user manual - Institute of Bioinformatics
Contents
1. gt rococo test x2 y2 similarity gauss r 0 1 alternative greater Robust Gamma Rank Correlation data x2 and y2 length 15 Similarity gauss rx 0 1 ry 0 1 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 7638568 estimated p value lt 2 2e 16 0 of 1000 values It is also possible to specify different values of r for x and y observations Analogously to the parameter similarity this can be done by supplying a two element vector to the parameter r gt rococo test x2 y2 similarity c linear gauss r c 0 05 0 1 alternative greater 4 Adjusting Similarities and t Norms 11 Robust Gamma Rank Correlation data x2 and y2 length 15 similarity for x linear rx 0 05 similarity for y gauss ry 0 1 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 742687 estimated p value 0 002 2 of 1000 values It should be clear from the formulas in Table 1 that r 0 is either invalid or does not make sense The rococo package still admits choosing a zero value In this case rococo makes an automatic adjustment of r to 10 of the interquartile range the difference between the 75 and the 25 quantile of the observation under consideration gt rococo test x2 y2 similarity c linear gauss r 0 alternative greater Robust Gamma Rank Correlation data x2 and y2 length 15 similarity for x linear rx 02. RoCoCo An R Package Implementing a Robust Rank Correlation Coefficient and a Corresponding Test Ulrich Bodenhofer and Martin Krone Institute of Bioinformatics Johannes Kepler University Altenberger Str 69 4040 Linz Austria Department of Computer Science Ostfalia University of Applied Sciences Salzdahlumer Str 46 48 38302 Wolfenb ttel Germany rococo bioint jku at Version 1 1 1 February 27 2014 Institute of Bioinformatics Tel 43 732 2468 4520 Johannes Kepler University Linz Fax 43 732 2468 4539 A 4040 Linz Austria http www bioinf jku at 2 Contents Scope and Purpose of this Document This document is a user manual for the R package rococo It is only meant as a gentle introduction into how to use the basic functions implemented in this package Not all features of the R package are described in full detail Such details can be obtained from the documentation enclosed in the R package Further note the following 1 this is not an introduction to robust rank correlation 2 this is not an introduction to R If you lack the background for understanding this manual you first have to read literature on these subjects Contents 1 Introduction 2 Installation 2 1 Installation via CRAN 2 2 a 2 2 Manual installation oa aa ee 2 3 Compatibility 1ssues scine ee eet ne a ana ea A Se ee 3 Getting Started 4 Adjusting Similarities and t Norms 4 1 Backfround Jou a Soho eel Bai SR ee Pod
3. 1357292 similarity for y gauss ry 0 02639592 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 6497884 estimated p value lt 2 2e 16 0 of 1000 values gt IQR x2 0 1 1 0 1357292 gt IQR y2 0 1 1 0 02639592 In the results above note the values rx and ry in the output of rococo test These are the values that have been specified or that have been determined automatically The computations of 10 of the interquartile range should demonstrate that this is what takes place when r is set to 0 Note that this is also done by default if the argument r is not specified 4 4 Choosing the t Norm for Aggregation As mentioned above the robust gamma rank correlation coefficient further requires to specify a t norm T that is used for aggregation of the ordering measures from x and y observations The rococo package offers three built in t norms that can be selected by specifying the tnorm argu ment when calling the functions rococo and rococo test Table 2 provides an overview Here is an example that uses the product t norm for aggregation 12 4 Adjusting Similarities and t Norms Table 2 Overview of strict fuzzy orderings implemented in the rococo package Setting t Norm min default Tu x y min z y prod Tp z y x y lukasiewicz T z y max 0 x y 1 gt rococo test x2 y2 similarity c linear gauss tnorm prod alter
4. a negative correlation This p value is contained in the slot p value of the output object the test returns This output object further contains a slot count that corresponds to the absolute number of times a the test statistics absolute value exceeded the absolute value of the test statistic for the unshuffled data in case we perform a two sided test a the test statistic was greater than the test statistic for the unshuffled data in case we perform a one sided test with alternative hypothesis that there is a positive correlation and the test statistic was less than the test statistic for the unshuffled data in case we perform a one sided test with alternative hypothesis that there is a negative correlation Please note that these p values are only estimates The smaller the number of shuffles the higher the variance of the estimates This number of shuffles performed by rococo test is controlled by the parameter numtests the default is 1000 We concur that 1000 samples are sufficient at least with high probability if one wants to test whether the association is significant with a significance threshold of 95 or 99 If the user however is interested in much more exact estimates of the p value or if he she wants to test against much more stringest significance thresholds it may be necessary to perform much higher numbers of shuffles Needless to mention this will also result in an increase of computation times where the computat
