Power and Sample Size for (Bio)Equivalence Studies

Contains functions to calculate power and sample size for various study designs used in bioequivalence studies. Use known.designs() to see the designs supported. Power and sample size can be obtained based on different methods, amongst them prominently the TOST procedure (two one-sided t-tests). See README and NEWS for further information.


News

Version 1.4-7 2018-04-12 (dedicated to the birthday of Alfie Schuetz :-)) - Bug corrected in functions using expected power - Function type1error.2TOST() no longer available since it suffers from insufficient precision to obtain the type 1 error (TIE) via simulations. Due to the intersection-union principle the TIE is always upper bounded to alpha by theory. - Function power.2TOST() now based on simulations to obtain the power of 2 TOSTs (statistical flaw in 4-dimensional t-distribution approach). - sampleN.2TOST and man page adapted according to this change. - regulator="FDA" implemented in scABEL.ad(). - New function power.RSABE2L.sds() which implements the 'exact' based method for RSABE (ncTOST) of the two Laszlos. Documentation included.

Version 1.4-6 Released 2017-08-17 - Bug fixed in functions relying on simulations if nsims>1e7. - Functions CVwRfromU() / U2CVwR() to calculate CVwR from the upper expanded limit of an ABEL study according to EMA's or Health Canada's rules. - Power and sample size for TOST: Argument alpha restricted to scalar. Internal functions now allow length = 2 (different alpha's for the two null hypotheses). - Man pages for non-inferiority again updated. - Deprecated argument dfCV in expected power functions now completely removed. - Data lazy loading (allows access by the name)

Version 1.4-5 Released 2017-05-19 - Man pages for non-inferiority functions updated to include a description of the underlying hypotheses. - Remaining man pages updated where term 'Null (true) ratio' was mentioned instead of 'True ratio'. - Two new test scrips in /test subdirectory added: test_ABEL.R and test_RSABE which recalculate the sample size tables given in Tothfalusi & Endrenyi "Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs". Contributed by Helmut. - power.scABEL.sdsims() now has an argument design_dta to specify the design via a data.frame. May be useful for considering missing data. Attention! This feature is experimental because the data.frame is not checked at moment. - Bug in power.scABEL.sdsims() fixed which gave power=NA if alpha=0. - Internal change of coding in design helpers (repeated creation of data.frame with design characteristics avoided). - Typo corrected in Expected_Power_for_TOST.pdf. - Updated functions scABEL.ad() and sampleN.scABEL.ad() to allow subject data simulations via power.scABEL.sdsims() if regulator = "EMA". Removed regulator = "ANVISA". Changed the order of sequences to be consistent with the other functions of PowerTOST. - Renamed the variable 'adj. alpha' in sampleN.scABEL.ad() to 'alpha.adj' for consistency with scABEL.ad(). Bug corrected if pre-specified alpha gives TIE <0.05. - Removed regulator = "ANVISA" from pa.scABE(). - Small bug in the S3 method plot for class 'pwrA' corrected (text in the 4th box was truncated on top if saved as a PDF). Replaced text() by TeachingDemos::shadowtext() to enhance legibility (interfered with underlying grid).

Version 1.4-4 Released 2017-03-15 - Administrative change for expected power: cubature::adaptIntegrate replaced by cubature::hcubature to reflect change of function name within package cubature. - Added function power.scABEL.sdsims() to calculate power for the BE decision via scaled (widened) BE acceptance limits (EMA recommended) based on subject data simulations. - Low CV (cosmetic) correction in power.scABEL() introduced. - Removed default CV = 0.3 from scABEL.ad() and sampleN.scABEL.ad(). Stops execution if CV not specified. - DOI for references added in many man pages. THX to Helmut. - Bens description of expected power added in /doc subdirectory - BE_power_sample_size_excerpt.pdf updated to reflect the changes in computation of OwensQ. - Use of removed functions power.scABEL2() and sampleN.scABEL2() in pa.scABE() corrected.

