tbk V1.0
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Package‘tcftt’October14,2022Type PackageTitle Two-Sample Tests for Skewed DataVersion0.1.0Author Huaiyu Zhang,Haiyan WangMaintainer Huaiyu Zhang<*************************>Description The classical two-sample t-test works well for the normally distributed data or data with large sample size.The tcfu()and tt()tests implemented in this package providebetter type-I-error control with more accurate power when testing the equality of two-samplemeans for skewed populations having unequal variances.These tests are especially usefulwhen the sample sizes are moderate.The tcfu()uses the Cornish-Fisher expansion to achievea better approximation to the true percentiles.The tt()provides transformations of the Welch'st-statistic so that the sampling distribution become more symmetric.For more technical details, please refer to Zhang(2019)</2097/40235>.License GPL-2Encoding UTF-8LazyData trueRoxygenNote7.1.0Depends R(>=3.1.0)Imports statsNeedsCompilation noRepository CRANDate/Publication2020-07-2314:50:02UTCR topics documented:adjust_power (2)boot_test (3)pauc (3)tcftt (4)tcfu (5)tt (6)t_cornish_fisher (7)t_edgeworth (8)12adjust_power Index10 adjust_power Adjusting power to assure actual size is within significance levelDescriptionIt is common to use Monte Carlo experiments to evaluate the performance of hypothesis tests and compare the empirical power among competing tests.High power is desirable but difficulty arises when the actual sizes of competing tests are not comparable.A possible way of tackling this issue is to adjust the empirical power according to the actual size.This function incorporates three types of power adjustment methods.Usageadjust_power(size,power,method="ZW")Argumentssize the empirical size of a test.power the empirical power of a test.method the power adjustment method.’ZW’is the method proposed by Zhang and Wang(2020),’CYS’is the method proposed by Cavus et al.(2019),and’probit’is the"method1:probit analysis"in Lloyd(2005).Valuethe power value after adjustment.ReferencesLloyd,C.J.(2005).Estimating test power adjusted for size.Journal of Statistical Computation and Simulation,75(11):921-933.Cavus,M.,Yazici,B.,&Sezer,A.(2019).Penalized power approach to compare the power of the tests when Type I error probabilities are munications in Statistics-Simulation and Computation,1-15.Zhang,H.and Wang,H.(2020).Transformation tests and their asymptotic power in two-sample comparisons Manuscript in review.Examplesadjust_power(size=0.06,power=0.8,method= ZW )adjust_power(size=0.06,power=0.8,method= CYS )adjust_power(size=0.06,power=0.8,method= probit )boot_test3 boot_test Bootstrap_t test for two-sample comparisonsDescriptionThis function provides bootstrap approximation to the sampling distribution of the the Welch’s t-statisticUsageboot_test(x1,x2,B=1000,alternative="greater")Argumentsx1thefirst sample.x2the second sample.B number of resampling rounds.Default value is1000.alternative the alternative hypothesis:"greater"for upper-tailed,"less"for lower-tailed,and "two.sided"for two-sided alternative.Valuethe p-value of the bootstrap_t test.Examplesx1<-rnorm(100,0,1)x2<-rnorm(100,0.5,2)boot_test(x1,x2)pauc Power-adjustment based on non-parametric estimation of the ROCcurveDescriptionIt is common to use Monte Carlo experiments to evaluate the performance of hypothesis tests and compare the empirical power among competing tests.High power is desirable but difficulty arises when the actual sizes of competing tests are not comparable.A possible way of tackling this issue is to adjust the empirical power according to the actual size.This function implements the"method 2:non-parametric estimation of the ROC curve"in Lloyd(2005).For more details,please refer to the paper.Usagepauc(stat_h0,stat_ha,target_range_lower,target_range_upper)4tcfttArgumentsstat_h0simulated test statistics under the null hypothesis.stat_ha simulated test statistics under the alternative hypothesis.target_range_lowerthe lower end of the size range.target_range_upperthe upper end of the size range.Valuethe adjusted power.ReferencesLloyd,C.J.