The binom Package. February 13, 2007

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1 The biom Package February 13, 2007 Title Biomial Cofidece Itervals For Several Parameterizatios Versio Author Sudar Dorai-Raj Costructs cofidece itervals o the probability of success i a biomial experimet via several parameterizatios Maitaier Sudar Dorai-Raj <sudar.dorai-raj@pdf.com> Depeds lattice, polyom Licese GPL R topics documeted: biom-iteral biom.bayes biom.cloglog biom.cofit biom.coverage biom.legth biom.logit biom.lrt biom.optim biom.plot biom.power biom.probit biom.profile biom.sim cloglog.sample.size tkbiom.power Idex 20 1

2 2 biom.bayes biom-iteral Iteral biom fuctios Iteral fuctios for biom package. var.asymp(p, = 1) var.cloglog(p, = 1) var.logit(p, = 1) var.probit(p, = 1) ldbiom(x, size, prob, log = TRUE) Iteral fuctios for biom package. biom.bayes Biomial cofidece itervals usig Bayesia iferece Uses a beta prior o the probability of success for a biomial distributio, determies a two-sided cofidece iterval from a beta posterior. biom.bayes(x,, cof.level = 0.95, type = c("highest", "cetral"), prior.shape1 = 0.5, prior.shape2 = 0.5, tol =.Machie$double.eps^0.5, maxit = 1000,...) x cof.level type Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. The type of cofidece iterval (see ). prior.shape1 The value of the first shape parameter to be used i the prior beta.

3 biom.bayes 3 prior.shape2 The value of the secod shape parameter to be used i the prior beta. tol maxit... Igored. A tolerace to be used i determiig the highest probability desity iterval. Maximum umber of iteratios to be used i determiig the highest probability iterval. Usig the cojugate beta prior o the distributio of p (the probability of success) i a biomial experimet, costructs a cofidece iterval from the beta posterior. From Bayes theorem the posterior distributio of p give the data x is: p x ~ Beta(x + prior.shape1, - x + prior.shape2) The default prior is Jeffrey s prior which is a Beta(0.5, 0.5) distributio. Thus the posterior mea is (x + 0.5)/( + 1). The default type of iterval costructed is "highest" which computes the highest probability desity (hpd) iterval which assures the shortest iterval possible. The hpd itervals will achieve a probability that is withi tol of the specified cof.level. Settig type to "cetral" costructs itervals that have equal tail probabilities. If 0 or successes are observed, a oe-sided cofidece iterval is retured. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval. Refereces Gelma, A., Carli, J. B., Ster, H. S., ad Rubi, D. B. (1997) Bayesia Data Aalysis, Lodo, U.K.: Chapma ad Hall. biom.cofit, biom.cloglog, biom.logit, biom.probit biom.bayes(x = 0:10, = 10, tol = 1e-9)

4 4 biom.cloglog biom.cloglog Biomial cofidece itervals usig the cloglog parameterizatio Uses the complemetary log (cloglog) parameterizatio o the observed proportio to costruct cofidece itervals. biom.cloglog(x,, cof.level = 0.95,...) x Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. cof.level The level of cofidece to be used i the cofidece iterval.... igored For derivatios see doc/biom.pdf. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval. biom.cofit, biom.bayes, biom.logit, biom.probit, biom.coverage biom.cloglog(x = 0:10, = 10)

5 biom.cofit 5 biom.cofit Biomial cofidece itervals Uses eight differet methods to obtai a cofidece iterval o the biomial probability. biom.cofit(x,, cof.level = 0.95, methods = "all",...) x cof.level methods Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. Which method to use to costruct the iterval. Ay combiatio of c("exact", "ac", "asymptotic", "wilso", "prop.test", "bayes", "logit", "cloglog", "probit") is allowed. Default is "all".... Additioal argumets to be passed to biom.bayes. Nie methods are allowed for costructig the cofidece iterval(s): exact - Pearso-Klopper method. asymptotic - the text-book defiitio for cofidece limits o a sigle proportio usig the Cetral Limit Theorem. agresti-coull - Agresti-Coull method. wilso - Wilso method. prop.test - equivalet to prop.test(x = x, =, cof.level = cof.level)$cof.it. bayes - see biom.bayes. logit - see biom.logit. cloglog - see biom.cloglog. probit - see biom.probit. profile - see biom.profile. By default all eight are estimated for each value of x ad/or. For the "logit", "cloglog", "probit", ad "profile" methods, the cases where x == 0 or x == are treated separately. Specifically, the lower boud is replaced by (alpha/2)^ ad the upper boud is replaced by (1-alpha/2)^. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval for all the methods i "methods".

