# Calculate Power in a Group Sequential Design Derives power in a two arm clinical trial under a group sequential design. Allows for arbitrary number of interim analyses, arbitrary specification of arm-0/arm-1 time to event distributions (via survival or hazard), arm-0/arm-1 censoring distribution, provisions for two types of continuous time non-compliance according to arm-0/arm-1 rate followed by switch to new hazard rate. Allows for analyses using (I) weighted log-rank statistic, with weighting function (a) a member of the Flemming-Harrington G-Rho class, or (b) a stopped version thereof, or (c) the ramp-plateau deterministic weights, or (II) the integrated survival distance (currently under method=="S" without futility only). Stopping boundaries are computed via the Lan-Demets method, Haybittle method, converted from the stochastic curtailment procedure, or be completely specified by the user. The Lan-Demets boundaries can be constructed usign either O'Brien-Flemming, Pocock or Wang-Tsiatis power alpha-spending. The C kernel is readily extensible, and further options will become available in the near future. ```r PwrGSD(EfficacyBoundary = LanDemets(alpha = 0.05, spending = ObrienFleming), FutilityBoundary = LanDemets(alpha = 0.1, spending = ObrienFleming), NonBindingFutility = TRUE, sided = c("2>", "2<", "1>", "1<"), method = c("S", "A"), accru, accrat, tlook, tcut0 = NULL, h0 = NULL, s0 = NULL, tcut1 = NULL, rhaz = NULL, h1 = NULL, s1 = NULL, tcutc0 = NULL, hc0 = NULL, sc0 = NULL, tcutc1 = NULL, hc1 = NULL, sc1 = NULL, tcutd0A = NULL, hd0A = NULL, sd0A = NULL, tcutd0B = NULL, hd0B = NULL, sd0B = NULL, tcutd1A = NULL, hd1A = NULL, sd1A = NULL, tcutd1B = NULL, hd1B = NULL, sd1B = NULL, tcutx0A = NULL, hx0A = NULL, sx0A = NULL, tcutx0B = NULL, hx0B = NULL, sx0B = NULL, tcutx1A = NULL, hx1A = NULL, sx1A = NULL, tcutx1B = NULL, hx1B = NULL, sx1B = NULL, noncompliance = c("none", "crossover", "mixed", "user"), gradual = FALSE, WtFun = c("FH", "SFH", "Ramp"), ppar = cbind(c(0, 0)), Spend.Info = c("Variance", "Events", "Hybrid(k)", "Calendar"), RR.Futility = NULL, qProp.one.or.Q = c("one", "Q"), Nsim = NULL, detail = FALSE, StatType = c("WLR", "ISD"), doProj=FALSE) ``` ## Arguments - `EfficacyBoundary`: This specifies the method used to construct the efficacy boundary. The available choices are: (i) `Lan-Demets`(`alpha`=, `spending` =). The Lan-Demets method is based upon a error probability spending approach. The spending function can be set to `ObrienFleming`, `Pocock`, or `Power(rho)`, where `rho` is the the power argument for the power spending function: rho=3 is roughly equivalent to the O'Brien-Fleming spending function and smaller powers result in a less conservative spending function. (ii) `Haybittle`(`alpha`=, `b.Haybittle`=). The Haybittle approach is conceptually the simplest of all methods for efficacy boundary construction. However, as it spends nearly no alpha until the end, is for all practical purposes equivalent to a single analysis design and to be considered overly conservative. This method sets all the boundary points equal to `b.Haybittle`, a user specified value (try 3) for all analyses except the last, which is calculated so as to result in the total type I error, set with the argument `alpha`. (iii) `SC`(`be.end`=, `prob`=). The stochastic curtailment method is based upon the conditional probability of type I error given the current value of the statistic. Under this method, a sequence of boundary points on the standard normal scale (as are boundary points under all other methods) is calculated so that the total probability of type I error is maintained. This is done by considering the joint probabilities of continuing to the current analysis and then exceeding the threshold at the current analysis. A good value for the threshold value for the conditional type I error, `prob` is 0.90 or greater. (iv) User supplied boundary points in the form `c(b1, b2, b3, ..., b_m)`, where `m` is the number of looks. - `FutilityBoundary`: This specifies the method used to construct the futility boundary. The available choices are: (i) `Lan-Demets`(`alpha`=, `spending`= ). The Lan-Demets method is based upon a error probability spending approach. The spending function can be set to `ObrienFleming`, `Pocock`, or `Power(rho)`, where `rho` is the the power argument for the power spending function: rho=3 is roughly equivalent to the O'Brien-Fleming