Version 1.4-3 Released 2016-11-01 - Little bug in power.TOST() removed which caused power <0 for large degrees of freedom. - Functions for expected power of the TOST procedure and sample size based on expected power reworked to deal also with uncertainty of theta0 or dealing with both uncertainties of CV and theta0. Contributed mainly by Benjamin Lang. - Functions for expected power of non-inferiority testing and sample size based on that expected power reworked to deal also with uncertainty of theta0 or dealing with both uncertainties of CV and theta0. Contributed mainly by Benjamin Lang. - Deprecated functions power.scABEL2() and sampleN.scABEL2() now removed. regulator = "ANVISA" no longer allowed in the scaled ABEL functions. - OwensQ() simpler/faster implemented. Now based soleyly on numerical integrataion in combination with non-central t. - Owen's T-function, used in OwensQOwen() now based on algorithm AS76 and remarks to that algorithm to avoid numerical errors of the implementation via integrate(). - Data.frame ct9.6.6 in data("data2x2x3") added which was missing a long time ago. - Little inconsistency resolved (by Helmut) in output of power analysis functions w.r.t. ratio or percent. Thanks to user myttiri of BEBA forum for pointing out this inconsistency.

Version 1.4-2 Released 2016-07-14 - Implementation_scaledABE_sims.pdf in \doc subdirectory updated to reflect the changes in code of the scaled ABE functions. - The S3 method plot for class 'pwrA' now has an argument ratiolabel for labeling the axis concerning theta0. Wish of B. Lang. - Various enhancements in man pages. - Power and sample size for FDA RSABE now take into account a CVcap if defined as finite. - Deprecated argument point in functions CI2CV()/CVfromCI() now removed - Misleading term 'Null (true) ratio' in output of sample size functions changed to 'True ratio'. - Minor bug in sampleN.2TOST() fixed.

Version 1.4-1 Published on GitHub 2016-06-14 - Objects of class 'regSet' now have an additional component 'est_method' which controls the simulations via key statistics of the evaluation using EMA's ANOVA or FDA's recommended ISC in xyz.scABEL() functions. - power.scABEL()/power.scABEL2() as well as sampleN.scABEL()/sampleN.scABEL2() are now unified. The regulator component 'est_method' is used for switching between sims based on EMA's ANOVA evaluation or ISC evaluation, respectively. - power.scABEL2()/sampleN.scABEL2() are therefore deprecated and will be removed in future versions. A corresponding warning is thrown. - URL of PowerTOST on GitHub added, URL for bug reports added.

Version 1.3-7 Published on GitHub 2016-6-10 - Bug fix in functions scABEL.ad() and sampleN.scABEL.ad() if former regulator="ANVISA" settings are used.

Version 1.3-6 Released 2016-06-06 via CRAN Released 2016-05-04 to beta-testers - Functions for power and sample size of BE decision of highly variable drugs or drug products via ABEL (average BE with expanding limits) updated to incorporate the reagulatory settings of Health Canada (new functions power.scABEL2() and sampleN.scABEL2(), based on simulations of intra-subject contrasts evaluation). - The same functions now accept as regulator argument an object of class 'regSet' and allow via this way User definitions of regulatory settings. - New function reg_const made visible to define objects of 'regSet'. Class 'regSet' has an S3 print method. - Default value of argument theta0 in functions power.scABEL(), power.scABEL2() and power.RSABE() changed to 0.90 to be in agreement with the setting in the corresponding sample size functions. - Function scABEL (calculation of widened acceptance limits) no longer accepts regulator="USER". This case may be handled via an object with class 'regSet', defined by help of function reg_const(). - Functions CVfromCI()/CI2CV now use pe instead of point as argument due to more consistency with their dual CI.BE(). For backward compatibility point may be used also but then a warning is thrown. Argument point will be removed in future versions. - Functions for expected power for TOST and non-inferiority updated to avoid numerical dificulties with integrate() if method="exact". - Documentation (typos in man pages) rigorously enhanced. Thanks to Helmut and Ben.

Version 1.3-5 Released 2016-04-12 - Functions for expected power for TOST and non-inferiority updated (including corresponding sample size functions). Functions now include an exact method that calculates the expected value of the power with respect to the (prior) distribution of sigma^2 (inverse gamma distribution). Former only an approximation according to Julious/Owen was implemented. Exact implementation contributed by B. Lang. - The functions for iteratively adjusting alpha for the EMA recommended ABEL method now have a new argument tol for convergence tolerance of the internal used uniroot(). Was former hardcoded as tol=1e-5. Contributed by H. Schütz. - plot method of 'pwrA' objects reworked by H. Schütz to avoid some ugly overlays.

Version 1.3-4 Released 2016-03-09 - Bug removed in all sample size estimation functions for scaled ABE which resulted for very low variability in NA for sample size. Thanks to Shuanghe for detecting this bug. - All functions for scaled ABE now have an argument imax=100 for the maximum number of steps for sample size search. Wish of Helmut Schütz.