6 6 biom.coverage Refereces A. Agresti ad B.A. Coull (1998), Approximate is better tha "exact" for iterval estimatio of biomial proportios, America Statisticia, 52: R.G. Newcombe, Logit cofidece itervals ad the iverse sih trasformatio (2001), America Statisticia, 55: L.D. Brow, T.T. Cai ad A. DasGupta (2001), Iterval estimatio for a biomial proportio (with discussio), Statistical Sciece, 16: Gelma, A., Carli, J. B., Ster, H. S., ad Rubi, D. B. (1997) Bayesia Data Aalysis, Lodo, U.K.: Chapma ad Hall. biom.bayes, biom.logit, biom.probit, biom.cloglog, biom.coverage, prop.test biom.cofit(x = c(2, 4), = 100, tol = 1e-8) biom.coverage Probability coverage for biomial cofidece itervals Determies the probability coverage for a biomial cofidece iterval. biom.coverage(p,, cof.level = 0.95, method = "all",...) p cof.level method The (true) probability of success i a biomial experimet. Vector of umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. Either a character strig to be passed to biom.cofit or a fuctio that computes the upper ad lower cofidece boud for a biomial proportio. If a fuctio is supplied, the first three argumets must be the same as biom.cofit ad the retur value of the fuctio must be a data.frame with colum headers "method", "lower", ad "upper". See biom.cofit for available methods. Default is "all".... Additioal parameters to be passed to biom.cofit. Oly used whe method is either "bayes" or "profile"

7 biom.legth 7 Derivatios are based o the results give i the refereces. Methods whose coverage probabilities are cosistetly closer to 0.95 are more desireable. Thus, Wilso s, logit, ad cloglog appear to be good for this sample size, while Jeffreys, asymptotic, ad prop.test are poor. Jeffreys is a variatio of Bayes usig prior shape parameters of 0.5 ad havig equal probabilities i the tail. The Jeffreys equal-tailed iterval was created usig biom.bayes usig (0.5,0.5) as the prior shape parameters ad type = "cetral". A data.frame cotaiig the "method" used, "", "p", ad the coverage probability, C(p,). Refereces L.D. Brow, T.T. Cai ad A. DasGupta (2001), Iterval estimatio for a biomial proportio (with discussio), Statistical Sciece, 16: L.D. Brow, T.T. Cai ad A. DasGupta (2002), Cofidece Itervals for a Biomial Proportio ad Asymptotic Expasios, Aals of Statistics, 30: biom.cofit, biom.legth biom.coverage(p = 0.5, = 50) biom.legth Expected legth for biomial cofidece itervals Determies the expected legth for a biomial cofidece iterval. biom.legth(p,, cof.level = 0.95, method = "all",...)

8 8 biom.legth p cof.level method The (true) probability of success i a biomial experimet. Vector of umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. Either a character strig to be passed to biom.cofit or a fuctio that computes the upper ad lower cofidece boud for a biomial proportio. If a fuctio is supplied, the first three argumets must be the same as biom.cofit ad the retur value of the fuctio must be a data.frame with colum headers "method", "lower" ad "upper". See biom.cofit for available methods. Default is "all".... Additioal parameters to be passed to biom.cofit. Oly used whe method is either "bayes" or "profile" Derivatios are based o the results give i the refereces. Methods whose legth probabilities are cosistetly closer to 0.95 are more desireable. Thus, Wilso s, logit, ad cloglog appear to be good for this sample size, while Jeffreys, asymptotic, ad prop.test are poor. Jeffreys is a variatio of Bayes usig prior shape parameters of 0.5 ad havig equal probabilities i the tail. The Jeffreys equal-tailed iterval was created usig biom.bayes usig (0.5,0.5) as the prior shape parameters ad type = "cetral". A data.frame cotaiig the "method" used, "", "p", ad the average legth, L(p,). Refereces L.D. Brow, T.T. Cai ad A. DasGupta (2001), Iterval estimatio for a biomial proportio (with discussio), Statistical Sciece, 16: L.D. Brow, T.T. Cai ad A. DasGupta (2002), Cofidece Itervals for a Biomial Proportio ad Asymptotic Expasios, Aals of Statistics, 30: biom.cofit, biom.coverage biom.legth(p = 0.5, = 50)

9 biom.logit 9 biom.logit Biomial cofidece itervals usig the logit parameterizatio Uses the logistic (logit) parameterizatio o the observed proportio to costruct cofidece itervals. biom.logit(x,, cof.level = 0.95,...) x Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. cof.level The level of cofidece to be used i the cofidece iterval.... igored For derivatios see doc/biom.pdf. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval. biom.cofit, biom.bayes, biom.logit, biom.probit, biom.coverage biom.logit(x = 0:10, = 10)

