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Notebook[{

Cell[CellGroupData[{
Cell[TextData[StyleBox["Exact p-values and bounds on p-values of the \
two-sided Steel multiple t-sample rank statistic for t=3,4,5.\nAuthor: Mark \
van de Wiel (markvdw@win.tue.nl)\nDepartment of Mathematics and Computing \
Science, Eindhoven University of Technology, The Netherlands",
  FontSize->24]], "Title"],

Cell[TextData[StyleBox["DOUBLE-CLICK BRACKET WITH HOOK TO OPEN MODULE (AND TO \
CLOSE IT)",
  FontColor->RGBColor[1, 0, 1]]], "Text"],

Cell[CellGroupData[{

Cell["Functions for branch-and-bound", "Section"],

Cell[TextData[StyleBox["PLEASE DO NOT EDIT THESE FUNCTIONS!!!",
  FontColor->RGBColor[1, 0, 1]]], "Text"],

Cell[BoxData[{
    \(\(Bounding[1, a_, b_, \((c_:  1)\)*x[1]^\((d_:  1)\)] := 
        If[Max[d - 1 + \((size - a)\)*\((size - b)\), size^2 - \((d - 1)\)] < 
            obsMW, 0, c*x[1]^d];\)\), "\n", 
    \(\(Bounding[2, a_, b_, 
          c_, \((k_:  1)\)*x[1]^\((d_:  1)\)*x[2]^\((e_:  1)\)] := 
        If[Min[Max[d - 1 + \((size - a)\)*\((size - b)\), 
                size^2 - \((d - 1)\)], \n\t\tMax[
                e - 1 + \((size - a)\)*\((size - c)\), 
                size^2 - \((e - 1)\)]] < obsMW, 0, 
          k*x[1]^d*x[2]^e];\)\), "\n", 
    \(\(Bounding[3, a_, b_, 
          c_, \((k_:  1)\)*x[1]^\((d_:  1)\)*x[2]^\((e_:  1)\)*
            x[3]^\((f_:  1)\)] := 
        If[Min[Max[d - 1 + \((size - a)\)*\((size - b)\), 
                size^2 - \((d - 1)\)], \n\t\tMax[
                e - 1 + \((size - a)\)*\((size - c)\), 
                size^2 - \((e - 1)\)], \n\t\t\tMax[
                f - 1 + \((size - b)\)*\((size - c)\), 
                size^2 - \((f - 1)\)]] < obsMW, 0, 
          k*x[1]^d*x[2]^e*x[3]^f];\)\), "\n", 
    \(\(Bounding[4, a_, b_, c_, 
          d_, \((k_:  1)\)*x[1]^\((l_:  1)\)*x[2]^\((e_:  1)\)*
            x[3]^\((f_:  1)\)] := 
        If[Min[Max[l - 1 + \((size - a)\)*\((size - b)\), 
                size^2 - \((l - 1)\)], \n\t\tMax[
                e - 1 + \((size - a)\)*\((size - c)\), 
                size^2 - \((e - 1)\)], \n\t\t\tMax[
                f - 1 + \((size - a)\)*\((size - d)\), 
                size^2 - \((f - 1)\)]] < obsMW, 0, 
          k*x[1]^l*x[2]^e*x[3]^f];\)\), "\n", 
    \(\(Bounding[5, a_, b_, c_, 
          d_, \((k_:  1)\)*x[1]^\((l_:  1)\)*x[2]^\((e_:  1)\)*
            x[3]^\((f_:  1)\)] := 
        If[Min[Max[l - 1 + \((size - a)\)*\((size - b)\), 
                size^2 - \((l - 1)\)], \n\t\tMax[
                e - 1 + \((size - a)\)*\((size - c)\), 
                size^2 - \((e - 1)\)], \n\t\t\tMax[
                f - 1 + \((size - b)\)*\((size - d)\), 
                size^2 - \((f - 1)\)]] < obsMW, 0, 
          k*x[1]^l*x[2]^e*x[3]^f];\)\)}], "Input",
  InitializationCell->True],