Version 1.3-3 Released 2016-01-15 via CRAN Released 2016-01-04 to alpha testers - New functions scABEL.ad() and sampleN.scABEL.ad() introduced to iteratively adjust alpha in order to maintain the consumer risk and adapt the sample size to compensate for the possible loss in power for the EMA method of scaled ABE (ABEL). Contributed by Helmut Schütz. - Default theta0 changed to 0.9 in sample size estimation for scaled ABE (as recommended by the Two Laszlos).

Version 1.3-02 Released 2015-12-02 - New functions for calculation of power, type 1 errror and sample size for 2 TOST instances. Contributed by Benjamin Lang.

Version 1.3-01 Released 2015-09-30 Released 2015-09-23 to alpha testers - New functions power.HVNTID() and sampleN.HVNTID() introduced to calculate power and sample size for the BE decision via the FDA procedure for highly variable NTID's (see FDA Dabigatran / rivaroxaban guidances). - power.HVNTID() and power.NTIDFDA() now return the power (p(BE)) and the components for the scaled ABE criterion, the conventional ABE test and the test for the ratio swT/swR <= 2.5 if the argument details=TRUE (wish of Helmut Schütz). - power.RSABE() now returns the power (p(BE)) and the components for the scaled ABE criterion, for the point estimate criterion and for the conventional ABE test alone if the argument details=TRUE. Analogous changes were made in power.scABEL(). See man pages.

Version 1.2-09 Released 2015-08-26 - power.TOST() with method="exact" reworked to give correct values in case alpha>0.5. OwensQOwen adapted to deal with upper integration limit R==Inf. - New power calculation method introduced, direct integration of the bivariate non-central t-distribution (pmvt() from package mvtnorm), also an exact calculation method but with somewhat lower precision and longer run-time. Contributed by Benjamin Lang. - New function power.TOST.sim() to obtain the power via simulations of the TOST. Only intended for checking purposes.

Version 1.2-08 Released 2015-07-10 - Maintenance release with NAMESPACE adapted to import the necessary functions from base R installation (CRAN request). - power.NTIDFDA() and sampleN.NTIDFDA() now have an design argument to choose between 2x2x4 (full replicate 4-period) and 2x2x3 (full replicate 3-period design). - Some minor documentation improvements.

Version 1.2-07 Released 2015-06-03 - Functions power.scABEL(), sampleN.scABEL() now allow the power and sample size calculations for scaled ABEL according to the (inofficial) regulatory settings of Brasilian ANVISA. - Function pa.scABE() also allows regulator="ANVISA". - New helper function scABEL for calculation of the (widened) ABE acceptance limits. - Function power2.TOST() now removed.

Version 1.2-06 Released 2015-01-23 - Function power.TOST() now handles balanced as well as unbalanced studies - Function power2.TOST() is therefore depreciated and will not be contained in later versions. - Internal change in interface to the (hidden) 'raw' power functions (sem used in function calls instead of se, n, bk). - BE_power_sample_size_excerpt.pdf in subdirectory doc changed to reflect the internal changes.

Version 1.2-05 Released 2015-01-07 - Bug in OwensQOwen() removed (debug code left over), THX to Helmut for discovering this. - Function CVfromCI() and alias CI2CV() now handle unbalanced studies, contributed by Benjamin Lang. - Some typos in BE_power_sample_size_excerpt.pdf, found by Ben, corrected.

Version 1.2-04 Released 2014-12-19 - Bug in OwensT(h,a) for the case of a=-Inf removed. Thanks to Benjamin Lang for discovering that nasty bug. - Functions power.noninf(), exppower.TOST(), exppower.noninf() adapted to deal with unbalanced studies. - Function pvalue.TOST() introduced to calculate the two p-values of the TOST procedure. Contributed by Benjamin Lang.

Version 1.2-03 Released 2014-11-13 - sampleN.dp() / power.dp() now cover "incomplete block designs" - minimum sample size in sampleN.dp() introduced to avoid errors for small CV <= 0.1 for the crossover design. - partitition of n(total) to (sequence) groups reworked in most power functions if design is unbalanced. - Liu's 2x2x2 design with 2 repeated measurements in each period added (see ?known.designs under Notes).

