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1 package Statistics::Distributions;
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2
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3 use strict;
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4 use vars qw($VERSION @ISA @EXPORT @EXPORT_OK);
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5 use constant PI => 3.1415926536;
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6 use constant SIGNIFICANT => 5; # number of significant digits to be returned
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7
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8 require Exporter;
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9
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10 @ISA = qw(Exporter AutoLoader);
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11 # Items to export into callers namespace by default. Note: do not export
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12 # names by default without a very good reason. Use EXPORT_OK instead.
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13 # Do not simply export all your public functions/methods/constants.
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14 @EXPORT_OK = qw(chisqrdistr tdistr fdistr udistr uprob chisqrprob tprob fprob);
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15 $VERSION = '1.02';
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16
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17 # Preloaded methods go here.
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18
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19 sub chisqrdistr { # Percentage points X^2(x^2,n)
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20 my ($n, $p) = @_;
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21 if ($n <= 0 || abs($n) - abs(int($n)) != 0) {
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22 die "Invalid n: $n\n"; # degree of freedom
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23 }
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24 if ($p <= 0 || $p > 1) {
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25 die "Invalid p: $p\n";
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26 }
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27 return precision_string(_subchisqr($n, $p));
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28 }
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29
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30 sub udistr { # Percentage points N(0,1^2)
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31 my ($p) = (@_);
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32 if ($p > 1 || $p <= 0) {
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33 die "Invalid p: $p\n";
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34 }
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35 return precision_string(_subu($p));
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36 }
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37
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38 sub tdistr { # Percentage points t(x,n)
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39 my ($n, $p) = @_;
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40 if ($n <= 0 || abs($n) - abs(int($n)) != 0) {
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41 die "Invalid n: $n\n";
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42 }
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43 if ($p <= 0 || $p >= 1) {
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44 die "Invalid p: $p\n";
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45 }
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46 return precision_string(_subt($n, $p));
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47 }
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48
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49 sub fdistr { # Percentage points F(x,n1,n2)
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50 my ($n, $m, $p) = @_;
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51 if (($n<=0) || ((abs($n)-(abs(int($n))))!=0)) {
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52 die "Invalid n: $n\n"; # first degree of freedom
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53 }
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54 if (($m<=0) || ((abs($m)-(abs(int($m))))!=0)) {
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55 die "Invalid m: $m\n"; # second degree of freedom
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56 }
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57 if (($p<=0) || ($p>1)) {
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58 die "Invalid p: $p\n";
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59 }
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60 return precision_string(_subf($n, $m, $p));
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61 }
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62
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63 sub uprob { # Upper probability N(0,1^2)
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64 my ($x) = @_;
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65 return precision_string(_subuprob($x));
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66 }
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67
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68 sub chisqrprob { # Upper probability X^2(x^2,n)
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69 my ($n,$x) = @_;
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70 if (($n <= 0) || ((abs($n) - (abs(int($n)))) != 0)) {
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71 die "Invalid n: $n\n"; # degree of freedom
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72 }
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73 return precision_string(_subchisqrprob($n, $x));
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74 }
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75
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76 sub tprob { # Upper probability t(x,n)
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77 my ($n, $x) = @_;
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78 if (($n <= 0) || ((abs($n) - abs(int($n))) !=0)) {
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79 die "Invalid n: $n\n"; # degree of freedom
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80 }
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81 return precision_string(_subtprob($n, $x));
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82 }
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83
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84 sub fprob { # Upper probability F(x,n1,n2)
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85 my ($n, $m, $x) = @_;
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86 if (($n<=0) || ((abs($n)-(abs(int($n))))!=0)) {
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87 die "Invalid n: $n\n"; # first degree of freedom
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88 }
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89 if (($m<=0) || ((abs($m)-(abs(int($m))))!=0)) {
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90 die "Invalid m: $m\n"; # second degree of freedom
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91 }
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92 return precision_string(_subfprob($n, $m, $x));
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93 }
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94
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95
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96 sub _subfprob {
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97 my ($n, $m, $x) = @_;
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98 my $p;
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99
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100 if ($x<=0) {
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101 $p=1;
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102 } elsif ($m % 2 == 0) {
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103 my $z = $m / ($m + $n * $x);
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104 my $a = 1;