Cell[BoxData[{
    \(\(freqlijst[
          3] = {{3, {1, 2}}, {\(-3\), {1, 2}, {1, 3}}, {1, {1, 2}, {1, 
              3}, {2, 3}}};\)\), "\n", 
    \(\(freqlijst[
          4] = {{6, 2, 3, 4, 5, 6}, {\(-12\), 3, 4, 5, 6}, {\(-3\), 2, 3, 4, 
            5}, \n\t\t{4, 3, 5, 6}, {4, 4, 5, 6}, {12, 3, 4, 6}, {\(-12\), 5, 
            6}, {\(-3\), 3, 4}, {6, 6}, {\(-1\), 
            1}};\)\), "\[IndentingNewLine]", 
    \(\(freqlijst[
          5] = {{10, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {30, 3, 4, 5, 6, 7, 8, 9, 
            10}, {15, 2, 3, 4, 5, 6, 7, 9, 10}, \[IndentingNewLine]{10, 3, 4, 
            6, 7, 8, 9, 10}, {20, 4, 5, 6, 7, 8, 9, 
            10}, \[IndentingNewLine]{60, 3, 4, 5, 7, 8, 9, 10}, {30, 3, 4, 5, 
            6, 7, 8, 9}};\)\)}], "Input",
  InitializationCell->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Three treatments", "Section"],

Cell[TextData[{
  "The following procedure computes bounds for exact p-values.  EDIT THE \
FIRST LINE OF THIS PROCEDURE ONLY! The procedure uses Wilcoxon scores. The \
observed value is the maximum over all treatment-pairs (i,j) of \
Max(Sij,sumofscores-Sij), where sumofscores equals size(2*size+1) and Sij is \
the sum of scores corresponding to treatment i when compared with treatment \
j.\nIn this procedure, `size' is the size of each sample, `maxcomp' is the \
cut-off value for computing bounds on the p-value and `obs' is the observed \
value. The parameter `maxcomp' equals either 2 or 3. If 'maxcomp' equals 2 a \
bounding interval is computed. If 'maxcomp' equals 3, the exact p-value is \
computed using more computing time. ",
  Cell[BoxData[
      \(We\ recommend\ to\ start\ with\ maxcomp = 
        2. \ One\ may\ increase\ this\ number\ when\ the\ bounding\ interval\ \
is\ too\ \(\(large\)\(.\)\)\)], "Input"],
  " "
}], "Text",
  FontColor->RGBColor[1, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Timing[size = 5; obs = 39; 
      maxcomp = 2; \[IndentingNewLine]obsMW = obs - size*\((size + 1)\)/2; 
      totalprob = 0; \nClear[palg]; \npalg[1, 0, 0] := x[1]; \n
      palg[1, \(-1\), a_] := 0; \npalg[1, a_, \(-1\)] := 0; \n
      palg[1, a_, b_] := \(palg[1, a, b] = 
          Module[{temp = 
                Expand[palg[1, a - 1, b]*x[1]^\((size - b)\) + \n\t\t\tpalg[
                      1, a, b - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[1, a, b, #] &, temp], 
              temp]]\); \npalg[2, 0, 0, 0] := x[1]*x[2]; \n
      palg[2, \(-1\), b_, c_] := 0; \npalg[2, a_, \(-1\), c_] := 0; \n
      palg[2, a_, b_, \(-1\)] := 0; \n
      palg[2, a_, b_, c_] := \(palg[2, a, b, c] = 
          Module[{temp = 
                Expand[palg[2, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[2, a, b - 1, c] + 
                    palg[2, a, b, c - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[2, a, b, c, #] &, temp], 
              temp]]\); \npalg[3, 0, 0, 0] := x[1]*x[2]*x[3]; \n
      palg[3, \(-1\), b_, c_] := 0; \npalg[3, a_, \(-1\), c_] := 0; \n
      palg[3, a_, b_, \(-1\)] := 0; \n
      palg[3, a_, b_, c_] := \(palg[3, a, b, c] = 
          Module[{temp = 
                Expand[palg[3, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[3, a, b - 1, c]*
                      x[3]^\((size - c)\) + 
                    palg[3, a, b, c - 1]]}, \n\t\t\tIf[Head[temp] === Plus, 
              Map[Bounding[3, a, b, c, #] &, temp], temp]]\)\t; \n\t\tFor[
        aant = 1, 
        aant \[LessEqual] maxcomp, \(aant++\), \n\t\tIf[aant == 1, 
          prob = \((palg[1, size, size]\  /. \ x[i_] -> 1)\)*3/
                Binomial[2*size, size]]; \n\t
        If[aant == 2, 
          prob = \((palg[2, size, size, size]\  /. \ 
                  x[i_] -> 1)\)*\((\(-3\))\)/Multinomial[size, size, size]; 
          extra = prob/\((\(-3\))\);]; \n\t
        If[aant == 3, 
          prob = \((palg[3, size, size, size] /. \ x[i_] -> 1)\)/
              Multinomial[size, size, size]]; 
        totalprob = totalprob + prob; \n\tPrint[N[prob]]]; 
      If[maxcomp == 2, Print["\<Interval which contains p-value:\>"]; 
        Print[N[{totalprob, totalprob + extra}]], Print["\<p-value:\>"]; 
        Print[N[totalprob]]]]\)\(\n\)
    \)\)], "Input"],

