TwoSampleTests            package:fBasics            R Documentation

_T_w_o _S_a_m_p_l_e _T_e_s_t_s

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description of functions for  two sample
     statistical tests. The functions allow  to test for distributional
     equivalence, for difference in location, variance and scale, and
     for correlations. 

     Distributional Equivalence:

       'ks2Test'  Two sample Kolmogorov-Smirnov test.

     Difference in Locations:

       'locationTest'  The location test suite,
       '.tTest'        The t test,
       '.kw2Test'      the Kruskal-Wallis test.

     Difference in Variance:

       'varianceTest'    The variance test suite,
       '.varfTest'       The variance F test,
       '.bartlett2Test'  the Bartlett test,
       '.fligner2Test'   the Fligner-Killeen test.

     Difference in Scale:

       'scaleTest'    The scale test suite,
       '.ansariTest'  The Ansari-Bradley test,
       '.moodTest'    the Mood test.

     Correlations:

       'correlationTest'  The correlation test suite,
       '.pearsonTest'     Pearson's coefficient,
       '.kendallTest'     Kendall's rho,
       '.spearmanTest'    Spearman's rho.

     Test Distributions:

       'dansariw'  Returns density of the Ansari W statistic,
       'pansariw'  Returns probabilities of the Ansari W statistic,
       'qansariw'  Returns quantiles of the Ansari W statistic.

_U_s_a_g_e:

     ks2Test(x, y, title = NULL, description = NULL)

     locationTest(x, y, method = c("t", "kw2"), title = NULL, 
         description = NULL) 

     varianceTest(x, y, method = c("varf", "bartlett", "fligner"), title = NULL, 
         description = NULL)

     scaleTest(x, y, method = c("ansari", "mood"), title = NULL, 
         description = NULL)

     correlationTest(x, y, method = c("pearson", "kendall", "spearman"), title = NULL, 
         description = NULL)

     dansariw(x = NULL, m, n = m)
     pansariw(q = NULL, m, n = m)
     qansariw(p, m, n = m)

_A_r_g_u_m_e_n_t_s:

description: optional description string, or a vector of character
          strings. 

    m, n: [*ansariw] - 


  method: a character string naming which test should be applied. 

       p: [qansariw] - 
           a numeric vector of quantiles. 

       q: [pansariw] - 
           a numeric vector of quantiles. 

   title: an optional title string, if not specified the inputs data 
          name is deparsed. 

    x, y: a numeric vector of data values. 
           [bartlett2Test][fligner2Test][kw2Test] - 
           here 'x' is a list, where each element is either a vector or
          an object of class 'timeSeries'. 'y' is only used for the
          two-sample test situation, where 'x' and 'y' are two vectors
          or objects of class 'timeSeries'. 
           [dansariw] - 
           a numeric vector of quantiles. 

_D_e_t_a_i_l_s:

     The tests may be of interest for many financial  and economic
     applications, especially for the  comparison of two time series.
     The tests are grouped  according to their functionalities. 

     *Distributional Equivalence:* 

        The test 'ks2Test' performs a Kolmogorov-Smirnov two sample
     test  that the two data samples 'x' and 'y' come from the same 
     distribution, not necessarily a normal distribution. That means
     that  it is not specified what that common distribution is.  

     *Differences in Location:* 

        The function 'tTest' can be used to determine if the two sample
      means are equal for unpaired data sets. Two variants are used,
     assuming equal or unequal variances. 

     The function 'kw2Test' performs a Kruskal-Wallis rank sum  test of
     the null hypothesis that the central tendencies or medians of  two
     samples are the same. The alternative is that they differ.  Note,
     that it is not assumed that the two samples are drawn from the 
     same distribution. It is also worth to know that the test assumes 
     that the variables under consideration have underlying continuous 
     distributions. 

     *Differences in Variances:* 

          The function 'varfTest' can be used to compare variances of
     two  normal samples performing an F test. The null hypothesis is
     that  the ratio of the variances of the populations from which
     they were  drawn is equal to one. 