Version 1.2-02 Released 2014-10-06 - functions pa.XYZ() adapted so that they also function under R < 3.1.1 (request of U. Ligges from CRAN) although with minpower>=0.5

Version 1.2-01 Released 2014-09-30 - Some minor bugs removed and spelling in man pages corrected. - print method for class pwrA now calls plot(). - power analysis in case of n=12 for the plan now drops n below 12 subjects.

Version 1.2-00 Build 2014-09-19, released to alpha/beta testers only - Contains now functions for power analysis of a sample size plan for ABE (pa.ABE()), scaled ABE (pa.sABE()) and scaled ABE for NTID's (pa.NTIDFDA()) analyzing power for deviations from assumptions for the sample size estimation. - Contains further experimental functions for power calculations / samplesize estimation for dose proportionality studies using the Power model.

Version 1.1-13 Released 2014-08-12 - Design constant bk(ni) for design="paired" corrected. THX to Helmut Schütz who detected that bug.

Version 1.1-12 Released 2014-07-02 - Some minor improvements in documentation. - Function power.TOST() now throws a warning if used with imbalanced designs (n not an even multiple of no. of sequences). - Internal change of start value of sample size search in sampleN.TOST() to avoid failed searches if variability is high and theta0 is near 1.

Version 1.1-11 Released 2014-04-30 - Utility function added which calculates 1-2alpha confidence interval(s) given point est., CV and n using log-tansformed evaluation. - Utility function added which calculates 1-2alpha Fieller confidence interval(s) given point est., CV (, CVb) and n for the ratio of un-transformed means.

Version 1.1-10 Released 2014-01-31 - The package is now pre-compiled to 'byte code' via the compiler package for speed reasons. F.i. OwensQOwen() gains a sixfold boost. - Some bulk code changes to make the power calculations for extreme cases more bullet proof without computation time burden. - Bug removed where noncentrality parameters in the power calculations became NaN (not a number). - OwensQ() now tries to return a value via nct-approximation if its value is due to numeric difficulties falsely equal zero. This approximation is up to 6 decimals correct as far as tested. OwensQ() issues a warning if the nct-approximation is used. - Some improvements in documentation (man pages and *.pdf) to reflect the code changes.

Version 1.1-09 Build 2014-01-03 Not released to the public. - OwensQ() now uses OwensQOwen() in case of high delta and/or high upper integration limit. Thus extreme cases can be handled properly where the former implementation via integrate() was prone to fail. Thanks to Jiri Hofmann and Helmut Schütz for pointing me to such extreme cases.

Version 1.1-08 Released 2013-12-27 - Dates typo in the history (NEWS) corrected from PowerTOST 1.1-00 on. THX to Julien Grassot. - CVfromCI() now accepts either both CLs or one CL and the point estimator, programmed by Helmut Schütz. - power.RatioF() and sampleN.RatioF() now have an argument setseed=TRUE which avoids the dependence if the power from the state of the random number generator (due to the calculation method of pmvt() from package mvtnorm). Thanks to Benjamin Lang.

Version 1.1-07 Released 2013-09-02 - design 2x2x3 (TRT|RTR) implemented in power.scABEL(), sampleN.scABEL() and in power.RSABE(), sampleN.RSABE(). - power.scABEL() now gives a warning if CVwT not equal CVwR in the design 2x3x3, partial replicate. - Simulation details for the full replicate design slightly changed to obtain better numeric agreement of power to subject data sims.

Version 1.1-06 Released 2013-06-21 - Fat bug corrected in sampleN.NTIDFDA() which lead to false sample size for cases where the test of equal variabilities of Test vs. Reference comes into effect.

Version 1.1-05 Released 2013-06-17 - The functions sampleN.scABEL(), sampleN.RSABE() and sampleN.NTIDFDA() return now a data.frame with the input and the sample size result. The "Sample size" column contains the total sample size. The "nlast" column contains the last n value handled. May be useful for restarting. - Start values for sample size search in functions sampleN.scABEL(), sampleN.RSABE() reworked. Failed sample size searches are now more seldom observed. - Flaw in sample size search removed which does not indicate a failed search (n=NA) if started with a too high n (Thanks to Helmut Schütz).

Version 1.1-04 Released 2013-05-15 to alpha testers only - Functions power.NTIDFDA() and sampleN.NTIDFDA() introduced to calculate power and sample size for the BE decision via the method of the FDA for narrow therapeutic index drugs (NTID's). Power and sample size are based on simulations.