spending function and smaller powers result in a less conservative spending function. NOTE: there is no implementation of the `Haybittle` method for futility boundary construction. Given that the futility boundary depends upon values of the drift function, this method doesn't apply. (ii) `SC`(`be.end`=, `prob`=, `drift.end`=). The stochastic curtailment method is based upon the conditional probability of type II error given the current value of the statistic. Under this method, a sequence of boundary points on the standard normal scale (as are boundary points under all other methods) is calculated so that the total probability of type II error, is maintained. This is done by considering the joint probabilities of continuing to the current analysis and then exceeding the threshold at the current analysis. A good value for the threshold value for the conditional type I error, `prob` is 0.90 or greater. (iii) User supplied boundary points in the form `c(b1, b2, b3, ..., b_m)`, where `m` is the number of looks. - `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'. - `sided`: Set to 2\> (quoted) for two sided tests of the null hypothesis when a positive drift corresponds to efficacy. Set to 2\< (quoted) for two sided tests of the null hypothesis when a negative drift corresponds to efficacy. Set to 1\> or 1\< for one sided tests of H0 when efficacy corresponds to a positive or negative drift, respectively. If `method`==S then this must be of the same length as `StatType` because the interpretation of `sided` is different depending upon whether `StatType`==WLR (negative is benefit) or `StatType`==ISD (positive is benefit) - `method`: Determines how to calculate the power. Set to A (Asymptotic method, the default) or S (Simulation method) - `accru`: The upper endpoint of the accrual period beginning with time 0. - `accrat`: The rate of accrual per unit of time. - `tlook`: The times of planned interim analyses. - `tcut0`: Left hand endpoints for intervals upon which the arm-0 specific mortality is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. - `h0`: A vector of the same length as `tcut0` which specifies the piecewise constant arm-0 mortality rate. - `s0`: Alternatively, the arm-0 mortality distribution can be supplied via this argument, in terms of of the corresponding survival function values at the times given in the vector `tcut0`. If `s0` is supplied, then `h0`is derived internally, assuming the piecewise exponential distrubiton. If you specify `s0`, the first element must be 1, and correspondingly, the first component of `tcut0` will be the lower support point of the distribution. You must supply either `h0` or `s0` but not both. - `tcut1`: Left hand endpoints for intervals upon which the arm-1 specific mortality is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. - `rhaz`: A vector of piecewise constant arm-1 versus arm-0 mortality rate ratios. If `tcut1` and `tcut0` are not identical, then `tcut1`, `h0`, and `rhaz` are internally rederived at the union of the sequences `tcut0` and `tcut1`. In all cases the arm-1 mortality rate is then derived at the time cutpoints `tcut1` as `rhaz` times`h0`. - `h1`: Alternatively, the arm-1 mortality distribution can be supplied via this argument by specifying the piecewise constant arm-1 mortality rate. See the comments above. - `s1`: Alternatively, the arm-1 mortality distribution can be supplied via this argument, in terms of of the corresponding survival function values at the times given in the vector `tcut1`. Comments regarding `s0` above apply here as well. You must supply exactly one of the following: `h1`, `rhaz`, or `s1`. - `tcutc0`: Left hand endpoints for intervals upon which the arm-0 specific censoring distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. - `hc0`: A vector of the same length as `tcutc0` which specifies the arm-0 censoring distribution in terms of a piecewise constant hazard function. - `sc0`: Alternatively, the arm-0 censoring distribution can be supplied via this argument, in terms of of the corresponding survival function values at the times given in the vector `tcutc0`. See comments above. You must supply either `hc0` or `sc0` but not both. - `tcutc1`: Left hand endpoints for intervals upon which the arm-1 specific censoring distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. - `hc1`: A vector of the same length as `tcutc1` which specifies