#81

Adding H-Test and Documentation
&
Correction of chi-squared-distribution (PDF&CDF)

HTest.fsx

@@ -14,18 +14,6 @@ let myAxisRange title range = LinearAxis.init(Title=title,Range=Range.MinMax ran
 let styleChart x y chart = chart |> Chart.withX_Axis (myAxis x) |> Chart.withY_Axis (myAxis y)
 let styleChartRange x y rx ry chart = chart |> Chart.withX_Axis (myAxisRange x rx) |> Chart.withY_Axis (myAxisRange y ry)
 
-(**
-#Statistical testing
-FSharp.Stats provides hypothesis tests for different applications.
-A hypothesis test is a statistical test that is used to determine whether there is enough evidence 
-in a sample of data to infer that a certain condition is true for the entire population. 
-A hypothesis test examines two opposing hypotheses about a population: the null hypothesis and the alternative hypothesis.
-<a name="TestStatistics"></a>
-##Test Statistics
-<a name="Anova"></a>
-##Anova
-*)
-
 open FSharp.Stats
 open FSharp.Stats.Testing
 
@@ -56,4 +44,19 @@ let groupB = seq {45.;55.;60.;70.;72.}
 let groupC = seq {18.;30.;34.;40.;44.} 
 let samples = seq{groupA;groupB;groupC}
 
-// calculation of p-Value
+// calculation of p-Value
+HTest.createHTest samples 
+
+(*
+        { Statistic = 6.72
+        DegreesOfFreedom = 2.0
+        PValueLeft = 0.9652647411
+        PValueRight = 0.03473525894
+        PValue = 0.06947051789 }
+*)
+// PValueRight equals the alpha level 
+
+(**
+The implemented H-test is testing for double values in the data. 
+Double values lead to ties in the ranking, and are corrected by using a correction factor. 
+*)

docsrc/content/Testing.fsx

@@ -325,6 +325,56 @@ let fTestFromParameters = FTest.testVariancesFromVarAndDof sampleF1 sampleF2
     Using a significance level of 0.05 the sample variances do differ significantly.
 *)
 
+(** 
+##H-Test
+The H test is also known as Kruskal-Wallis one-way analysis-of-variance-by-ranks and is the nonparametric equivalent of one-way ANOVA. 
+It is a non-parametric test for comparing the means of more than two independent samples (equal or different sample size), and therefor is an extension of Wilcoxon-Mann-Whitney two sample test.
+Testing with H test gives information whether the samples are from the same distribution.
+A benefit of the H-test is that it does not require normal distribution of the samples.
+The downside is that there is no information which samples are different from each other, or how many differences occur. For further investigation a Post Hoc test is recommended. 
+The distribution of the H test statistic is approximated by chi-square distribution with degrees of freedom - 1. 
+
+Prerequisites:
+
+  - random and independent samples
+
+  - observations are from populations with same shape of distribution
+
+  - nominal scale, ordinal scale, ratio scale or interval scale data
+
+
+References:
+
+  - E. Ostertagová,  Methodology and Application of the Kruskal-Wallis Test (2014)
+
+  - Y. Chan, RP Walmsley, Learning and understanding the Kruskal-Wallis one-way analysis-of-variance-by-ranks test for differences among three or more independent groups (1997)
+
+*H-test*
+*)
+
+// input : seq{seq<float>} 
+let groupA = seq {23.;41.;54.;66.;78.} 
+let groupB = seq {45.;55.;60.;70.;72.}
+let groupC = seq {18.;30.;34.;40.;44.} 
+let samples = seq{groupA;groupB;groupC}
+
+// calculation of p-Value
+HTest.createHTest samples 
+
+(*
+        { Statistic = 6.72
+        DegreesOfFreedom = 2.0
+        PValueLeft = 0.9652647411
+        PValueRight = 0.03473525894
+        PValue = 0.06947051789 }
+*)
+// PValueRight equals the alpha level 
+
+(**
+The implemented H-test is testing for double values in the data. 
+Double values lead to ties in the ranking, and are corrected by using a correction factor. 
+*)
+
 
 (**
 <a name="ChiSquareTest"></a>

src/FSharp.Stats/Distributions/Continuous.fs

@@ -49,16 +49,13 @@ module Continuous =
             if x < 0.0 || dof < 1. then
                 0.0
             else
-                let k = float dof * 0.5
-                let x = x * 0.5
-                if dof = 2. then
-                    exp (-1. * x)
-                else
-                    let pValue = SpecialFunctions.Gamma.lowerIncomplete k x // incGamma -> gamma lower incomplete
-                    if (isNan pValue) || (isInf pValue) ||  (pValue <= 1e-8) then
-                        1e-14
-                    else
-                        1.- pValue / (SpecialFunctions.Gamma.gamma k)  
+                let gammaF = Gamma.gamma (dof/2.)
+                let k = 2.**(dof/2.)
+                let fraction = (1./((k)*gammaF))
+                let ex1 = (x**((dof/2.)-1.))
+                let ex2 = exp(-x/2.)
+                let pdffunction = fraction*(ex1*ex2)
+                pdffunction 
 
         /// Computes the logarithm of probability density function.
         static member PDFLn dof x = 
@@ -73,7 +70,7 @@ module Continuous =
             if dof = 0. then 
                 if x > 0. then 1.
                 else 0.
-            else (Gamma.lowerIncomplete (dof /2.0) (x/2.0) )/ (Gamma.gamma (dof/2.0))
+            else Gamma.lowerIncomplete (dof/2.) (x/2.)
 
         /// Returns the support of the exponential distribution: [0, Positive Infinity).
         static member Support dof =

src/FSharp.Stats/HTest.fs

@@ -6,7 +6,7 @@ module HTest =
     open FSharp.Stats
     // H-test / one-way ANOVA of ranks 
     // input : seq{seq<float>} 
-    let htest (samples : seq<#seq<float>>) = 
+    let createHTest (samples : seq<#seq<float>>) = 
         // calculating n for each group 
         let n = Seq.map Seq.length samples |> Seq.map float