5. which are identified by the similarity that is used to define the strict fuzzy ordering for more details see 4 5 Table 1 provides an overview Obviously in all five cases the strict fuzzy ordering R is computed from the similarity in the following way 1 E x x ifa gt a 0 otherwise R a 2 4 Adjusting Similarities and t Norms Table 1 Overview of strict fuzzy orderings implemented in the rococo package Setting Similarity Strict fuzzy ordering linear E x 2 max 0 1 x R x 2 max 0 min 1 4 2 x exp E x 2 exp 4 2 2 R x x max 0 1 exp 4 a 2 1 exp six e 2 ife lt a gauss E x x exp 522 x 2 R z x exp g72 t 2 ifas f 0 otherwise 1 if r r lt 1 ifr epstol E x x ae x Isr R x 2 ne as a 0 otherwise 0 otherwise 1 ife a 1 ifr classical E x x i K R x x w au 0 otherwise 0 otherwise Further note that the choices epstol and classical are not continuous The former is the well known intolerant similarity that has been discussed widely in the context of the Poincar paradox note that this similarity is not a transitive relation The latter setting classical corresponds to the classical Goodman and Kruskal gamma The functions rococo and rococo test choose the first variant linear by default both for x and y observa
6. 2 764 1954 10 U H hle and N Blanchard Partial ordering in L underdeterminate sets Inform Sci 35 133 144 1985 11 M G Kendall A new measure of rank correlation Biometrika 30 81 93 1938 12 M G Kendall Rank Correlation Methods Charles Griffin amp Co London third edition 1962 13 E P Klement R Mesiar and E Pap Triangular Norms volume 8 of Trends in Logic Kluwer Academic Publishers Dordrecht 2000 14 W H Kruskal Ordinal measures of association J Amer Statist Assoc 53 284 814 861 1958 15 K Pearson Notes on the history of correlation Biometrika 13 25 45 1920 16 R Sedgewick Permutation generation methods ACM Comput Surv 9 2 137 164 1977 17 C Spearman The proof and measurement of association between two things Am J Psy chol 15 1 72 101 1904 18 C Spearman Demonstration of formulae for true measurement of correlation Am J Psy chol 18 2 161 169 1907 19 R R Yager On a general class of fuzzy connectives Fuzzy Sets and Systems 4 235 242 1980
7. OG gd Sala ene Oe Sg 4 2 Choosing the Family of Similarities 000 4 3 Parametrizing Similarities 2 00 20 0 00 00004 4 4 Choosing the t Norm for Aggregation 02 000 4 5 A Note on Permutation Testing 6 The Gaussian Rank Correlation Estimator 7 Change Log 7 1 Upgrading to Version 1 1 1 0 0 2000000000000 8 How to Cite This Package 16 17 18 18 1 Introduction 3 1 Introduction Correlation measures are among the most basic tools in statistical data analysis and machine learn ing They are applied to pairs of observations to measure to which extent the two observations comply with a certain model The most prominent representative is surely Pearson s product moment coefficient 1 15 often nonchalantly called correlation coefficient for short Pearson s product moment coefficient can be applied to numerical data and assumes a linear relationship as the underlying model therefore it can be used to detect linear relationships but no non linear ones Rank correlation measures 9 12 14 are intended to measure to which extent a monotonic function is able to model the inherent relationship between the two observables They neither assume a specific parametric model nor specific distributions of the observables They can be applied to ordinal data and if some ordering relation is given to numerical data too Therefore rank correlation measures are ideally suited
8. Robust Gamma Rank Correlation data x2 and y2 length 15 similarity linear rx 0 1357292 ry 0 02639592 t norm Yager t norm with lambda 0 2 alternative hypothesis true gamma is not equal to 0 sample gamma 0 6367255 estimated p value lt 2 2e 16 0 of 1000 values Note that the rococo package performs only a few basic checks on a user defined t norm It remains the duty of the user to ensure that the function is actually a valid t norm The three built in t norms see Table 2 are efficiently implemented in C and called with the help of the Rcpp package 8 while user defined t norms have to be evaluated in R loops Even though we use the compiler package if available to pre compile the user defined t norm this may still result in a drastic slowdown of computations particularly for larger data sets or when using rococo test with exact TRUE see end of next section 5 A Note on Permutation Testing Classical rank correlation measures only depend on the sorting of x and y observations Thus for a given number of samples n one can deduce the distribution of the test statistic under the Ho hypothesis as a simple function of the number of samples n We neither want to make any prior assumption about the distribution of data nor does the complex inner structure of the robust gamma rank correlation coefficient and the variety of possible parameter settings allow for an analytic deduction of the test statistic s distrib