the arm-1 censoring distribution in terms of a piecewise constant hazard function. - `sc1`: Alternatively, the arm-1 censoring distribution can be supplied via this argument, in terms of of the corresponding survival function values at the times given in the vector `tcutc1`. See comments above. You must supply either `hc1` or `sc1` but not both. - `noncompliance`: (i) Seting `noncompliance` to none for no non-compliance will automatically set the non-compliance arguments, below, to appropriate values for no compliance. This requires no additional user specification of non-compliance parameters. (ii) Setting `noncompliance` to crossover will automatically set crossover values in the arm 0/1 specific **post-cause-B-delay-mortality** for cross-over, i.e. simple interchange of the arm 0 and arm 1 mortalities. The user is required to specify all parameters corresponding to the arm 0/1 specific **cause-B-delay** distributions. The **cause-A-delay** and **post-cause-A-delay-mortality** are automatically set so as not to influence the calculations. Setting `noncompliance` to mixed will set the arm 0/1 specific **post-cause-B-delay-mortality** distributions for crossover as defined above. The user specifies the arm 0/1 specific **cause-B-delay** distribution as above, and in addition, all parameters related to the arm 0/1 specific **cause-A-delay** distributions and corresponding arm 0/1 specific **post-cause-A-delay-mortality** distributions. (iii) Setting `noncompliance` to user requires the user to specify all non-compliance distribution parameters. - `tcutd0A`: Left hand endpoints for intervals upon which the arm-0 specific **cause-A delay** distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Required only when `noncompliance` is set to mixed or user . - `hd0A`: A vector of the same length as `tcutd0A` containing peicewise constant hazard rates for the arm-0 **cause-A delay** distribution. Required only when `noncompliance` is set to mixed or user . - `sd0A`: When required, the arm-0 **cause-A-delay** distribution is alternately specified via a survival function. A vector of the same length as `tcutd0A`. - `tcutd0B`: Left hand endpoints for intervals upon which the arm-0 specific **cause-B delay** distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Always required when `noncompliance` is set to any value other than none . - `hd0B`: A vector of the same length as `tcutd0B` containing peicewise constant hazard rates for the arm-0 **cause-B delay** distribution. Always required when `noncompliance` is set to any value other than none . - `sd0B`: When required, the arm-0 **cause-B-delay** distribution is alternately specified via a survival function. A vector of the same length as `tcutd0B`. - `tcutd1A`: Left hand endpoints for intervals upon which the arm-1 specific **cause-A delay** distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Required only when `noncompliance` is set to mixed or user . - `hd1A`: A vector of the same length as `tcutd1A` containing peicewise constant hazard rates for the arm-1 **cause-A delay** distribution. Required only when `noncompliance` is set to mixed or user . - `sd1A`: When required, the arm-1 **cause-A-delay** distribution is alternately specified via a survival function. A vector of the same length as `tcutd1A`. - `tcutd1B`: Left hand endpoints for intervals upon which the arm-1 specific **cause-B delay** distribution hazard function is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Always required when `noncompliance` is set to any value other than none . - `hd1B`: A vector of the same length as `tcutd1B` containing peicewise constant hazard rates for the arm-1 **cause-B delay** distribution. Always required when `noncompliance` is set to any value other than none . - `sd1B`: When required, the arm-1 **cause-A-delay** distribution is alternately specified via a survival function. A vector of the same length as `tcutd1A`. - `tcutx0A`: Left hand endpoints for intervals upon which the arm-0 specific **post-cause-A-delay-mortality** rate is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Required only when `noncompliance` is set to mixed or user . - `hx0A`: A vector of the same length as `tcutx0A` containing the arm-0 **post-cause-A-delay mortality** rates. Required only when `noncompliance` is set to mixed or user . - `sx0A`: When required, the arm-0 **post-cause-A-delay mortality** distribution