(2005).Estimating test power adjusted for size.Journal of Statistical Computation andSimulation,75(11):921-933.Examplesstath0<-rnorm(100)statha<-rnorm(100,mean=1)pauc(stath0,statha,0.01,0.1)tcftt tcftt:Two-Sample Tests for Skewed DataDescriptionThe classical two-sample t-test works well for the normally distributed data or data with largesample size.The tcfu()and tt()tests implemented in this package provide better type I error con-trol with more accurate power when testing the equality of two-sample means for skewed popu-lations having unequal variances.The approximation is especially useful when the sample sizesare moderate.The tcfu()uses the Cornish-Fisher expansion to achieve a better approximation tothe true percentiles.The tt()provides transformations of the Welch’s t-statistic so that the sam-pling distribution become more symmetric.For more technical details,please refer to Zhang(2019)</2097/40235>.tcftt functionsThe function‘tcfu()‘implements the Cornish-Fisher based two-sample test(TCFU)and‘tt()‘im-plements the transformation based two-sample test(TT).The function‘t_edgeworth()‘provides theEdgeworth expansion for cumulative distribution function for the Welch’s t-statistic,and‘t_cornish_fisher()‘provides the Cornish-Fisher expansion for the percentiles.The functions‘adjust_power()‘and‘pauc()‘provide power adjustment for simulation studies so that the actual size of the tests arewithin the significance level.tcfu5 tcfu The TCFU testDescriptionThis test is suitable for testing the equality of two-sample means for the populations having unequal variances.When the populations are not normally distributed,this test can provide better type I error control and more accurate power than a large-sample t-test using normal approximation.The critical values of the test are computed based on the Cornish-Fisher expansion of the Welch’s t-statistic.The order of the Cornish-Fisher expansion is allowed to be0,1,or2.More details please refer to Zhang and Wang(2020).Usagetcfu(x1,x2,effectSize=0,alternative="greater",alpha=0.05,order=2)Argumentsx1thefirst sample.x2the second sample.effectSize the effect size of the test.The default value is0.alternative the alternative hypothesis:"greater"for upper-tailed,"less"for lower-tailed,and "two.sided"for two-sided alternative.alpha the significance level.The default value is0.05.order the order of the Cornish-Fisher expansion.Valuetest statistic,critical value,p-value,reject decision at the given significance level.ReferencesZhang,H.and Wang,H.(2020).Transformation tests and their asymptotic power in two-sample comparisons.Manuscript in review.Examplesx1<-rnorm(20,1,3)x2<-rnorm(21,2,3)tcfu(x1,x2,alternative= two.sided )6tt tt The transformation based testDescriptionThis test is suitable for testing the equality of two-sample means for the populations having unequal variances.When the populations are not normally distributed,the sampling distribution of the Welch’s t-statistic may be skewed.This test conducts transformations of the Welch’s t-statistic to make the sampling distribution more symmetric.For more details,please refer to Zhang and Wang (2020).Usagett(x1,x2,alternative="greater",effectSize=0,alpha=0.05,type=1)Argumentsx1thefirst sample.x2the second sample.alternative the alternative hypothesis:"greater"for upper-tailed,"less"for lower-tailed,and "two.sided"for two-sided alternative.effectSize the effect size of the test.The default value is0.alpha the significance level.The default value is0.05.type the type of transformation to be used.Possible choices are1to4.They cor-respond to the TT1to TT4in Zhang and Wang(2020).Which type providesthe best test depends on the relative skewness parameter A in Theorem2.2ofZhang and Wang(2020).In general,if A is greater than3,type=3is recom-mended.Otherwise,type=1or4is recommended.The type=2transformationmay be more conservative in some cases and more liberal in some other casesthan the type=1and4transformations.For more details,please refer to Zhangand Wang(2020).Valuetest statistic,critical value,p-value,reject decision at the given significance level.ReferencesZhang,H.and Wang,H.(2020).Transformation tests and their asymptotic power in two-sample comparisons Manuscript in review.t_cornish_fisher7Examplesx1<-rnorm(20,1,3)x2<-rnorm(21,2,3)tt(x1,x2,alternative= two.sided ,type=1)#Negative lognormal versus normal datan1=50;n2=33x1=-rlnorm(n1,meanlog=0,sdlog=sqrt(1))-0.3*sqrt((exp(1)-1)*exp(1))x2=rnorm(n2,-exp(1/2),0.5)tt(x1,x2,alternative= less ,type=1)tt(x1,x2,alternative= less ,type=2)tt(x1,x2,alternative= less ,type=3)tt(x1,x2,alternative= less ,type=4)#Lognormal versus normal datan1=50;n2=33x1=rlnorm(n1,meanlog=0,sdlog=sqrt(1))+0.3*sqrt((exp(1)-1)*exp(1))x2=rnorm(n2,exp(1/2),0.5)tt(x1,x2,alternative= greater ,type=1)tt(x1,x2,alternative= greater ,type=2)tt(x1,x2,alternative= greater ,type=3)tt(x1,x2,alternative= greater ,type=4)t_cornish_fisher Cornish-Fisher expansion for Welch’s t-statisticDescriptionThis function provides approximation for the quantile function of the sampling distribution of the Welch’s t-statistic using Cornish-Fisher expansion(up to second order).Usaget_cornish_fisher(p,order=2,n1,n2,mu1,mu2,sigma1,sigma2,gamma1,gamma2,tau1,tau2)Argumentsp a probability value.order the order of Cornish-Fisher expansion.Valid options are0,1,and2.If set to0,it reduces to a normal approximation and it returns the p-th percentile of standardnormal