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105 for (my $i = $m - 2; $i >= 2; $i -= 2) {
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106 $a = 1 + ($n + $i - 2) / $i * $z * $a;
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107 }
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108 $p = 1 - ((1 - $z) ** ($n / 2) * $a);
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109 } elsif ($n % 2 == 0) {
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110 my $z = $n * $x / ($m + $n * $x);
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111 my $a = 1;
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112 for (my $i = $n - 2; $i >= 2; $i -= 2) {
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113 $a = 1 + ($m + $i - 2) / $i * $z * $a;
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114 }
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115 $p = (1 - $z) ** ($m / 2) * $a;
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116 } else {
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117 my $y = atan2(sqrt($n * $x / $m), 1);
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118 my $z = sin($y) ** 2;
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119 my $a = ($n == 1) ? 0 : 1;
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120 for (my $i = $n - 2; $i >= 3; $i -= 2) {
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121 $a = 1 + ($m + $i - 2) / $i * $z * $a;
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122 }
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123 my $b = PI;
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124 for (my $i = 2; $i <= $m - 1; $i += 2) {
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125 $b *= ($i - 1) / $i;
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126 }
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127 my $p1 = 2 / $b * sin($y) * cos($y) ** $m * $a;
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128
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129 $z = cos($y) ** 2;
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130 $a = ($m == 1) ? 0 : 1;
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131 for (my $i = $m-2; $i >= 3; $i -= 2) {
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132 $a = 1 + ($i - 1) / $i * $z * $a;
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133 }
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134 $p = max(0, $p1 + 1 - 2 * $y / PI
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135 - 2 / PI * sin($y) * cos($y) * $a);
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136 }
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137 return $p;
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138 }
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139
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140
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141 sub _subchisqrprob {
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142 my ($n,$x) = @_;
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143 my $p;
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144
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145 if ($x <= 0) {
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146 $p = 1;
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147 } elsif ($n > 100) {
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148 $p = _subuprob((($x / $n) ** (1/3)
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149 - (1 - 2/9/$n)) / sqrt(2/9/$n));
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150 } elsif ($x > 400) {
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151 $p = 0;
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152 } else {
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153 my ($a, $i, $i1);
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154 if (($n % 2) != 0) {
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155 $p = 2 * _subuprob(sqrt($x));
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156 $a = sqrt(2/PI) * exp(-$x/2) / sqrt($x);
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157 $i1 = 1;
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158 } else {
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159 $p = $a = exp(-$x/2);
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160 $i1 = 2;
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161 }
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162
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163 for ($i = $i1; $i <= ($n-2); $i += 2) {
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164 $a *= $x / $i;
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165 $p += $a;
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166 }
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167 }
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168 return $p;
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169 }
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170
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171 sub _subu {
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172 my ($p) = @_;
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173 my $y = -log(4 * $p * (1 - $p));
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174 my $x = sqrt(
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175 $y * (1.570796288
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176 + $y * (.03706987906
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177 + $y * (-.8364353589E-3
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178 + $y *(-.2250947176E-3
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179 + $y * (.6841218299E-5
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180 + $y * (0.5824238515E-5
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181 + $y * (-.104527497E-5
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182 + $y * (.8360937017E-7
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183 + $y * (-.3231081277E-8
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184 + $y * (.3657763036E-10
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185 + $y *.6936233982E-12)))))))))));
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186 $x = -$x if ($p>.5);
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187 return $x;
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188 }
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189
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190 sub _subuprob {
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191 my ($x) = @_;
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192 my $p = 0; # if ($absx > 100)
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193 my $absx = abs($x);
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194
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195 if ($absx < 1.9) {
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196 $p = (1 +
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197 $absx * (.049867347
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198 + $absx * (.0211410061
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199 + $absx * (.0032776263
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200 + $absx * (.0000380036
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201 + $absx * (.0000488906
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202 + $absx * .000005383)))))) ** -16/2;
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203 } elsif ($absx <= 100) {
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204 for (my $i = 18; $i >= 1; $i--) {
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205 $p = $i / ($absx + $p);
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206 }
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207 $p = exp(-.5 * $absx * $absx)
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208 / sqrt(2 * PI) / ($absx + $p);
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209 }
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210
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211 $p = 1 - $p if ($x<0);
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212 return $p;
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213 }
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214
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215
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216 sub _subt {