is alternately specified via a survival function. A vector of the same length as `tcutx0A`. - `tcutx0B`: Left hand endpoints for intervals upon which the arm-0 specific **post-cause-B-delay-mortality** rate is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Always required when `noncompliance` is set to any value other than none . - `hx0B`: A vector of the same length as `tcutx0B` containing the arm-0 **post-cause-B-delay mortality** rates. Always required when `noncompliance` is set to any value other than none . - `sx0B`: When required, the arm-0 **post-cause-B-delay mortality** distribution is alternately specified via a survival function. A vector of the same length as `tcutx0B`. - `tcutx1A`: Left hand endpoints for intervals upon which the arm-1 specific **post-cause-A-delay-mortality** rate is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Required only when `noncompliance` is set to mixed or user . - `hx1A`: A vector of the same length as `tcutx1A` containing the arm-1 **post-cause-A-delay mortality** rates. Required only when `noncompliance` is set to mixed or user . - `sx1A`: When required, the arm-1 **post-cause-A-delay mortality** distribution is alternately specified via a survival function. A vector of the same length as `tcutx1A`. - `tcutx1B`: Left hand endpoints for intervals upon which the arm-1 specific **post-cause-B-delay-mortality** rate is constant. The last given component is the left hand endpoint of the interval having right hand endpoint infinity. Always required when `noncompliance` is set to any value other than none . - `hx1B`: A vector of the same length as `tcutx1B` containing the arm-1 **post-cause-B-delay mortality** rates. Always required when `noncompliance` is set to any value other than none . - `sx1B`: When required, the arm-1 **post-cause-B-delay mortality** distribution is alternately specified via a survival function. A vector of the same length as `tcutx1B`. - `gradual`: Should the conversion to post-noncompliance mortality be gradual. Under the default behavior, `gradual`=`FALSE`, there is an immediate conversion to the post-noncompliance mortality rate function. If `gradual` is set to `TRUE` then this conversion is done gradually . In truth, at the individual level what is done is that the new mortality rate function is a convex combination of the pre-noncompliance mortality and the post-noncompliance mortality, with the weighting in proportion to the time spent in compliance with the study arm protocal. - `WtFun`: Specifies the name of a weighting function (of time) for assigning relative weights to events according to the times at which they occur. The default, FH , a two parameter weight function, specifies the Fleming-Harrington `g-rho` family of weighting functions defined as the pooled arm survival function (Kaplan-Meier estimate) raised to the `g` times its complement raised to the `rho`. Note that `g`=`rho`=0 corresponds to the unweighted log-rank statistic. A second choice is the SFH function, (for Stopped Fleming-Harrington ), meaning that the FH weights are capped at their value at a user specified time, which has a total of 3 parameters. A third choice is `Ramp(tcut)`. Under this choice, weights are assigned in a linearly manner from time 0 until a user specified cut-off time, `tcut`, after which events are weighted equally. It is possible to conduct computations on `nstat` candidate statistics within a single run. In this case, `WtFun` should be a character vector of length `nstat` having components set from among the available choices. - `ppar`: A vector containing all the weight function parameters, in the order determined by that of WtFun . For example, if `WtFun` is set to `c("FH","SFH","Ramp")` then `ppar` should be a vector of length six, with the FH parameters in the first two elements, SFH parameters in the next 3 elements, and Ramp parameter in the last element. - `RR.Futility`: The relative risk corresponding to the alternative alternative hypothesis that is required in the construction of the futility boundary. Required if `Boundary.Futility` is set to a non-null value. - `Spend.Info`: When the test statistic is something other than the unweighted log-rank statistic, the variance information, i.e. the ratio of variance at interim analysis to variance at the end of trial, is something other than the ratio of events at interim analysis to the events at trial end. The