     The function 'bartlett2Test' performs the Bartlett's test of the 
     null hypothesis that the variances in each of the samples are the 
     same. This fact of equal variances across samples is also called 
     _homogeneity of variances_. Note, that Bartlett's test is 
     sensitive to departures from normality. That is, if the samples 
     come from non-normal distributions, then Bartlett's test may
     simply  be testing for non-normality. The Levene test (not yet
     implemented) is an alternative to the Bartlett test that is less
     sensitive to  departures from normality. 

     The function 'fligner2Test' performs the Fligner-Killeen test of 
     the null that the variances in each of the two samples are the
     same.  

     *Differences in Scale:* 

        The function 'ansariTest' performs the Ansari-Bradley
     two-sample  test for a difference in scale parameters. Note, that
     we have completely  reimplemented this test based on the statistcs
     and p-values computed  from algorithm AS 93. The test returns for
     any sizes of the series  'x' and 'y' the exact p value together
     with its asymptotic  limit. The test procedure is not limited to
     sizes shorter of length 50  as this is the case for the function
     'ansari.Test' implemented in  R's 'stats' package. For the test
     statistics the following functions are available: 'dansariw',
     'pansariw', and  'qansariw'.

     The function code{moodTest}, is another test which performs a 
     two-sample test for a difference in scale parameters. The
     underlying  model is that the two samples are drawn from _f(x-l)_
     and  _f((x-l)/s)/s_, respectively, where _l_ is a common  location
     parameter and _s_ is a scale parameter. The null  hypothesis is
     _s=1_.  

     *Correlations:* 

        The function 'correlationTest' tests for association  between
     paired samples, using Pearson's product moment  correlation
     coefficient, 

     The function 'kendallTest' performs Kendall's tau test

     The function 'spearmanTest' performs Spearman's rho test.

_V_a_l_u_e:

     In contrast to R's output report from S3 objects of class
     '"htest"' a different output report is produced. The classical
     tests presented here return an S4 object of class '"fHTEST"'. The
     object contains the following slots:

   @call: the function call.   

   @data: the data as specified by the input argument(s). 

   @test: a list whose elements contail the results from the
          statistical test. The information provided is similar to a
          list object of class{"htest"}. 

  @title: a character string with the name of the test. This can be 
          overwritten specifying a user defined input argument. 

@description: a character string with an optional user defined
          description.  By default just the current date when the test
          was applied will be returned. 

statistic: the value(s) of the test statistic. 

 p.value: the p-value(s) of the test. 

parameters: a numeric value or vector of parameters. 

estimate: a numeric value or vector of sample estimates. 

conf.int: a numeric two row vector or matrix of 95 

  method: a character string indicating what type of test was
          performed. 

data.name: a character string giving the name(s) of the data. 

_N_o_t_e:

     Some of the test implementations are selected from R's 'ctest' 
     package.

_A_u_t_h_o_r(_s):

     R-core team for the tests from R's ctest package,
      Diethelm Wuertz for the Rmetrics R-port.

_R_e_f_e_r_e_n_c_e_s:

     Conover, W. J. (1971); _Practical nonparametric statistics_, New
     York: John Wiley & Sons.

     Durbin J. (1961);  _Some Methods of Constructing Exact Tests_,
     Biometrika 48, 41-55. 

     Durbin,J. (1973); _Distribution Theory Based on the Sample
     Distribution Function_, SIAM, Philadelphia.

     Lehmann E.L. (1986);  _Testing Statistical Hypotheses_,  John
     Wiley and Sons, New York.

     Moore, D.S. (1986); _Tests of the chi-squared type_,  In:
     D'Agostino, R.B. and Stephens, M.A., eds.,  Goodness-of-Fit
     Techniques, Marcel Dekker, New York.

_E_x_a_m_p_l_e_s:

     ## SOURCE("fBasics.5C-TwoSampleTests")

     ## x, y -
        x = rnorm(50)
        y = rnorm(50)
       
     ## ks2Test - 
        ks2Test(x, y)
        
     ## locationTest | .tTest | .kw2Test - 
        locationTest(x, y)
        
     ## varianceTest | .varfTest, .bartlett2Test | .fligner2Test -
        varianceTest(x, y)

     ## scaleTest | .ansariTest | .moodTest -
        scaleTest(x, y)
        
     ## correlationTest | .pearsonTest | .kendallTest | .spearmanTest -
        correlationTest(x, y)