Version 1.1-03 Released 2013-05-03 - Flaw in implementing the sims of 2x3x3 design with different intra-subject variabilities in functions power.RSABE() and sampleN.RSABE() as well as in functions power.scABEL() and sampleN.scABEL() disposed. - Methods and implementation details of the simulations for scaled ABE documented in a PDF file in the /doc subdirectory of the package. - Simulation method for the EMA scaled ABEL changed to conform better with power values from simulations via subject data. - Warning section in the help file of power.scABEL() introduced to appoint the fact that simulations via subject data and via the methods implemented in power.scABEL() gave empirical power values that are only approximately in agreement.

Version 1.1-02 Released 2013-02-28 - Functions power.RSABE() and sampleN.RSABE() introduced to calculate power and sample size for the BE decision via linearized scaled ABE criterion as favored method of the FDA. Power and sample size are based on simulations. - Default of nsims changed to 1E5 (suggested by Helmut Schütz). - Argument setseed introduced in the scaled ABE functions to avoid different outcomes depending on the state of the (pseudo) random number generator. If setseed=TRUE a set.seed(123456) is issued before each call of the simulation functions. - Documentation improved.

Version 1.1-00 Released 2013-02-08 - Functions power.scABEL() and sampleN.scABEL() introduced to calculate power and sample size for the BE decision via scaled (widened) BE acceptance limits based on simulations. Thanks to Helmut Schütz for power testing these functions. http://forum.bebac.at/mix_entry.php?id=9997 - False News file included. Sorry for that.

Version 1.0-01 not released, integrated in 1.1-00 - Upper one-sided CL of the CV and therefore argument alpha2 removed from expsampleN.noninf() and expsampleN.TOST() because it lead to some confusion in users thinking this had to do with the algorithm via expected power. - CI2CV() as alias to CVfromCI() introduced because my self always typed this name if aimed to calculate the CV from a given CI. - Function CVCL() now returns a 2 element vector also if an one sided interval is requested.

Version 1.0-00 Released 2012-10-26 - Bugfix in power.noninf() to get the correct power if theta0 is below margin (if margin <1) or theta0 is above margin (if margin >1). power.noninf() calculated up to now the power of an inferiority test. Thanks to Helmut Schütz. - New helper functions CV2mse() and mse2CV. - New function CVCL() to calculate a confidence interval of a CV.

Version 0.9-11 Released 2012-08-07 - Bug fix in power.TOST(), power2.TOST() and power.noninf() to use the right df depending on argument robust. robust=FALSE had used the robust df's. Thanks to Ben.

Version 0.9-10 Released 2012-07-20 - Bug-fix in internal function .Q.integrand() - Bug fix in nmin - must be a multiple of steps to assure balance in sequence groups. Again thanks to Helmut Schütz for detecting both.

Version 0.9-9 Released 2012-07-18 - Workaround introduced to handle numeric problems in integrate() if CV<5.3E-6. Thanks to Helmut Schütz. - Minimum sample size adapted to design used (f.i. n=2 if design="paired").

Version 0.9-8 Released 2012-04-05 (Easter egg) - Functions added for 'expected' power and sample size calculations based on it for the non-inferiority test for sake of completeness. - Functions CVpooled(), exppower.TOST() and expsampleN.TOST() now also implemented for logscale=FALSE, i.e. contain that argument in their calls. - Function OwensQOwen() made public. This is an implementation of the algorithm given by Owen itself in the original paper (Biometrica 1965) via repeated integration by parts. This function is only for comparative purposes. - Function OwensT() made public. It is needed internal in OwensQOwen() but may be useful for other purposes.

Version 0.9-6/7 Released 2012-03-26 - PowerTOST 0.9-7 is a small bug fix PowerTOST required from CRAN. Contains the old PowerTOST number 0.9-6 in the package man page. - Functions added for power and sample size calculations based on non-inferiority t-test. This is not a TOST procedure but eventually useful if the question of 'non-superiority' within a BE study must be evaluated. Hint: Evaluation of Fluctuation in the EMA MR NfG (1999) between modified release formulation and immediate release product.

Version 0.9-4 Released 2012-03-05 - Little bug in sampleN.TOST removed which causes extra doubled output of n and power if n=4. Thanks to Ben: http://forum.bebac.at/mix_entry.php?id=8206

Version 0.9-3 Released 2012-02-13 - Bug in power.TOST removed which prevented calculation of Power according to method = "exact". - Sample size tables for replicate design 2x2x3 in data sub-directory added. - Sample size tables for replicate design 2x4x4 in data sub-directory added. - Scripts in the /test sub-directory made available.