9. Similarities and t Norms 4 1 Background The robust gamma rank correlation coefficient requires the definition of two strict fuzzy orderings 4 Rx and Ry A strict fuzzy ordering is a two place function that measures to which degree its second argument is strictly larger than its first argument Given a data set consisting of n pairs of observations where n gt 2 Hi En 1 Rx is used for comparing x observations and Ry is used for comparing y observations Given a data set as in Eq 1 the strict fuzzy orderings Rx and Ry are used to compute two important numbers the number of concordant pairs C and the number of discordant pairs D n C 3 gt T Rx xi T3 Ry yi yj i 1 j i 7 n D X X T Rx zi 25 Ry vj y i 1 j i The function T is a triangular t norm 13 that is used to aggregate degrees of relationships between pairs of x observations and the corresponding degrees for y observations see below The final robust gamma rank correlation coefficient is then computed as Cap CAL y in perfect analogy to Goodman s and Kruskal s gamma 9 4 2 Choosing the Family of Similarities It should be clear from the description above that the robust gamma rank correlation coefficient requires the following ingredients 1 A fuzzy ordering Rx for the x observations 2 A fuzzy ordering Ry for the y observations 3 A t norm T for aggregation For Rx and Ry the rococo package provides five possible choices
10. es rococo 2 2 Manual installation If for what reason ever you prefer to install the package manually download the package file suitable for your computer system and copy it to your harddisk Open the package s page at CRAN and then proceed as follows http cran r project org http cran r pro ject org web packages rococo index html 4 3 Getting Started Manual installation under Windows 1 Download rococo_1 1 1 zip and save it to your harddisk 2 Open the R GUI and select the menu entry Packages Install package s from local zip files if you use R in a different language search for the analogous menu entry In the file dialog that opens go to the folder where you placed rococo_1 1 1 zip and select this file The package should be installed now Manual installation under Linux UNIX MacOS 1 Download rococo_1 1 1 tar gz and save it to your harddisk 2 Open a shell window and change to the directory where you put rococo_1 1 1 tar gz Enter R CMD INSTALL rococo_1 1 1 tar gz to install the package 2 3 Compatibility issues Both the Windows and the Linux UNIX MacOS version available from CRAN have been built using the latest version R 3 0 2 However the package should work without severe problems on R versions gt 2 12 0 3 Getting Started To load the package enter the following in your R session gt library rococo You will probably see a message that package Rcpp has been loaded i
11. f shuffles 6 The Gaussian Rank Correlation Estimator The Gaussian rank correlation estimator 7 is a simple and well performing alternative to our proposed robust gamma rank correlation coefficient Since to our best knowledge no R im plementation has been available so far we implemented this rank correlation measure and the corresponding test in the rococo package Example gt gauss cor x1 y1 1 0 9783698 gt gauss cor test x1 y1 alternative two sided 7 Change Log 17 Gaussian rank correlation estimator data xi and yl t 17 0526 df 13 p value 2 81e 10 alternative hypothesis true correlation is not equal to 0 95 percent confidence interval 0 9344238 0 9929723 sample estimates cor 0 9783698 gt gauss cor test Sepal Length Petal Length iris alternative two sided Gaussian rank correlation estimator data Sepal Length and Petal Length t 21 4175 df 148 p value lt 2 2e 16 alternative hypothesis true correlation is not equal to 0 95 percent confidence interval 0 8240938 0 9038302 sample estimates cor 0 8695177 7 Change Log Version 1 1 1 adapted dependencies and linking to Repp version 0 11 0 a cleared up package dependencies Version 1 1 0 estimation of p values now based on relative frequencies Gaussian rank correlation estimator and test added m corresponding adaptations of man pages and vignette a updated citation moved vignette to vignettes acco