10 10 biom.lrt biom.lrt Biomial cofidece itervals usig the lrt likelihood Uses the lrt likelihood o the observed proportio to costruct cofidece itervals. biom.lrt(x,, cof.level = 0.95, bayes = FALSE, cof.adj = FALSE, plot = FALSE,...) x cof.level bayes cof.adj plot... igored Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. logical; if TRUE use a Bayesia correctio at the edges. Specfically, a beta prior with shape parameters 0.5 is used. If bayes is umeric, it is assumed to be the parameters to beta distributio. logical; if TRUE 0 or 100% successes retur a oe-sided cofidece iterval logical; if TRUE a plot showig the the square root of the biomial deviace with referece lies for mea, lower, ad upper bouds. This argumet ca also be a list of plottig parameters to be passed to xyplot. Cofidece itervals are based o profilig the biomial deviace i the eighbourhood of the MLE. If x == 0 or x == ad bayes is TRUE, the a Bayesia adjustmet is made to move the loglikelihood fuctio away from If. Specifically, these values are replaced by (x + 0.5)/( + 1), which is the posterier mode of f(p x) usig Jeffrey s prior o p. Furthermore, if cof.adj is TRUE, the the upper (or lower) boud uses a 1 - alpha cofidece level. Typically, the observed mea will ot be iside the estimated cofidece iterval. If bayes is FALSE, the the Clopper-Pearso exact method is used o the edpoits. This teds to make cofidece itervals at the ed too coservative, though the observed mea is guarateed to be withi the estimated cofidece limits. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval.

11 biom.optim 11 biom.cofit, biom.bayes, biom.cloglog, biom.logit, biom.probit, biom.coverage, cofit i package MASS, family, glm biom.lrt(x = 0:10, = 10) biom.optim Optimal biomial cofidece itervals Uses optimizatio to miimize the itegrated mea squared error betwee the calculated coverage ad the desired cofidece level for a give biomial cofidece iterval. biom.optim(, cof.level = 0.95, method = biom.lrt, k = %/%2 + 1, p0 = 0, trasform = TRUE, plot = FALSE, tol =.Machie$double.eps^0.5, start = NULL,...) cof.level method k p0 trasform plot tol start The umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. The method used to estimate the cofidece iterval. See. The miimum probability of success to allow i the optimizatio. See. logical; If TRUE the optimizer will do a ucostraied optimizatio o the sigficace probability i the logit space. logical; If TRUE the results are set to biom.plot. The miimum sigificace level to allow i the optimizatio. See. A startig value o the optimal cofidece level.... Additioal argumets to pass to optim.

12 12 biom.optim This fuctio miimizes the squared error betwee the expected coverage probability ad the desired cofidece level. α opt = arg mi α 1 0 [C(p, ) (1 α) 2 dp The optimizer will adjust cofidece itervals for all x = 0 to depedig o the value of k provided. If k is oe, oly the cofidece levels for x = 0 ad are adjusted. If k = [/2] the all cofidece itervals are adjusted. This assumes the cofidece itervals are the same legth for x = x[k] ad x[ - k + 1], which is the case for all methods provided i this package except biom.cloglog. A list with the followig elemets: par Fial cofidece levels. The legth of this vector is k. value couts covergece message cofit The fial miimized value from optim. The umber of fuctio ad gradiet calls from optim. Covergece code from optim. Ay message retured by the L-BFGS-B or BFGS optimizer. A data.frame retured from a call to method usig the optimized cofidece levels. ormal-bracket77bracket-ormal biom.cofit, biom.plot, biom.coverage, optim biom.optim(10, k = 1) ## determie optimal sigificace for x = 0, 10 oly biom.optim(3, method = biom.wilso) ## determie optimal sigificace for all x