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217 my ($n, $p) = @_;
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218
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219 if ($p >= 1 || $p <= 0) {
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220 die "Invalid p: $p\n";
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221 }
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222
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223 if ($p == 0.5) {
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224 return 0;
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225 } elsif ($p < 0.5) {
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226 return - _subt($n, 1 - $p);
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227 }
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228
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229 my $u = _subu($p);
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230 my $u2 = $u ** 2;
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231
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232 my $a = ($u2 + 1) / 4;
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233 my $b = ((5 * $u2 + 16) * $u2 + 3) / 96;
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234 my $c = (((3 * $u2 + 19) * $u2 + 17) * $u2 - 15) / 384;
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235 my $d = ((((79 * $u2 + 776) * $u2 + 1482) * $u2 - 1920) * $u2 - 945)
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236 / 92160;
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237 my $e = (((((27 * $u2 + 339) * $u2 + 930) * $u2 - 1782) * $u2 - 765) * $u2
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238 + 17955) / 368640;
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239
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240 my $x = $u * (1 + ($a + ($b + ($c + ($d + $e / $n) / $n) / $n) / $n) / $n);
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241
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242 if ($n <= log10($p) ** 2 + 3) {
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243 my $round;
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244 do {
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245 my $p1 = _subtprob($n, $x);
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246 my $n1 = $n + 1;
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247 my $delta = ($p1 - $p)
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248 / exp(($n1 * log($n1 / ($n + $x * $x))
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249 + log($n/$n1/2/PI) - 1
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250 + (1/$n1 - 1/$n) / 6) / 2);
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251 $x += $delta;
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252 $round = sprintf("%.".abs(int(log10(abs $x)-4))."f",$delta);
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253 } while (($x) && ($round != 0));
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254 }
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255 return $x;
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256 }
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257
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258 sub _subtprob {
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259 my ($n, $x) = @_;
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260
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261 my ($a,$b);
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262 my $w = atan2($x / sqrt($n), 1);
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263 my $z = cos($w) ** 2;
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264 my $y = 1;
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265
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266 for (my $i = $n-2; $i >= 2; $i -= 2) {
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267 $y = 1 + ($i-1) / $i * $z * $y;
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268 }
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269
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270 if ($n % 2 == 0) {
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271 $a = sin($w)/2;
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272 $b = .5;
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273 } else {
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274 $a = ($n == 1) ? 0 : sin($w)*cos($w)/PI;
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275 $b= .5 + $w/PI;
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276 }
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277 return max(0, 1 - $b - $a * $y);
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278 }
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279
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280 sub _subf {
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281 my ($n, $m, $p) = @_;
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282 my $x;
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283
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284 if ($p >= 1 || $p <= 0) {
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285 die "Invalid p: $p\n";
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286 }
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287
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288 if ($p == 1) {
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289 $x = 0;
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290 } elsif ($m == 1) {
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291 $x = 1 / (_subt($n, 0.5 - $p / 2) ** 2);
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292 } elsif ($n == 1) {
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293 $x = _subt($m, $p/2) ** 2;
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294 } elsif ($m == 2) {
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295 my $u = _subchisqr($m, 1 - $p);
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296 my $a = $m - 2;
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297 $x = 1 / ($u / $m * (1 +
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298 (($u - $a) / 2 +
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299 (((4 * $u - 11 * $a) * $u + $a * (7 * $m - 10)) / 24 +
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300 (((2 * $u - 10 * $a) * $u + $a * (17 * $m - 26)) * $u
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301 - $a * $a * (9 * $m - 6)
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302 )/48/$n
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303 )/$n
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304 )/$n));
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305 } elsif ($n > $m) {
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306 $x = 1 / _subf2($m, $n, 1 - $p)
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307 } else {
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308 $x = _subf2($n, $m, $p)
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309 }
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310 return $x;
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311 }
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312
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313 sub _subf2 {
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314 my ($n, $m, $p) = @_;
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315 my $u = _subchisqr($n, $p);
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316 my $n2 = $n - 2;
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317 my $x = $u / $n *
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318 (1 +
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319 (($u - $n2) / 2 +
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320 (((4 * $u - 11 * $n2) * $u + $n2 * (7 * $n - 10)) / 24 +
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321 (((2 * $u - 10 * $n2) * $u + $n2 * (17 * $n - 26)) * $u
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322 - $n2 * $n2 * (9 * $n - 6)) / 48 / $m) / $m) / $m);
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323 my $delta;
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324 do {
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325 my $z = exp(