12. for detecting monotonic relationships in particular if more specific information about the data is not available The two most common approaches are Spearman s rank correlation coefficient short Spearman s rho 17 18 and Kendall s tau rank correlation coefficient 2 11 12 Another simple rank correlation measure is the gamma rank correlation measure according to Goodman and Kruskal 9 The rank correlation measures cited above are designed for ordinal data However as argued in 5 they are not ideally suited for measuring rank correlation for numerical data that are per turbed by noise Consequently 5 introduces a family of robust rank correlation measures The idea is to replace the classical ordering of real numbers used in Goodman s and Kruskal s gamma 9 by some fuzzy ordering 10 3 4 with smooth transitions thereby ensuring that the corre lation measure is continuous with respect to the data This package provides this family of rank correlation measures along with corresponding statistical tests as introduced in 6 Moreover the package also implements the Gaussian rank correlation estimator 7 and a corresponding sta tistical test 2 Installation 2 1 Installation via CRAN The R package rococo current version 1 1 1 is part of the Comprehensive R Archive Network CRAN The simplest way to install the package therefore is to enter the following command into your R session gt install packag
13. gamma is not equal to 0 sample gamma 0 9856833 estimated p value lt 2 2e 16 0 of 1000 values 6 3 Getting Started The argument alternative works in the same way as for the standard function cor test The function rococo test is a generic method that can be called on two numeric vectors as above or alternatively using a formula to conveniently extract two columns from a data frame gt data iris gt plot Sepal Length Petal Length iris gt rococo test Sepal Length Petal Length iris alternative two sided Robust Gamma Rank Correlation data Sepal Length and Petal Length length 150 Similarity linear rx 0 13 ry 0 35 t norm min alternative hypothesis true gamma is not equal to 0 sample gamma 0 7940949 estimated p value lt 2 2e 16 0 of 1000 values Pe o o0 5 o o o Ona o o go o amp o 94 80 Sog 00 B26 wow 4 9008 8 2 6 8 5 92G 6 490 0 D o Oo00 02009980 5 ggo 3 7q o 90800 T o8 S a oo 4 o ALE o 2 o o 98g 8 808608888096 3 o 4 o T T T T T T T l 45 50 55 60 65 70 75 8 0 Sepal Length All examples above use default settings for the fuzzy orderings that are used to define the rank correlation coefficient In the following section we introduce the concept behind the robust gamma rank correlation coefficient in greater depth and describe how to adjust the corresponding settings properly 4 Adjusting Similarities and t Norms 7 4 Adjusting
14. herefore the support of this project is gratefully acknowledged References 1 H Abdi Coefficients of correlation alienation and determination In N J Salkind editor Encyclopedia of Measurement and Statistics Sage Thousand Oaks CA 2007 2 H Abdi The Kendall rank correlation coefficient In N J Salkind editor Encyclopedia of Measurement and Statistics Sage Thousand Oaks CA 2007 3 U Bodenhofer A similarity based generalization of fuzzy orderings preserving the classical axioms Internat J Uncertain Fuzziness Knowledge Based Systems 8 5 593 610 2000 4 U Bodenhofer and M Demirci Strict fuzzy orderings with a given context of similarity Internat J Uncertain Fuzziness Knowledge Based Systems 16 2 147 178 2008 5 U Bodenhofer and F Klawonn Robust rank correlation coefficients on the basis of fuzzy orderings initial steps Mathware Soft Comput 15 1 5 20 2008 References 19 6 U Bodenhofer M Krone and F Klawonn Testing noisy numerical data for monotonic association Inform Sci 2013 accepted 7 K Boudt J Cornelissen and C Croux The Gaussian rank correlation estimator robustness properties Stat Comput 22 2 471 483 2012 8 D Eddelbuettel and R Francois Repp seamless R and C integration J Stat Softw 40 8 1 18 2011 9 L A Goodman and W H Kruskal Measures of association for cross classifications J Amer Statist Assoc 49 268 73
15. ion time grows linearly with the number of shuffles Here is an example with 100 000 shuffles where we use the option storeValue TRUE to get access to the rank correlation values for all random shuffles gt res lt rococo test x2 y2 numtests 100000 storeValues TRUE gt res Robust Gamma Rank Correlation 5 A Note on Permutation Testing 15 data x2 and y2 length 15 Similarity linear rx 0 1357292 ry 0 02639592 t norm min alternative hypothesis true gamma is not equal to 0 sample gamma 0 62615 estimated p value 0 00107 107 of 100000 values For small data sets not more than 10 samples the package further provides computations of exact p values This can be enforced by passing the argument exact TRUE to rococo test In this case all possible permutations are considered using the Steinhaus Johnson Trotter algorithm see e g 16 and the exact p value is computed as the quotient of the above count slot count over the number of trials the factorial of the length of x and y gt rococo test x2 1 8 y2 1 8 exact TRUE Robust Gamma Rank Correlation data x2 1 8 and y2 1 8 length 8 Similarity linear rx 0 1253808 ry 0 02205911 t norm min alternative hypothesis true gamma is not equal to 0 sample gamma 0 4110654 exact p value 0 1771329 7142 of 40320 values Using the exact test for 10 samples results in 10 3628800 permutations that have to be consid ered F
