edu/landemets

Reboussin DM, DeMets DL, Kim K, Lan KKG (2000): Computations for group sequential boundaries using the Lan-DeMets spending function method. Controlled Clinical Trials 21:190-207.

### See Also

`gbayes`

### Examples

    ## Not run:
    # Get boundaries for O'Brien-Fleming spending function, 5 looks, alpha=.05
    b <- ldBands(5, pr=FALSE)
    plot(b)
    # Same but times are irregular, and information times are different than
    # test times.  Use Pocock spending function.
    b <- ldBands(times=      c(.4, .6, .8, .9, .95),
                 information=c(.42,.65,.83,.89,.94), spending='Pocock')

    # Get power calculations
    u <- ldBands(times=c(.4, .6, .8, .9, .95),  power=.9)
    u$drift                  # standardize difference * sqrt(n per arm)
                             # needed to provide power=.9
    summary(u, n=50)         # obtain detectable standardized difference
    summary(u, p1=.4, p2=.5) # get sample size per arm, two-sample binomial
    summary(u, hr=1.5)       # get number of events per arm needed
                             # to detect a hazard ratio of 1.5

    # Asymmetric boundaries with different spending functions, truncate
    b <- ldBands(5, sided=3, spending='alpha*t^phi', phi=1, phi2=1.5,
                 alphaLower=.01, alphaUpper=.04, truncate=4)
    b
    plot(b)
    # Compute differences in proportions and odds ratios needed to cross
    # the boundaries, given a mean probability in two treatment arms of 0.1
    # and given a vector of sample sizes per arm corresponding to the looks
    s <- summary(b, n=seq(200,1000,by=200), pbar=.1)
    s
    d <- s$data
    plot(d$n, d$or.lower, xlab='N Per Arm',
         ylab='Critical Odds Ratio', type='b',
         ylim=range(d$or.lower, d$or.upper), log='y')
    lines(d$n, d$or.upper, type='b')
    abline(h=1, lty=2)
    ## End(Not run)

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