Cell[BoxData[
    \(0.047619047619047616`\)], "Print"],

Cell[BoxData[
    \(\(-0.005137719423433709`\)\)], "Print"],

Cell[BoxData[
    \("Interval containing p-value:"\)], "Print"],

Cell[BoxData[
    \({0.04248132819561391`, 0.04419390133675848`}\)], "Print"],

Cell[BoxData[
    \({0.35999999999999943`\ Second, Null}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Four treatments", "Section"],

Cell[TextData[{
  "The following procedure computes bounds for exact p-values.  EDIT THE \
FIRST LINE OF THIS PROCEDURE ONLY! The procedure uses Wilcoxon scores. The \
observed value is the maximum over all treatment-pairs (i,j) of \
Max(Sij,sumofscores-Sij), where sumofscores equals size(2*size+1) and Sij is \
the sum of scores corresponding to treatment i when compared with treatment \
j.\nIn this procedure, `size' is the size of each sample, `obs' is the \
observed value and `maxcomp' is the cut-off value for computing bounds on the \
p-value. The parameter `maxcomp' equals either 2 or 3. In both cases a \
bounding interval is computed. If 'maxcomp' equals 3, the bounding interval \
is smaller (hence more precise), but computing time is larger than for \
maxcomp=2. ",
  Cell[BoxData[
      \(We\ recommend\ to\ start\ with\ maxcomp = 
        2. \ One\ may\ increase\ this\ number\ when\ the\ bounding\ interval\ \
is\ too\ \(\(large\)\(.\)\)\)], "Input"],
  "  "
}], "Text",
  FontColor->RGBColor[1, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Timing[size = 5; obs = 38; 
      maxcomp = 2; \[IndentingNewLine]obsMW = obs - size*\((size + 1)\)/2; 
      totalprob = 0; 
      If[maxcomp \[Equal] 2, maxaant = 3, 
        maxaant = 5]; \[IndentingNewLine]Clear[palg]; \n
      palg[1, 0, 0] := x[1]; \npalg[1, \(-1\), a_] := 0; \n
      palg[1, a_, \(-1\)] := 0; \n
      palg[1, a_, b_] := \(palg[1, a, b] = 
          Module[{temp = 
                Expand[palg[1, a - 1, b]*x[1]^\((size - b)\) + \n\t\t\tpalg[
                      1, a, b - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[1, a, b, #] &, temp], 
              temp]]\); \npalg[2, 0, 0, 0] := x[1]*x[2]; \n
      palg[2, \(-1\), b_, c_] := 0; \npalg[2, a_, \(-1\), c_] := 0; \n
      palg[2, a_, b_, \(-1\)] := 0; \n
      palg[2, a_, b_, c_] := \(palg[2, a, b, c] = 
          Module[{temp = 
                Expand[palg[2, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[2, a, b - 1, c] + 
                    palg[2, a, b, c - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[2, a, b, c, #] &, temp], 
              temp]]\); \npalg[3, 0, 0, 0] := x[1]*x[2]*x[3]; \n
      palg[3, \(-1\), b_, c_] := 0; \npalg[3, a_, \(-1\), c_] := 0; \n
      palg[3, a_, b_, \(-1\)] := 0; \n
      palg[3, a_, b_, c_] := \(palg[3, a, b, c] = 
          Module[{temp = 
                Expand[palg[3, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[3, a, b - 1, c]*