16. n case it has not been loaded previously in the current R session Apart from this you can be sure that the package has been installed successfully if this command terminates without any further error message or warning If so the package is ready for use now The package includes both a user manual this document and a reference manual help pages for each function To view the user manual enter gt vignette rococo Help pages can be viewed using the help command It is recommended to start with gt help rococo 3 Getting Started 5 For demonstration purposes let us first create an artificial toy data set gt x1 lt rnorm 15 gt yl lt 2 x1 rnorm length x1 sd 0 25 gt plot x1 y1 st PaA N 0 i 0 o 4 0 oO 0 0 N 0 i 0 O T T T T 1 0 1 2 x1 Obviously these are linearly correlated Gaussian data so Pearson s product moment correlation coefficient 1 15 would be the optimal choice We use these data anyway only for illustration purposes The function rococo can be used to compute the robust rank correlation coefficient as follows gt rococo x1 y1 1 0 9856833 To perform a robust rank correlation test use the function rococo test gt rococo test x1 y1 alternative two sided Robust Gamma Rank Correlation data x1 and y1 length 15 Similarity linear rx 0 1575895 ry 0 3151384 t norm min alternative hypothesis true
17. native greater Robust Gamma Rank Correlation data x2 and y2 length 15 similarity for x linear rx 0 1357292 similarity for y gauss ry 0 02639592 t norm prod alternative hypothesis true gamma is greater than 0 sample gamma 0 6513292 estimated p value lt 2 2e 16 0 of 1000 values The choice of the t norm T is not particularly critical and should not have a strong influence on the significance of results Most users will suffice with the default setting min The choice prod produces similar but slightly smoother results which may be suitable in combination with the only differentiable fuzzy ordering similarity gauss Even though the choice is not critical the rococo package also offers the choice of user defined t norms by supplying a two place function as tnorm argument Here is an example with some Yager t norm 13 19 gt DrastictNorm lt function x y if x 1 y else if y 1 x else 0 EGF gt YagertNorm lt function lambda ort fun lt function x y if lambda 0 DrastictNorm x y else if is infinite lambda 5 A Note on Permutation Testing 13 min x y else max 0 1 1 x lambda 1 y lambda 1 lambda attr fun name lt paste Yager t norm with lambda lambda P fun gt rococo x2 y2 tnorm YagertNorm 0 5 1 0 6367255 gt rococo test x2 y2 tnorm YagertNorm 0 2
18. or the built in t norms such a computation should finish within a few seconds However in conjunction with a user defined t norm computations may be significantly longer in the range of several minutes Kendall s rank correlation test is based on a normal approximation of the null distribution The following picture suggests that a normal approximation may also be suitable for the robust rank correlation test data from the above example with 100 000 shuffles gt hist res perm gamma breaks 100 probability TRUE xlab gamma main Distribution of gamma for random shuffles gt plot function x dnorm x mean res HOgamma mu sd res HOgamma sd min res perm gamma max res perm gamma col red lwd 2 add TRUE 16 6 The Gaussian Rank Correlation Estimator Distribution of gamma for random shuffles OQ _ D D gt _ d D Q To 2 oO oS I 0 5 0 0 0 5 gamma However statistical analyses have shown that the null distributions are seldom normal Any way the rococo package also provides p values based on the normal approximation in the slot p value approx of the resulting output object gt res p value approx 1 0 002012103 The user should be warned however that these p values are systematically biased and that he she is recommended to use the standard p values as described above for a sufficiently large number o