Version 0.9-2 released 2011-12-24 - Function power2.TOST() added to allow power calculations for studies with unbalanced (sequence) groups. - Argument exact replaced by method in power.TOST(), sampleN.TOST(). See help for these functions. - Attention! The sample size for the parallel group design is now the TOTAL sample size (to be consistent across all functions). - Sample size tables added for the 2x2 crossover and for the parallel group design to alleviate validation/qualification of the package. See data(package="PowerTOST"). - Scripts added in the \tests sub-directory that create the sample size tables from the data section. - Updated BE_power_sample_size_excerpt.pdf in the doc sub-directory.

Version 0.9-0 released 2011-12-15 - Paired means design introduced - 'robust' argument added to nearly all functions. With robust=TRUE the degrees of freedom for the so-called robust evaluation (df2 in known.designs() output) will be used. This may be helpful if planning is done for higher order designs evaluated via mixed model or via intra-subject contrasts (aka Senn's basic estimator). - Due to the necessary NAMESPACE from R14.0 on the internal functions (names starting with .) are no longer exported.

Version 0.8-7 released 2011-10-20 - Problem with slash in \name field of manual resolved (requested by B. Ripley)

Version 0.8-6 released 2011-05-18 - Bug removed which gave incorrect exact power values in case of alpha>=0.5 (very unusual setting). Thanks again to Craig Zupke. Cross checked results of power at equivalence margins against SAS Proc Power.

Version 0.8-5 released 2011-05-16 - Code in Owen's Q adapted to account for large delta or large b leading to integrand function almost zero over the whole range which then gave an error in integrate(). Thanks to Craig Zupke.

Version 0.8-4 released 2011-03-11 - Number of maximal steps in sample size search in the sample size functions, formerly hard coded as 50, made accessible to users via argument imax. but needs only adapted in rare extremal cases. - Start value for sample size search improved around theta0=1 (logscale) or theta0=0 (untransformed).

Version 0.8-3 released 2011-01-18 - Bug fix in known.designs().

Version 0.8-2 released 2011-01-10 - error in df for 3x3 and 4x4 crossover removed. - function for pooling CV's from different studies made public see ?CVpooled

Version 0.8-1 released 2010-11-25 - helper function CVfromCI() added to estimate the CV from a confidence interval. Useful if no CV but CI was given in literature.

Version 0.7-3 released 2010-10-09 - bugs in documentation removed. - input argument diff removed from sampleN.TOST, expsampleN.TOST, power.TOST and exppower.TOST

Version 0.7-2 released 2010-08-27 - little bug causing warnings in case of 2x2 alias 2x2x2 removed

Version 0.7-1 released 2010-08-12 - functions added for the power and sample size for the ratio of two means with normally distributed data on the original scale (based on Fieller's confidence (fiducial) interval). AFAIK until now only implemented in the commercial nQuery. - Note that the parameter diff (Null ratio / Null diff.) is now named theta0 since it was annoying for users to call it diff in case of ratios (logscale=TRUE). The parameter diff is still supported but will be ceased in the next release. Therefore a warning is issued if diff is used.

Version 0.6-2 released 2010-07-21 - some internal code consolidation - internal hidden functions .CV2se() and .se2CV() made public - minor enhancements in help pages - more examples - short documentation of used statistical apparatus BE_power_sample_size_excerpt.pdf for classical power / sample size in directory /doc added

Version 0.5-1 - first release to CRAN on 2010-05-07

Reference manual

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install.packages("PowerTOST")

1.4-9 by Detlew Labes, a month ago


https://github.com/Detlew/PowerTOST


Report a bug at https://github.com/Detlew/PowerTOST/issues


Browse source code at https://github.com/cran/PowerTOST


Authors: Detlew Labes [aut, cre] , Helmut Schütz [aut] , Benjamin Lang [aut]


Documentation:   PDF Manual  


Task views: Clinical Trial Design, Monitoring, and Analysis


GPL (>= 2) license


Imports mvtnorm, stats, utils, graphics, grDevices, cubature, TeachingDemos, tufte

Suggests crossdes, knitr, rmarkdown, devtools


Imported by Power2Stage, replicateBE.


See at CRAN