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326 (($n+$m) * log(($n+$m) / ($n * $x + $m))
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327 + ($n - 2) * log($x)
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328 + log($n * $m / ($n+$m))
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329 - log(4 * PI)
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330 - (1/$n + 1/$m - 1/($n+$m))/6
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331 )/2);
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332 $delta = (_subfprob($n, $m, $x) - $p) / $z;
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333 $x += $delta;
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334 } while (abs($delta)>3e-4);
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335 return $x;
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336 }
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337
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338 sub _subchisqr {
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339 my ($n, $p) = @_;
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340 my $x;
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341
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342 if (($p > 1) || ($p <= 0)) {
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343 die "Invalid p: $p\n";
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344 } elsif ($p == 1){
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345 $x = 0;
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346 } elsif ($n == 1) {
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347 $x = _subu($p / 2) ** 2;
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348 } elsif ($n == 2) {
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349 $x = -2 * log($p);
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350 } else {
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351 my $u = _subu($p);
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352 my $u2 = $u * $u;
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353
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354 $x = max(0, $n + sqrt(2 * $n) * $u
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355 + 2/3 * ($u2 - 1)
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356 + $u * ($u2 - 7) / 9 / sqrt(2 * $n)
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357 - 2/405 / $n * ($u2 * (3 *$u2 + 7) - 16));
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358
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359 if ($n <= 100) {
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360 my ($x0, $p1, $z);
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361 do {
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362 $x0 = $x;
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363 if ($x < 0) {
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364 $p1 = 1;
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365 } elsif ($n>100) {
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366 $p1 = _subuprob((($x / $n)**(1/3) - (1 - 2/9/$n))
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367 / sqrt(2/9/$n));
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368 } elsif ($x>400) {
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369 $p1 = 0;
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370 } else {
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371 my ($i0, $a);
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372 if (($n % 2) != 0) {
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373 $p1 = 2 * _subuprob(sqrt($x));
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374 $a = sqrt(2/PI) * exp(-$x/2) / sqrt($x);
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375 $i0 = 1;
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376 } else {
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377 $p1 = $a = exp(-$x/2);
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378 $i0 = 2;
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379 }
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380
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381 for (my $i = $i0; $i <= $n-2; $i += 2) {
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382 $a *= $x / $i;
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383 $p1 += $a;
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384 }
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385 }
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386 $z = exp((($n-1) * log($x/$n) - log(4*PI*$x)
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387 + $n - $x - 1/$n/6) / 2);
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388 $x += ($p1 - $p) / $z;
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389 $x = sprintf("%.5f", $x);
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390 } while (($n < 31) && (abs($x0 - $x) > 1e-4));
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391 }
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392 }
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393 return $x;
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394 }
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395
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396 sub log10 {
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397 my $n = shift;
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398 return log($n) / log(10);
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399 }
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400
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401 sub max {
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402 my $max = shift;
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403 my $next;
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404 while (@_) {
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405 $next = shift;
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406 $max = $next if ($next > $max);
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407 }
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408 return $max;
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409 }
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410
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411 sub min {
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412 my $min = shift;
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413 my $next;
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414 while (@_) {
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415 $next = shift;
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416 $min = $next if ($next < $min);
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417 }
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418 return $min;
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419 }
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420
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421 sub precision {
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422 my ($x) = @_;
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423 return abs int(log10(abs $x) - SIGNIFICANT);
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424 }
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425
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426 sub precision_string {
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427 my ($x) = @_;
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428 if ($x) {
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429 return sprintf "%." . precision($x) . "f", $x;
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430 } else {
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431 return "0";
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432 }
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433 }
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434
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435
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436 # Autoload methods go after =cut, and are processed by the autosplit program.
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437
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438 1;
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439 __END__
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440 # Below is the stub of documentation for your module. You better edit it!