19. rding to the new guidelines Version 1 0 1 improved estimation of p values added exact computation of p values only for 10 or less samples m corresponding adaptations of man pages and vignette Version 1 0 0 first official release released October 3 2011 18 References 7 1 Upgrading to Version 1 1 1 Users who upgrade to Version 1 1 1 from an older version should be aware that the package now requires Rcpp Version 0 11 0 or newer This issue can simply be solved by re installing Repp from CRAN using install packages Rcpp 8 How to Cite This Package If you use this package for research that is published later you are kindly asked to cite it as follows U Bodenhofer M Krone and F Klawonn 2012 Testing noisy numerical data for monotonic association Inform Sci DOI 10 1016 j ins 2012 11 026 U Bodenhofer and F Klawonn 2008 Robust rank correlation coefficients on the basis of fuzzy orderings initial steps Mathware Soft Comput 15 1 5 20 To obtain a BibT X entry of the reference you can enter the following into your R session gt toBibtex citation rococo Acknowledgment The core of this package was implemented during a short term scientific mission of Martin Krone at the Institute of Bioinformatics Johannes Kepler University within the framework of COST Action IC0702 SoftStat Combining Soft Computing Techniques and Statistical Methods to Im prove Data Analysis Solutions T
20. ting Similarities and t Norms gt rococo test x2 y2 similarity c classical gauss alternative greater Robust Gamma Rank Correlation data x2 and y2 length 15 Similarity for x classical similarity for y gauss ry 0 02639592 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 6467861 estimated p value lt 2 2e 16 0 of 1000 values Particularly note the latter of the two examples different settings for x and y observations can be done by supplying a vector to the similarity argument where the first element determines the choice for x observations and the second element determines the choice for y observations 4 3 Parametrizing Similarities So far we have neglected that four of the five similarities fuzzy orderings listed in Table 1 require an additional parameter r In all four cases r controls the importance of observations that are close to each other The smaller r the more similar observations are taken into account when computing the numbers of concordant and discordant pairs This entails that the smaller r the easier noise can corrupt the result The larger r the less similar observations are considered i e the more noise tolerant the result will be However that does not mean that the largest possible r is the best choice An overly large r can result in unspecific and unsignificant results The rococo package allows for setting one r for both x and y observations
21. tions If one wants to choose a different setting or two different strict fuzzy orderings for x and y observations the similarity argument can be used For demonstrating this we create a noisy data set from a function f that has a large flat area y2 lt sapply x2 f2 rnorm length x2 sd 0 1 plot x2 y2 gt x2 lt rnorm 15 gt f2 lt function x 4 if x gt 0 8 x 0 8 else if x lt 0 8 x 0 8 else 0 gt gt 4 Adjusting Similarities and t Norms 9 1 0 0 5 Oo y2 0 0 1 0 0 5 0 0 0 5 1 0 1 5 As said before Linear is the default choice gt rococo test x2 y2 alternative greater Robust Gamma Rank Correlation data x2 and y2 length 15 similarity linear rx 0 1357292 ry 0 02639592 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 62615 estimated p value 0 002 2 of 1000 values This default setting makes sense in particular if no information about the data their distribution or not even the noise distribution is known Now let us try some different settings gt rococo test x2 y2 similarity gauss alternative greater Robust Gamma Rank Correlation data x2 and y2 length 15 similarity gauss rx 0 1357292 ry 0 02639592 t norm min alternative hypothesis true gamma is greater than 0 sample gamma 0 7052882 estimated p value lt 2 2e 16 0 of 1000 values 10 4 Adjus
22. ution Therefore we use permutation testing for estimating the null distribution of the test statistic under the assumption of independence This is done in the following way For a given data set x y _ we first compute the robust rank correlation coefficient according to the specified parameters Then we create K random shuffles 14 5 A Note on Permutation Testing yi of yi and compute the robust rank correlation coefficient for x y _ according to the specified parameters Due to the shuffling x _ and y _ are independent where y has the same marginal distribution as y _ Thus the robust rank correlation coefficients obtained for a sufficiently large number of shuffles allows for estimating the distribution of the test statistics under the Ho hypothesis with given marginal distributions and we can estimate the test s p value in the following simple way a as the relative frequency the test statistics absolute value exceeded the absolute value of the test statistic for the unshuffled data in case we perform a two sided test a as the relative frequency the test statistic was greater than the test statistic for the unshuffled data in case we perform a one sided test with alternative hypothesis that there is a positive correlation and as the relative frequency the test statistic was less than the test statistic for the unshuffled data in case we perform a one sided test with alternative hypothesis that there is
Download Pdf Manuals
Related Search