13 biom.plot 13 biom.plot Coverage plots for biomial cofidece itervals Costructs coverage plots for biomial cofidece itervals. biom.plot(, method = biom.lrt, p = 500, cof.level = 0.95, actual = cof.level, type = c("xyplot", "levelplot"), tol =.Machie$double.eps^0.5,...) cof.level p method actual type tol The umber of idepedet trials i the biomial experimet. The level of cofidece to be used i the cofidece iterval. Number of poits to use i the plot. The method used to estimate the cofidece iterval. The actual cofidece iterval used i the cofidece iterval. See. See. The miimum probability of success to use i the plot.... Additioal argumets to pass to pael.xyplot or pael.levelplot. If type is "xyplot", a lie plot is created with coverage o the y-axis ad biomial probability o the x-axis. A separate pael for every is provided. If actual is provided the a horizotal referece lie is added to the plot. This is oly useful whe actual is differet from cof.level, as is the case whe callig biom.optim. If type is "levelplot", a image plot is created with x = 0 to o the vertical axis ad biomial probability o the horizotal axis. Each row i the plot will be the cofidece level for a give x. The color of the cofidece iterval is determied by the coverage probability. The argumet must oly be of legth oe. If ot, oly the first will be used ad a warig is issued. I either plot type, the umber of poits at which the coverage probability is determied is specified by p. Icreasig p gives a fier graularity but performace will suffer. A object of class trellis.

14 14 biom.power biom.cofit, biom.optim, xyplot, levelplot biom.plot(5, type = "levelplot") biom.plot(c(3, 5, 10, 25), type = "xyplot") biom.power Power curves for biomial parameterizatios Uses Wald statistics to compute power curves for several parameterizatios. biom.power(p.alt, = 100, p = 0.5, alpha = 0.05, phi = 1, alterative = c("two.sided", "greater", "less"), method = c("cloglog", "logit", "probit", "asymp", "lrt", "exact")) p.alt p alpha phi alterative method Probability of success uder the alterative hypothesis. Vector of umber of idepedet trials i the biomial experimet. Probability of success uder the ull hypothesis. Type-I error rate. Overdispersio parameter. Type of alterative hypothesis. The method used to compute power. For derivatios see doc/biom.pdf. The estimated probability of detectig the differece betwee p.alt ad p. biom.cofit, biom.bayes, biom.logit, biom.probit, biom.coverage

15 biom.probit 15 biom.power(0.95, alterative = "greater") biom.probit Biomial cofidece itervals usig the probit parameterizatio Uses the probit parameterizatio o the observed proportio to costruct cofidece itervals. biom.probit(x,, cof.level = 0.95,...) x Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. cof.level The level of cofidece to be used i the cofidece iterval.... igored For derivatios see doc/biom.pdf. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval. biom.cofit, biom.bayes, biom.probit, biom.logit, biom.coverage biom.probit(x = 0:10, = 10)

16 16 biom.profile biom.profile Biomial cofidece itervals usig the profile likelihood Uses the profile likelihood o the observed proportio to costruct cofidece itervals. biom.profile(x,, cof.level = 0.95, maxsteps = 50, del = zmax/5, bayes = TRUE, plot = FALSE,...) x Vector of umber of successes i the biomial experimet. Vector of umber of idepedet trials i the biomial experimet. cof.level The level of cofidece to be used i the cofidece iterval. maxsteps The maximum umber of steps to take i the profiles. del The size of the step to take bayes logical; if TRUE use a Bayesia correctio at the edges. plot logical; if TRUE plot the profile with a splie fit.... igored Cofidece itervals are based o profilig the biomial deviace i the eighbourhood of the MLE. If x == 0 or x == ad bayes is TRUE, the a Bayesia adjustmet is made to move the loglikelihood fuctio away from If. Specifically, these values are replaced by (x + 0.5)/( + 1), which is the posterier mode of f(p x) usig Jeffrey s prior o p. Typically, the observed mea will ot be iside the estimated cofidece iterval. If bayes is FALSE, the the Clopper- Pearso exact method is used o the edpoits. This teds to make cofidece itervals at the ed too coservative, though the observed mea is guarateed to be withi the estimated cofidece limits. A data.frame cotaiig the observed proportios ad the lower ad upper bouds of the cofidece iterval.

17 biom.sim 17 biom.cofit, biom.bayes, biom.cloglog, biom.logit, biom.probit, biom.coverage, cofit i package MASS, family, glm biom.profile(x = 0:10, = 10) biom.sim Simulates cofidece itervals for biomial data Simulates biomial data for testig cofidece iterval coverage. biom.sim(m = 200, = 100, p = 0.5, cof.level = 0.95, methods = "all",...) M p cof.level methods Number of simulatios to create. Vector of umber of idepedet trials i the biomial experimet. Probability of success uder the ull hypothesis. The level used i computig the cofidece iterval. The method used to compute power.... Additioal argumets to pass to biom.cofit M biomial observatios are created usig rbiom(m,, p). The average umber of times a cofidece iterval covers p is retured. The estimated coverage based o which method is requested. biom.cofit, biom.bayes, biom.logit, biom.probit, biom.coverage