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441
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442 =head1 NAME
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443
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444 Statistics::Distributions - Perl module for calculating critical values and upper probabilities of common statistical distributions
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445
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446 =head1 SYNOPSIS
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447
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448 use Statistics::Distributions;
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449
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450 $chis=Statistics::Distributions::chisqrdistr (2,.05);
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451 print "Chi-squared-crit (2 degrees of freedom, 95th percentile "
|
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452 ."= 0.05 level) = $chis\n";
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453
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454 $u=Statistics::Distributions::udistr (.05);
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455 print "u-crit (95th percentile = 0.05 level) = $u\n";
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456
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457 $t=Statistics::Distributions::tdistr (1,.005);
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458 print "t-crit (1 degree of freedom, 99.5th percentile = 0.005 level) "
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|
|
459 ."= $t\n";
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460
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|
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461 $f=Statistics::Distributions::fdistr (1,3,.01);
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|
462 print "F-crit (1 degree of freedom in numerator, 3 degrees of freedom "
|
|
|
463 ."in denominator, 99th percentile = 0.01 level) = $f\n";
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|
|
464
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|
|
465 $uprob=Statistics::Distributions::uprob (-0.85);
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|
|
466 print "upper probability of the u distribution (u = -0.85): Q(u) "
|
|
|
467 ."= 1-G(u) = $uprob\n";
|
|
|
468
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|
|
469 $chisprob=Statistics::Distributions::chisqrprob (3,6.25);
|
|
|
470 print "upper probability of the chi-square distribution (3 degrees "
|
|
|
471 ."of freedom, chi-squared = 6.25): Q = 1-G = $chisprob\n";
|
|
|
472
|
|
|
473 $tprob=Statistics::Distributions::tprob (3,6.251);
|
|
|
474 print "upper probability of the t distribution (3 degrees of "
|
|
|
475 ."freedom, t = 6.251): Q = 1-G = $tprob\n";
|
|
|
476
|
|
|
477 $fprob=Statistics::Distributions::fprob (3,5,.625);
|
|
|
478 print "upper probability of the F distribution (3 degrees of freedom "
|
|
|
479 ."in numerator, 5 degrees of freedom in denominator, F = 6.25): "
|
|
|
480 ."Q = 1-G = $fprob\n";
|
|
|
481
|
|
|
482 =head1 DESCRIPTION
|
|
|
483
|
|
|
484 This Perl module calculates percentage points (5 significant digits) of the u (standard normal) distribution, the student's t distribution, the chi-square distribution and the F distribution. It can also calculate the upper probability (5 significant digits) of the u (standard normal), the chi-square, the t and the F distribution.
|
|
|
485 These critical values are needed to perform statistical tests, like the u test, the t test, the F test and the chi-squared test, and to calculate confidence intervals.
|
|
|
486
|
|
|
487 If you are interested in more precise algorithms you could look at:
|
|
|
488 StatLib: http://lib.stat.cmu.edu/apstat/ ;
|
|
|
489 Applied Statistics Algorithms by Griffiths, P. and Hill, I.D., Ellis Horwood: Chichester (1985)
|
|
|
490
|
|
|
491 =head1 BUGS
|
|
|
492
|
|
|
493 This final version 1.02 has been released after more than one year without a bug report on the previous version 0.07.
|
|
|
494 Nevertheless, if you find any bugs or oddities, please do inform the author.
|
|
|
495
|
|
|
496 =head1 INSTALLATION
|
|
|
497
|
|
|
498 See perlmodinstall for information and options on installing Perl modules.
|
|
|
499
|
|
|
500 =head1 AVAILABILITY
|
|
|
501
|
|
|
502 The latest version of this module is available from the Distribution Perl Archive Network (CPAN). Please visit http://www.cpan.org/ to find a CPAN site near you or see http://www.cpan.org/authors/id/M/MI/MIKEK/ .
|
|
|
503
|
|
|
504 =head1 AUTHOR
|
|
|
505
|
|
|
506 Michael Kospach <mike.perl@gmx.at>
|
|
|
507
|
|
|
508 Nice formating, simplification and bug repair by Matthias Trautner Kromann <mtk@id.cbs.dk>
|
|
|
509
|
|
|
510 =head1 COPYRIGHT
|
|
|
511
|
|
|
512 Copyright 2003 Michael Kospach. All rights reserved.
|
|
|
513
|
|
|
514 This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
|
|
|
515
|
|
|
516 =head1 SEE ALSO
|
|
|
517
|
|
|
518 Statistics::ChiSquare, Statistics::Table::t, Statistics::Table::F, perl(1).
|
|
|
519
|
|
|
520 =cut
|
|
|
521
|
|
|
522
|
|
|
523
|
|
|
524
|
|
|
525
|
|
|
526
|
|
|
527
|
|
|
528
|
|
|
529
|