distribution.n1sample size for the sample from thefirst population.n2sample size for the sample from the second population.mu1mean of thefirst population.mu2mean of the second population.sigma1standard deviation of thefirst population.sigma2standard deviation of the second population.gamma1skewness of thefirst population.gamma2skewness of the second population.tau1kurtosis of thefirst population.tau2kurtosis of the second population.ValueCornish-Fisher expansion value evaluated at p.Examplest_cornish_fisher(0.9,order=2,n1=60,n2=30,mu1=0,mu2=0,sigma1=1,sigma2=0.5,gamma1=1,gamma2=0,tau1=6,tau2=0)t_cornish_fisher(0.3,order=1,n1=60,n2=30,mu1=0,mu2=0,sigma1=1,sigma2=0.5,gamma1=1,gamma2=0,tau1=6,tau2=0)t_edgeworth Edgeworth expansion for Welch’s t-statisticDescriptionThis function provides approximation for the cumulative distribution function of the sampling dis-tribution of the Welch’s t-statistic using Normal distribution,first order or second order Edgeworth expansion.Usaget_edgeworth(x,order=2,n1,n2,mu1,mu2,sigma1,sigma2,gamma1,gamma2,tau1,tau2)Argumentsx a real number.order the order of edgeworth expansion.Valid options are0,1,and2.If set to0, it reduces to approximation based on the central limit theorem and returns theCDF of standard normal distribution evaluated at x.n1sample size for the sample from thefirst population.n2sample size for the sample from the second population.mu1mean of thefirst population.mu2mean of the second population.sigma1standard deviation of thefirst population.sigma2standard deviation of the second population.gamma1skewness of thefirst population.gamma2skewness of the second population.tau1kurtosis of thefirst population.tau2kurtosis of the second population.ValueEdgeworth expansion evaluated at x.Examplest_edgeworth(1.96,order=2,n1=20,n2=30,mu1=0,mu2=0,sigma1=1,sigma2=0.5,gamma1=1,gamma2=0,tau1=6,tau2=0)Indexadjust_power,2boot_test,3pauc,3t_cornish_fisher,7t_edgeworth,8tcftt,4tcfu,5tt,610。
IntroductionWelcome to this support pack on assessment for learning tools in the classroom.This are a selection of assessment tools designed for Level 1 and below learners for use in the classroom by teachers who want to assess progress or want to help students to engage in self assessment.Almost every learning activity can present opportunities for assessment. The ideas contained in this pack can be used on a regular basis to support and inform teaching and learning and to help to assess a learners progress.Both teachers and students can use the feedback gained from the activities in this pack to improve learning. The kind of feedback which can be gather includes:•How students are learning•Their progress•The nature of their understanding•Any difficulties they are havingThese activities are intended to be a guide as to activities that can be done to assess learning in the classroom as well as motivate learners by helping them to identify their progress, any barriers to that progress and therefore start to become more independent during their learning.These can and should be adapted to the centres own learners and we encourage centres to do so.These activities represent some activities which could be used to assess learner progress. The activities are not exhaustive and are designed to compliment rather than replace a centres own usual practices.WHERE AM I NOW?What am i good at? What have I done well? What am I learning?What am I going to do next?What are my goals? What ways can I use to get better at this?What do i need to do? What people can support me?What other supports do i need? What ways can i use tohelp myself? WHERE AM I GOING?WHAT DO I NEED TO GET THERE?I am able tocomplete work I listened well I help others if I listen to others I raise my hand Green= Excellent Amber= OK Red= Have to improveunderstandI feel confident toRequest a Selfie If you are proudof your work, Complete theproud tokentake a lolly stick and clip these to your workand we will take a photo of you with yourworkPearson Proud TokenI am proud of this workbecause...........................................................................................................Pearson AirwaysAirwaysAirwaysBoarding CardLanding CardWhat I already know about this topic:What I discovered today:In this lesson I want to achieve:A question I want to know the answer to is:Did you achieve your goal? How?A question I still have about this learning is?Tell one person in the room what you like about theirwork.Write down the answer here.Ask one person in the room a question about their work. Write down the answer here.Give the person a positive suggestion on how they can improve their work.Write it down here.Person receiving feedback: ______________________________ Person giving feedback:__________________________________。