                      x[3]^\((size - c)\) + 
                    palg[3, a, b, c - 1]]}, \n\t\t\tIf[Head[temp] === Plus, 
              Map[Bounding[3, a, b, c, #] &, temp], temp]]\)\t; \n\t\n
      palg[4, 0, 0, 0, 0] := x[1]*x[2]*x[3]; \n
      palg[4, \(-1\), b_, c_, d_] := 0; \npalg[4, a_, \(-1\), c_, d_] := 0; \n
      palg[4, a_, b_, \(-1\), d_] := 0; \npalg[4, a_, b_, c_, \(-1\)] := 0; \n
      palg[4, a_, b_, c_, d_] := \(palg[4, a, b, c, d] = 
          Module[{temp = 
                Expand[palg[4, a - 1, b, c, d]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\)*\n\t\t\t\t\t\t\t\tx[
                          3]^\((size - d)\) + \n\t\t\tpalg[4, a, b - 1, c, 
                      d] + palg[4, a, b, c - 1, d] + 
                    palg[4, a, b, c, d - 1]]}, \n\t\t\tIf[
              Head[temp] === Plus, Map[Bounding[4, a, b, c, d, #] &, temp], 
              temp]]\)\t; \n\npalg[5, 0, 0, 0, 0] := x[1]*x[2]*x[3]; \t\n
      palg[5, \(-1\), b_, c_, d_] := 0; \npalg[5, a_, \(-1\), c_, d_] := 0; \n
      palg[5, a_, b_, \(-1\), d_] := 0; \npalg[5, a_, b_, c_, \(-1\)] := 0; \n
      palg[5, a_, b_, c_, d_] := \(palg[5, a, b, c, d] = 
          Module[{temp = 
                Expand[palg[5, a - 1, b, c, d]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[5, a, b - 1, c, d]*
                      x[3]^\((size - d)\) + palg[5, a, b, c - 1, d] + 
                    palg[5, a, b, c, d - 1]]}, \n\t\t\tIf[
              Head[temp] === Plus, Map[Bounding[5, a, b, c, d, #] &, temp], 
              temp]]\)\t; \n\t
      For[aant = 1, 
        aant \[LessEqual] maxaant, \(aant++\), \n\t\tIf[aant == 1, 
          probhulp = \((palg[1, size, size]\  /. \ x[i_] -> 1)\)/
              Binomial[2*size, size]; \n\t\tprob = 
            6*probhulp - 3*probhulp^2]; \n\t\t\n\t
        If[aant == 2, 
          prob = \((palg[2, size, size, size]\  /. \ 
                  x[i_] -> 1)\)*\((\(-12\))\)/Multinomial[size, size, size]; 
          extra2 = 12*probhulp^2 - prob/3]; \n\t
        If[aant == 3, 
          prob = \((palg[3, size, size, size] /. \ x[i_] -> 1)\)*4/
                Multinomial[size, size, size]; extra = \(-3\)*prob]; \n\t
        If[aant == 4, 
          prob = \((palg[4, size, size, size, size] /. \ x[i_] -> 1)\)*4/
                Multinomial[size, size, size, size]]; \n\t
        If[aant == 5, 
          prob = \((palg[5, size, size, size, size] /. \ x[i_] -> 1)\)*12/
                Multinomial[size, size, size, size]; 
          extra = extra - 0.25*prob]; \n\ttotalprob = totalprob + prob; 
        Print[N[prob]]]; Print["\<Interval which contains p-value:\>"]; 
      If[maxcomp \[Equal] 2, Print[N[{totalprob, totalprob + extra2}]], 
        Print[N[{totalprob + extra, totalprob}]]]]\)\(\n\)
    \)\)], "Input"],