problem is that in practice one doesn't necessarily have a good idea what the end of trial variance should be. In this case the user may wish to spend the type I and type II error probabilities according to a different time scale. Possible choices are Variance , (default), which just uses the variance ratio scale, Events , which uses the events ratio scale, Hybrid(k) , which makes a linear transition from the Variance scale to the Events scale beginning with analysis number `k`. The last choice, Calendar , uses the calendar time scale - `qProp.one.or.Q`: If a futility boundary is specified, what assumption should be made about the drift function (the mean value of the weighted log-rank statistic at analysis `j` normalized by the square root of the variance function at analysis `k`). In practice we don't presume to know the shape of the drift function. Set to one or Q . The choice one results in a more conservative boundary. - `Nsim`: If you specify `method`==S , then you must specify the number of simulations. 1000 should be sufficient. - `detail`: If you specify `method`==S , and want to see the full level of detail regarding arguments returned from the C level code, specify `detail`==TRUE - `StatType`: If you specify `method`==S , then the available choices are WLR (weighted log-rank) and ISD (integrated survival difference). - `doProj`: Works only when `method`==S . If a weighted log-rank statistic is specified without maximum information having been stipulated in the design then certain functionals, the `Q` first and second moments, must be projected. Setting this argument to `TRUE` includes this projection into the simulation runs. ## Returns Returns a value of class `PwrGSD` which has components listed below. Note that the print method will display a summary table of estimated powers and type I errors as a `nstat` by 2 matrix. The summary method returns the same object invisibly, but after computing the summary table mentioned above, and it is included in the returned value as a commponent `TBL`. See examples below. - **dPower**: A `length(tlook)` by `nstat` matrix containing in each column, an increment in power that resulted at that analysis time for the given statistic. - **dErrorI**: A `length(tlook)` by `nstat` matrix containing in each column, an increment in type I error that resulted at that analysis time for the given statistic. Always sums to the total alpha specified in `alphatot` - **detail**: A list with components equal to the arguments of the C-call, which correspond in a natural way to the arguments specified in the R call, along with the computed results in `palpha0vec`, `palpha1vec`, `inffrac`, and `mu`. The first two are identical to `dErrorI` and `dPower`, explained above. The last two are `length(tlook)` by `nstat` matrices. For each statistic specified in `par`, the corresponding columns of `pinffrac` and `mu` contain the information fraction and drift at each of the analysis times. - **call**: the call ## References Gu, M.-G. and Lai, T.-L. Determination of power and sample size in the design of clinical trialswith 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 Izmirlian G. (2014). Estimation of the relative risk following group sequential procedure based upon the weighted log-rank statistic. Statistics and its Interface 7(1), 27-42 ## Author(s) Grant Izmirlian izmirlian@nih.gov ## See Also `cpd.PwrGSD` ## Examples ```r library(PwrGSD) test.example <- PwrGSD(EfficacyBoundary = LanDemets(alpha = 0.05, spending = ObrienFleming), FutilityBoundary = LanDemets(alpha = 0.1, spending = ObrienFleming), RR.Futility = 0.82, sided="1<",method="A",accru =7.73, accrat =9818.65, tlook =c(7.14, 8.14, 9.14, 10.14, 10.64, 11.15, 12.14, 13.14, 14.14, 15.14, 16.14, 17.14, 18.14, 19.14, 20.14), tcut0 =0:19, h0 =c(rep(3.73e-04, 2), rep(7.45e-04, 3), rep(1.49e-03, 15)), tcut1 =0:19, rhaz =c(1, 0.9125, 0.8688, 0.7814, 0.6941, 0.6943, 0.6072, 0.5202, 0.4332, 0.6520, 0.6524, 0.6527, 0.6530, 0.6534, 0.6537, 0.6541, 0.6544, 0.6547, 0.6551, 0.6554), tcutc0 =0:19, hc0 =c(rep(1.05e-02, 2), rep(2.09e-02, 3), rep(4.19e-02, 15)), tcutc1 =0:19, hc1 =c(rep(1.05e-02, 2), rep(2.09e-02, 3), rep(4.19e-02, 15)), tcutd0B =c(0, 13), hd0B =c(0.04777, 0), tcutd1B =0:6, hd1B =c(0.1109, 0.1381, 0.1485, 0.1637, 0.2446, 0.2497, 0), noncompliance =crossover, gradual =TRUE, WtFun =c("FH", "SFH", "Ramp"), ppar =c(0, 1, 0, 1, 10, 10)) ```

Ch01-PwrGSD function