ScaleTests               package:coin               R Documentation

_I_n_d_e_p_e_n_d_e_n_t _T_w_o- _a_n_d _K-_S_a_m_p_l_e _S_c_a_l_e _T_e_s_t_s

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

     Testing the equality of the distributions of a numeric response in
     two or more independent groups against scale alternatives.

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

     ## S3 method for class 'formula':
     ansari_test(formula, data, subset = NULL, weights = NULL, ...)
     ## S3 method for class 'IndependenceProblem':
     ansari_test(object, 
         alternative = c("two.sided", "less", "greater"),
         distribution = c("asymptotic", "approximate", "exact"),
         ties.method = c("mid-ranks", "average-scores"),
         conf.int = FALSE, conf.level = 0.95, ...)

     ## S3 method for class 'formula':
     fligner_test(formula, data, subset = NULL, weights = NULL, ...)
     ## S3 method for class 'IndependenceProblem':
     fligner_test(object, 
         ties.method = c("mid-ranks", "average-scores"),
         distribution = c("asymptotic", "approximate"), 
         ...)

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

 formula: a formula of the form 'y ~ x | block' where 'y' is a numeric
          variable giving the data values and 'x' a factor with two or
          more levels giving the corresponding groups. 'block' is an
          optional factor for stratification.

    data: an optional data frame containing the variables in the model
          formula.

  subset: an optional vector specifying a subset of observations to be
          used.

 weights: an optional formula of the form '~ w' defining integer valued
          weights for the observations.

  object: an object of class 'IndependenceProblem'.

alternative: a character, the alternative hypothesis must be one of
          '"two.sided"' (default), '"greater"' or     '"less"'.  You
          can specify just the initial letter.

distribution: a character, the null distribution of the test statistic
          can be computed 'exact'ly or can be approximated by its
          asymptotic distribution ('asymptotic')    or via Monte-Carlo
          resampling ('approximate'). Alternatively, the functions 
          'exact', 'approximate' or 'asymptotic' can be used to specify
          how the exact conditional distribution of the test statistic
          should be calculated or approximated.

ties.method: a character, two methods are available to adjust scores
          for ties, either the score generating function is applied to
          'mid-ranks' or the scores computed based on random ranks are
          averaged for all tied values ('average-scores').

conf.int: a logical indicating whether a confidence interval for the
          difference in location should be computed.

conf.level: confidence level of the interval.

     ...: further arguments to be passed to or from methods.

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

     The null hypothesis of the equality of the distribution of 'y' in
     the groups given by 'x' is tested. In particular, the methods
     documented here are designed to detect scale alternatives. For a
     general description of the test procedures documented here we
     refer to Hollander & Wolfe (1999).

     The asymptotic null distribution is computed by default for both
     procedures. Exact p-values may be computed for the Ansari-Bradley
     test  can be approximated via Monte-Carlo for the Fligner-Killeen
     procedure.  Exact p-values are computed either by the shift
     algorithm  (Streitberg & Roehmel, 1986, 1987) or by the split-up
     algorithm  (van de Wiel, 2001).

     The Ansari-Bradley test can be used to test the two-sided
     hypothesis var(Y_1) / var(Y_2) = 1, where var(Y_i)  is the
     variance of the responses in the ith group. Confidence intervals 
     for the ratio of scales are available for the Ansari-Bradley test
     and are computed according to Bauer (1972). In case 'alternative =
     "less"', the null hypothesis var(Y_1) / var(Y_2) >= 1 is tested
     and 'alternative = "greater"' corresponds to var(Y_1) / var(Y_2)
     <= 1.

     For the adjustment of scores for tied values see Hajek, Sidak and
     Sen (1999), page 131ff.

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

     An object inheriting from class 'IndependenceTest-class' with
     methods 'show', 'statistic', 'expectation', 'covariance' and
     'pvalue'. The null distribution can be inspected by 'pperm',
     'dperm',   'qperm' and 'support' methods. Confidence intervals can
     be extracted by 'confint'.

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

     Myles Hollander & Douglas A. Wolfe (1999),   _Nonparametric
     Statistical Methods, 2nd Edition_. New York: John Wiley & Sons.

     Bernd Streitberg & Joachim R\"ohmel (1986), Exact distributions
     for permutations and rank tests: An introduction to some recently
     published algorithms. _Statistical Software Newsletter_ *12*(1),
     10-17.

     Bernd Streitberg & Joachim R\"ohmel (1987), Exakte Verteilungen
     f\"ur Rang- und Randomisierungstests im allgemeinen
     $c$-Stichprobenfall. _EDV in Medizin und Biologie_ *18*(1), 12-19.

     Mark A. van de Wiel (2001), The split-up algorithm: a fast
     symbolic method for computing p-values of rank statistics.
     _Computational Statistics_ *16*, 519-538.

     David F. Bauer (1972), Constructing confidence sets using rank
     statistics. _Journal of the American Statistical Association_
     *67*, 687-690.

     Jaroslav Hajek, Zbynek Sidak & Pranab K. Sen (1999), _Theory of
     Rank Tests_. San Diego, London: Academic Press.

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

     ### Serum Iron Determination Using Hyland Control Sera
     ### Hollander & Wolfe (1999), page 147
     sid <- data.frame(
         serum = c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
                   101, 96, 97, 102, 107, 113, 116, 113, 110, 98,
                   107, 108, 106, 98, 105, 103, 110, 105, 104,
                   100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99),
         method = factor(gl(2, 20), labels = c("Ramsay", "Jung-Parekh")))

     ### Ansari-Bradley test, asymptotical p-value
     ansari_test(serum ~ method, data = sid)

     ### exact p-value
     ansari_test(serum ~ method, data = sid, distribution = "exact")

     ### Platelet Counts of Newborn Infants
     ### Hollander & Wolfe (1999), Table 5.4, page 171
     platalet_counts <- data.frame(
         counts = c(120, 124, 215, 90, 67, 95, 190, 180, 135, 399, 
                    12, 20, 112, 32, 60, 40),
         treatment = factor(c(rep("Prednisone", 10), rep("Control", 6))))

     ### Lepage test, Hollander & Wolfe (1999), page 172 
     lt <- independence_test(counts ~ treatment, data = platalet_counts,
         ytrafo = function(data) trafo(data, numeric_trafo = function(x)       
             cbind(rank(x), ansari_trafo(x))),
         teststat = "quadtype", distribution = approximate(B = 9999))

     lt

     ### where did the rejection come from? Use maximum statistic
     ### instead of a quadratic form
     ltmax <- independence_test(counts ~ treatment, data = platalet_counts,
         ytrafo = function(data) trafo(data, numeric_trafo = function(x) 
             matrix(c(rank(x), ansari_trafo(x)), ncol = 2,
                    dimnames = list(1:length(x), c("Location", "Scale")))),
         teststat = "maxtype")

     ### points to a difference in location
     pvalue(ltmax, adjusted = TRUE)

     ### Funny: We could have used a simple Bonferroni procedure
     ### since the correlation between the Wilcoxon and Ansari-Bradley 
     ### test statistics is zero
     covariance(ltmax)