Cell[BoxData[
    \(0.1874527588813303`\)], "Print"],

Cell[BoxData[
    \(\(-0.054041196898339756`\)\)], "Print"],

Cell[BoxData[
    \(0.0005074290788576503`\)], "Print"],

Cell[BoxData[
    \("Interval which contains p-value:"\)], "Print"],

Cell[BoxData[
    \({0.1339189910618482`, 0.16402644974073546`}\)], "Print"],

Cell[BoxData[
    \({1.2900000000000205`\ Second, Null}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Five treatments", "Section"],

Cell["\<\
The following procedure computes bounds for exact p-values.  EDIT THE FIRST \
LINE OF THIS PROCEDURE ONLY! The procedure uses Wilcoxon scores. The observed \
value is the maximum over all treatment-pairs (i,j) of \
Max(Sij,sumofscores-Sij), where sumofscores equals size(2*size+1) and Sij is \
the sum of scores corresponding to treatment i when compared with treatment \
j.
In this procedure, `size' is the size of each sample and `obs' is the \
observed value. This procedure can be rather time-consuming and should be \
used with care. \
\>", "Text",
  FontColor->RGBColor[1, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[size = 7; 
      obs = 45; \[IndentingNewLine]obsMW = obs - size*\((size + 1)\)/2; 
      totalp = 0; \nClear[palg]; \npalg[1, 0, 0] := x[1]; \n
      palg[1, \(-1\), a_] := 0; \npalg[1, a_, \(-1\)] := 0; \n
      palg[1, a_, b_] := \(palg[1, a, b] = 
          Module[{temp = 
                Expand[palg[1, a - 1, b]*x[1]^\((size - b)\) + \n\t\t\tpalg[
                      1, a, b - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[1, a, b, #] &, temp], 
              temp]]\); \npalg[2, 0, 0, 0] := x[1]*x[2]; \n
      palg[2, \(-1\), b_, c_] := 0; \npalg[2, a_, \(-1\), c_] := 0; \n
      palg[2, a_, b_, \(-1\)] := 0; \n
      palg[2, a_, b_, c_] := \(palg[2, a, b, c] = 
          Module[{temp = 
                Expand[palg[2, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[2, a, b - 1, c] + 
                    palg[2, a, b, c - 1]]}, 
            If[Head[temp] === Plus, Map[Bounding[2, a, b, c, #] &, temp], 
              temp]]\); \npalg[3, 0, 0, 0] := x[1]*x[2]*x[3]; \n
      palg[3, \(-1\), b_, c_] := 0; \npalg[3, a_, \(-1\), c_] := 0; \n
      palg[3, a_, b_, \(-1\)] := 0; \n
      palg[3, a_, b_, c_] := \(palg[3, a, b, c] = 
          Module[{temp = 
                Expand[palg[3, a - 1, b, c]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[3, a, b - 1, c]*
                      x[3]^\((size - c)\) + 
                    palg[3, a, b, c - 1]]}, \n\t\t\tIf[Head[temp] === Plus, 
              Map[Bounding[3, a, b, c, #] &, temp], temp]]\)\t; \n\t\n
      palg[4, 0, 0, 0, 0] := x[1]*x[2]*x[3]; \n
      palg[4, \(-1\), b_, c_, d_] := 0; \npalg[4, a_, \(-1\), c_, d_] := 0; \n
      palg[4, a_, b_, \(-1\), d_] := 0; \npalg[4, a_, b_, c_, \(-1\)] := 0; \n
      palg[4, a_, b_, c_, d_] := \(palg[4, a, b, c, d] = 
          Module[{temp = 
                Expand[palg[4, a - 1, b, c, d]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\)*\n\t\t\t\t\t\t\t\tx[
                          3]^\((size - d)\) + \n\t\t\tpalg[4, a, b - 1, c, 
                      d] + palg[4, a, b, c - 1, d] + 
                    palg[4, a, b, c, d - 1]]}, \n\t\t\tIf[
              Head[temp] === Plus, Map[Bounding[4, a, b, c, d, #] &, temp], 
              temp]]\)\t; \n\npalg[5, 0, 0, 0, 0] := x[1]*x[2]*x[3]; \t\n
      palg[5, \(-1\), b_, c_, d_] := 0; \npalg[5, a_, \(-1\), c_, d_] := 0; \n
      palg[5, a_, b_, \(-1\), d_] := 0; \npalg[5, a_, b_, c_, \(-1\)] := 0; \n
      palg[5, a_, b_, c_, d_] := \(palg[5, a, b, c, d] = 
          Module[{temp = 
                Expand[palg[5, a - 1, b, c, d]*x[1]^\((size - b)\)*
                      x[2]^\((size - c)\) + \n\t\t\tpalg[5, a, b - 1, c, d]*
                      x[3]^\((size - d)\) + palg[5, a, b, c - 1, d] + 
                    palg[5, a, b, c, d - 1]]}, \n\t\t\tIf[
              Head[temp] === Plus, Map[Bounding[5, a, b, c, d, #] &, temp], 
              temp]]\); \[IndentingNewLine]\[IndentingNewLine]\ \ \ \ \ For[
        aant = 1, 
        aant \[LessEqual] 6, \(aant++\), \n\t\tIf[aant == 1, 
          probhulp = \((palg[1, size, size]\  /. \ x[i_] -> 1)\)/
              Binomial[2*size, size]; \n\t\tprob = 
            10*probhulp - 15*probhulp^2]; \n\t\tIf[aant == 2, 
          probhulp2 = \((palg[2, size, size, size]\  /. \ x[i_] -> 1)\)/
              Multinomial[size, size, size]; prob = \(-30\)*probhulp2]; \n\t
        If[aant == 3, 
          prob = \((palg[3, size, size, size] /. \ x[i_] -> 1)\)*10/
                Multinomial[size, size, size]; extra = \(-7.5\)*prob]; \n\t
        If[aant == 4, 
          prob = \((palg[4, size, size, size, size] /. \ x[i_] -> 1)\)*20/
                Multinomial[size, size, size, size]]; \n\t
        If[aant == 5, 
          prob = \((palg[5, size, size, size, size] /. \ x[i_] -> 1)\)*60/
                Multinomial[size, size, size, 
                  size]]; \[IndentingNewLine]\ \ If[aant \[Equal] 6, 
          prob = 30*probhulp*probhulp2; 
          extra = extra - 135*probhulp*probhulp2]; \n\t
        totalp = totalp + prob; Print[N[prob]]]; 
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