18 18 cloglog.sample.size cloglog.sample.size Power ad sample size Power ad sample size for a biomial proportio usig the cloglog parameterizatio. cloglog.sample.size(p.alt, = NULL, p = 0.5, power = 0.8, alpha = 0.05, alterative = c("two.sided", "greater", "less"), exact. = FALS recompute.power = FALSE, phi = 1) p.alt The alterative proportio i a oe-sample test. The sample size i a oe-sample test. p The ull proportio i a oe-sample test. Default is 0.5. power The desired power level. Default is alpha The desired alpha level - probability of a Type I error. Default is alterative Nature of alterative hypothesis. Oe of "two.sided", "greater", "less". exact. logical; If TRUE, the computed sample size will ot be rouded up. Default is FALSE. recompute.power logical; If TRUE, after the sample size is computed, the power will be recomputed. This is oly advatageous whe the sample size is rouded up. Default is FALSE. phi Dispersio parameter by which to iflate (phi > 1) or deflate (phi < 1) variace. Default is 1. This fuctio ca be used to calculate sample size, power or miimum detectable differece. It determies what to compute base o the argumets provided. If p.alt is give, but is ot, the sample size is computed. If p.alt is give alog with, the the power is computed. If oly is provided, the miimum detectable differece is computed usig the default power of A data.frame cotaiig the power, sample size ad all of the iput which was used to perform the computatios.

19 tkbiom.power 19 biom.cofit cloglog.sample.size(p.alt = 0.8) cloglog.sample.size( = 20) cloglog.sample.size( = 20, power = 0.9) tkbiom.power Power curves for biomial parameterizatios A Tcl/Tk graphics wrapper for biom.power. tkbiom.power() A wrapper for biom.power that creates power curves based o user iput. Noe. biom.power, biom.cofit, biom.bayes, biom.logit, biom.probit, biom.coverage biom.power(0.95, alterative = "greater")

20 Idex Topic hplot biom.plot, 12 Topic htest biom.bayes, 2 biom.cloglog, 3 biom.cofit, 4 biom.coverage, 5 biom.legth, 7 biom.logit, 8 biom.lrt, 9 biom.optim, 10 biom.plot, 12 biom.power, 13 biom.probit, 14 biom.profile, 15 biom.sim, 16 cloglog.sample.size, 17 tkbiom.power, 18 Topic iteral biom-iteral, 1 Topic models biom.bayes, 2 biom.cloglog, 3 biom.cofit, 4 biom.coverage, 5 biom.legth, 7 biom.logit, 8 biom.lrt, 9 biom.optim, 10 biom.plot, 12 biom.power, 13 biom.probit, 14 biom.profile, 15 biom.sim, 16 cloglog.sample.size, 17 tkbiom.power, 18 Topic optimize biom.optim, 10 Topic uivar biom.bayes, 2 biom.cloglog, 3 biom.cofit, 4 biom.coverage, 5 biom.legth, 7 biom.logit, 8 biom.lrt, 9 biom.optim, 10 biom.plot, 12 biom.power, 13 biom.probit, 14 biom.profile, 15 biom.sim, 16 cloglog.sample.size, 17 tkbiom.power, 18 biom-iteral, 1 biom.agresti.coull (biom.cofit), 4 biom.asymp (biom.cofit), 4 biom.bayes, 2, 4, 5, 8, 10, 13, 14, 16, 18 biom.cloglog, 3, 3, 5, 10, 11, 16 biom.cofit, 3, 4, 4, 6 8, 10, 11, 13, 14, 16, 18 biom.coverage, 4, 5, 5, 8, 10, 11, 13, 14, 16, 18 biom.exact (biom.cofit), 4 biom.legth, 6, 7 biom.logit, 3 5, 8, 8, 10, 13, 14, 16, 18 biom.lrt, 9 biom.optim, 10, 12, 13 biom.plot, 10, 11, 12 biom.power, 13, 18 biom.probit, 3 5, 8, 10, 13, 14, 14, 16, 18 biom.profile, 5, 15 biom.prop.test (biom.cofit), 4 biom.sim, 16 biom.wilso (biom.cofit), 4 20

21 INDEX 21 cloglog.sample.size, 17 cofit, 10, 16 family, 10, 16 glm, 10, 16 ldbiom (biom-iteral), 1 levelplot, 13 list, 9 optim, 10, 11 pael.levelplot, 12 pael.xyplot, 12 prop.test, 5 tkbiom.power, 18 var.asymp (biom-iteral), 1 var.cloglog (biom-iteral), 1 var.logit (biom-iteral), 1 var.probit (biom-iteral), 1 xyplot, 9, 13

Sample Size Determination (Two or More Samples)

Sample Size Determination (Two or More Samples) Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie