spower {Hmisc} | R Documentation

## Simulate Power of 2-Sample Test for Survival under Complex Conditions

### Description

Given functions to generate random variables for survival times and censoring times, `spower` simulates the power of a user-given 2-sample test for censored data. By default, the logrank (Cox 2-sample) test is used, and a `logrank` function for comparing 2 groups is provided. For composing S-Plus functions to generate random survival times under complex conditions, the `Quantile2` function allows the user to specify the intervention:control hazard ratio as a function of time, the probability of a control subject actually receiving the intervention (dropin) as a function of time, and the probability that an intervention subject receives only the control agent as a function of time (non-compliance, dropout). `Quantile2` returns a function that generates either control or intervention uncensored survival times subject to non-constant treatment effect, dropin, and dropout. There is a `plot` method for plotting the results of `Quantile2`, which will aid in understanding the effects of the two types of non-compliance and non-constant treatment effects. `Quantile2` assumes that the hazard function for either treatment group is a mixture of the control and intervention hazard functions, with mixing proportions defined by the dropin and dropout probabilities. It computes hazards and survival distributions by numerical differentiation and integration using a grid of (by default) 7500 equally-spaced time points.

The `logrank` function is intended to be used with `spower` but it can be used by itself as long as the `group` variable has only the values `1` and `2` and there are no missing data. It returns the 1 degree of freedom chi-square statistic.

The `Weibull2` function accepts as input two vectors, one containing two times and one containing two survival probabilities, and it solves for the scale and shape parameters of the Weibull distribution (`S(t)=exp(-alpha*t^ gamma)`) which will yield those estimates. It creates an S-Plus function to evaluate survival probabilities from this Weibull distribution. `Weibull2` is useful in creating functions to pass as the first argument to `Quantile2`.

The `Lognorm2` and `Gompertz2` functions are similar to `Weibull2` except that they produce survival functions for the log-normal and Gompertz distributions.

### Usage

    spower(rcontrol, rinterv, rcens, nc, ni,
           test=logrank, nsim=500, alpha=0.05, pr=TRUE)

    Quantile2(scontrol, hratio,
              dropin=function(times)0, dropout=function(times)0,
              m=7500, tmax, qtmax=.001, mplot=200, pr=TRUE, ...)

    ## S3 method for class 'Quantile2':
    print(x, ...)

    ## S3 method for class 'Quantile2':
    plot(x,
         what=c('survival','hazard','both','drop','hratio','all'),
         dropsep=FALSE, lty=1:4, col=1, xlim, ylim=NULL,
         label.curves=NULL, ...)

    logrank(S, group)

    Gompertz2(times, surv)
    Lognorm2(times, surv)
    Weibull2(times, surv)

### Arguments

`rcontrol` | a function of `n` which returns `n` random uncensored failure times for the control group. `spower` assumes that non-compliance (dropin) has been taken into account by this function.
`rinterv` | similar to `rcontrol` but for the intervention group
`rcens` | a function of `n` which returns `n` random censoring times. It is assumed that both treatment groups have the same censoring distribution.
`nc` | number of subjects in the control group
`ni` | number in the intervention group
`scontrol` | a function of a time vector which returns the survival probabilities for the control group at those times assuming that all patients are compliant
`hratio` | a function of time which specifies the intervention:control hazard ratio (treatment effect)
`x` | an object of class `"Quantile2"` created by `Quantile2`
`S` | a `Surv` object or other two-column matrix for right-censored survival times
`group` | group indicators have length equal to the number of rows in `S`. Only values allowed are 1 and 2.
`times` | a vector of two times
`surv` | a vector of two survival probabilities
`test` | any function of a `Surv` object and a grouping variable which computes a chi-square for a two-sample censored data test. The default is `logrank`.
`nsim` | number of simulations to perform (default=500)
`alpha` | type I error (default=.05)
`pr` | set to `FALSE` to cause `spower` to suppress progress notes for simulations. Set to `FALSE` to prevent `Quantile2` from printing `tmax` when it calculates `tmax`.
`dropin` | a function of time specifying the probability that a control subject actually becomes an intervention subject at the corresponding time
`dropout` | a function of time specifying the probability of an intervention subject dropping out to control conditions
`m` | number of time points used for approximating functions (default is 7500)
`tmax` | maximum time point to use in the grid of `m` times. Default is the time such that `scontrol(time)` is `qtmax`.
`qtmax` | survival probability corresponding to the last time point used for approximating survival and hazard functions. Default is `.001`. For `qtmax` of the time for which a simulated time is needed which corresponds to a survival probability of less than `qtmax`, the simulated value will be `tmax`.
`mplot` | number of points used for approximating functions for use in plotting (default is 200 equally spaced points)
`...` | optional arguments passed to the `scontrol` function when it's evaluated by `Quantile2`
`what` | a single character constant (may be abbreviated) specifying which functions to plot. The default is `"both"` meaning both survival and hazard functions. Specify `what="drop"` to just plot the dropin and dropout functions, `what="hratio"` to plot the hazard ratio functions, or `"all"` to make 4 separate plots showing all functions (6 plots if `dropsep=TRUE`).
`dropsep` | set `dropsep=TRUE` to make `plot.Quantile2` separate pure and contaminated functions onto separate plots
`lty` | vector of line types
`col` | vector of colors
`xlim` | optional x-axis limits
`ylim` | optional y-axis limits
`label.curves` | optional list which is passed as the `opts` argument to `labcurve`.

### Value

`spower` returns the power estimate (fraction of simulated chi-squares greater than the alpha-critical value). `Quantile2` returns an S-Plus function of class `"Quantile2"` with attributes that drive the `plot` method. The major attribute is a list containing several lists. Each of these sub-lists contains a `Time` vector along with one of the following: survival probabilities for either treatment group and with or without contamination caused by non-compliance, hazard rates in a similar way, intervention:control hazard ratio function with and without contamination, and dropin and dropout functions. `logrank` returns a single chi-square statistic, and `Weibull2`, `Lognorm2` and `Gompertz2` return an S function with three arguments, only the first of which (the vector of `times`) is intended to be specified by the user.

### Side Effects

`spower` prints the interation number every 10 iterations if `pr=TRUE`.

### Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
f.harrell@vanderbilt.edu

### References

Lakatos E (1988): Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics 44:229–241 (Correction 44:923).

Cuzick J, Edwards R, Segnan N (1997): Adjusting for non-compliance and contamination in randomized clinical trials. Stat in Med 16:1017–1029.

Cook, T (2003): Methods for mid-course corrections in clinical trials with survival outcomes. Stat in Med 22:3431–3447.

### See Also

`cpower`, `ciapower`, `bpower`, `cph`, `coxph`, `labcurve`

### Examples

    # Simulate a simple 2-arm clinical trial with exponential survival so
    # we can compare power simulations of logrank-Cox test with cpower()
    # Hazard ratio is constant and patients enter the study uniformly
    # with follow-up ranging from 1 to 3 years
    # Drop-in probability is constant at .1 and drop-out probability is
    # constant at .175.  Two-year survival of control patients in absence
    # of drop-in is .8 (mortality=.2).  Note that hazard rate is -log(.8)/2
    # Total sample size (both groups combined) is 1000
    # % mortality reduction by intervention (if no dropin or dropout) is 25
    # This corresponds to a hazard ratio of 0.7283 (computed by cpower)

    cpower(2, 1000, .2, 25, accrual=2, tmin=1,
           noncomp.c=10, noncomp.i=17.5)

    ranfun <- Quantile2(function(x)exp(log(.8)/2*x),
                        hratio=function(x)0.7283156,
                        dropin=function(x).1,
                        dropout=function(x).175)

    rcontrol <- function(n) ranfun(n, what='control')
    rinterv  <- function(n) ranfun(n, what='int')
    rcens    <- function(n) runif(n, 1, 3)

    set.seed(11)   # So can reproduce results
    spower(rcontrol, rinterv, rcens, nc=500, ni=500,
           test=logrank, nsim=50)  # normally use nsim=500 or 1000

    # Simulate a 2-arm 5-year follow-up study for which the control group's
    # survival distribution is Weibull with 1-year survival of .95 and
    # 3-year survival of .7.  All subjects are followed at least one year,
    # and patients enter the study with linearly increasing probability  after that
    # Assume there is no chance of dropin for the first 6 months, then the
    # probability increases linearly up to .15 at 5 years
    # Assume there is a linearly increasing chance of dropout up to .3 at 5 years
    # Assume that the treatment has no effect for the first 9 months, then
    # it has a constant effect (hazard ratio of .75)

    # First find the right Weibull distribution for compliant control patients
    sc <- Weibull2(c(1,3), c(.95,.7))
    sc

    # Inverse cumulative distribution for case where all subjects are followed
    # at least a years and then between a and b years the density rises
    # as (time - a) ^ d is a + (b-a) * u ^ (1/(d+1))

    rcens <- function(n) 1 + (5-1) * (runif(n) ^ .5)
    # To check this, type hist(rcens(10000), nclass=50)

    # Put it all together

    f <- Quantile2(sc,
          hratio=function(x)ifelse(x<=.75, 1, .75),
          dropin=function(x)ifelse(x<=.5, 0, .15*(x-.5)/(5-.5)),
          dropout=function(x).3*x/5)

    par(mfrow=c(2,2))
    # par(mfrow=c(1,1)) to make legends fit
    plot(f, 'all', label.curves=list(keys='lines'))

    rcontrol <- function(n) f(n, 'control')
    rinterv  <- function(n) f(n, 'intervention')

    set.seed(211)
    spower(rcontrol, rinterv, rcens, nc=350, ni=350,
           test=logrank, nsim=50)  # normally nsim=500 or more
    par(mfrow=c(1,1))

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