parconv              package:adaptTest              R Documentation

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

     This function converts between two different parameterizations of
     a family of conditional error functions: a (more traditional)
     parameter c, and a (more convenient) parameter alpha2 specifying
     the local level of the test after the second stage.

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

     parconv(typ, a2 = NA, c = NA)

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

     typ: type of test: '"b"' for Bauer and Koehne (1994), '"l"' for
          Lehmacher and Wassmer (1999), '"v"' for Vandemeulebroecke
          (2006) and '"h"' for the horizontal conditional error
          function

      a2: alpha2, the local level of the test after the second stage
          (see details)

       c: the parameter c (see details)

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

     Traditionally, a family of conditional error functions is often
     parameterized by some parameter c that, in turn, depends on the
     local level alpha2 of the test after the second stage. However, it
     can be convenient to parameterize the family directly by alpha2.
     The function 'parconv' converts one parameter into the other:
     provide one, and it returns the other.

     Essentially, the relation between the two parameterizations is
     implemented as:

        *  c = exp(-chi2_{4,alpha2}/2) for Fisher's combination test
           (Bauer and Koehne, 1994)

        *  c = Phi^{-1}(1-alpha2) for the inverse normal method
           (Lehmacher and Wassmer, 1999)

        *  alpha2 = {(Gamma(1+1/r))^2}/{Gamma(1+2/r)} for
           Vandemeulebroecke (2006)

        *  c = alpha2 for the family of horizontal conditional error
           functions

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

     'parconv' returns alpha2 corresponding to the supplied c, or c
     corresponding to the supplied alpha2.

_N_o_t_e:

     Provide either 'a2' or 'c', not both!

     alpha2 is the local level of the test after the second stage, and
     it equals the integral under the corresponding conditional error
     function: 

               alpha2 = int_0^1 cef_{alpha2}(p1) d p1,

     where cef_{alpha2} is the conditional error function (of a
     specified family) with parameter alpha2.

     Note that in this implementation of adaptive two-stage tests,
     early stopping bounds are _not_ part of the conditional error
     function. Rather, they are specified separately (see also 'tsT').

     alpha2 can take any value in [0,1]; c can take values in

        *  [0,1] for Fisher's combination test (Bauer and Koehne, 1994)

        *  (-infty, infty) for the inverse normal method (Lehmacher and
           Wassmer, 1999)

        *  [0,infty) for Vandemeulebroecke (2006)

        *  [0,1] for the family of horizontal conditional error
           functions

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

     Marc Vandemeulebroecke

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

     Bauer, P., Koehne, K. (1994). Evaluation of experiments with
     adaptive interim analyses. _Biometrics_ 50, 1029-1041.

     Lehmacher, W., Wassmer, G. (1999). Adaptive sample size
     calculations in group sequential trials. _Biometrics_ 55,
     1286-1290.

     Vandemeulebroecke, M. (2006). An investigation of two-stage tests.
     _Statistica Sinica_ 16, 933-951.

_S_e_e _A_l_s_o:

     'adaptTest' package description, 'getpar', 'CEF'

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

     ## Obtain the parameter c for Fisher's combination test, using the local level 0.05 for the test after the second stage
     parconv(typ="b", a2=0.05)

