/// <summary>
/// hypothesis testing for more than two classes using ANOVA
///
/// Given that:
/// H_0 : mu_1 = mu_2 = ... mu_k (where k is the number of classes)
/// H_A : mu != null_value
///
/// p-value = Pr(> f) is the probability of at least as large a ratio between the "between" and "within" group variablity if in fact the means of all groups are equal
/// if(p-value < significance_level) reject H_0
/// </summary>
/// <param name="groupedSample">The sample groupped based on the classes</param>
/// <param name="pValue"></param>
/// <param name="significance_level">p-value = Pr(> f) is the probability of at least as large a ratio between the "between" and "within" group variablity if in fact the means of all groups are equal</param>
/// <returns>True if H_0 is rejected; False if H_0 is failed to be rejected</returns>
public static bool RunANOVA(double[] totalSample, int[] grpCat, out ANOVA output, double significance_level = 0.05)
{
output = new ANOVA();
Dictionary <int, List <double> > groupedSample = new Dictionary <int, List <double> >();
for (int i = 0; i < totalSample.Length; ++i)
{
int grpId = grpCat[i];
double sampleVal = totalSample[i];
List <double> grp = null;
if (groupedSample.ContainsKey(grpId))
{
grp = groupedSample[grpId];
}
else
{
grp = new List <double>();
groupedSample[grpId] = grp;
}
grp.Add(sampleVal);
}
double grand_mean;
//Sum of squares measures the total variablity
output.SST = GetSST(totalSample, out grand_mean); //sum of squares total
output.SSG = GetSSG(groupedSample, grand_mean); //sum of squares group, which is known as explained variablity (explained by the group variable)
output.SSE = output.SST - output.SSG; //sum of squares error, which is known as unexplained variability (unexplained by the group variable, due to other reasons)
//Degrees of freedom
output.dfT = totalSample.Length - 1; //degrees of freedom total
output.dfG = groupedSample.Count - 1; //degrees of freedom group
output.dfE = output.dfT - output.dfG; // degrees of freedom error
//Mean squares measures variability between and within groups, calculated as the total variability (sum of squares) scaled by the associated degrees of freedom
output.MSG = output.SSG / output.dfG; // mean squares group : between group variability
output.MSE = output.SSE / output.dfE; // mean squares error : within group variablity
output.Intercepts = GetIntercepts(GetMeanWithinGroup(groupedSample));
//f statistic: ratio of the between group variablity and within group variablity
output.F = output.MSG / output.MSE;
try
{
//p-value = Pr(> f) is the probability of at least as large a ratio between the "between" and "within" group variablity if in fact the means of all groups are equal
output.pValue = 1 - FDistribution.GetPercentile(output.F, output.dfG, output.dfE);
}
catch
{
}
return(output.RejectH0 = output.pValue < significance_level);
}
/// <summary>
/// Return a matrix of reject H_0, for which rejectH0Matrix[i][j] = true if the pairwise comparison provide enough evidence that group[i] and group[j] does not have the same mean
///
/// Hypotheses for a pair of groups, group[i] and group[j]:
/// H_0 : mu_i = mu_j
/// H_A : mu_i != mu_j
/// </summary>
/// <param name="groupedSample">sampled groupped by classes</param>
/// <param name="significance_level">significance level for the test</param>
/// <returns>RejectH0 matrix: rejctH0Matrix[i][j] = true if the test provide enough evidence that group[i] and group[j] does not have the same mean</returns>
public static bool[][] RejectH0(double[] sample, int[] grpCat, double significance_level = 0.05)
{
ANOVA anova_output;
ANOVA.RunANOVA(sample, grpCat, out anova_output, significance_level);
Dictionary <int, List <double> > groupedSample = new Dictionary <int, List <double> >();
for (int i = 0; i < sample.Length; ++i)
{
int grpId = grpCat[i];
double sampleVal = sample[i];
List <double> grp = null;
if (groupedSample.ContainsKey(grpId))
{
grp = groupedSample[grpId];
}
else
{
grp = new List <double>();
groupedSample[grpId] = grp;
}
grp.Add(sampleVal);
}
int k = groupedSample.Count; // number of groups
double alpha_adj = BonferroniCorrection(significance_level, k);
bool[][] rejectH0Matrix = new bool[k][];
for (int i = 0; i < k; ++k)
{
rejectH0Matrix[i] = new bool[k];
}
List <int> groupIdList = groupedSample.Keys.ToList();
for (int i = 0; i < k - 1; ++i)
{
List <double> group1 = groupedSample[groupIdList[i]];
for (int j = i + 1; j < k; ++j)
{
List <double> group2 = groupedSample[groupIdList[j]];
double pValue = PairwiseCompare(group1, group2, anova_output);
bool reject_H0 = pValue < alpha_adj;
rejectH0Matrix[i][j] = reject_H0;
rejectH0Matrix[j][i] = reject_H0;
}
}
return(rejectH0Matrix);
}
/// <summary>
/// Pairwise comparison of group1 and group2
/// </summary>
/// <param name="group1">random sample from class 1</param>
/// <param name="group2">random sample from class 2</param>
/// <param name="anova">parameters obtained after ANOVA</param>
/// <returns>p-value = P(observed or more extreme values | H_0 is true)</returns>
public static double PairwiseCompare(List <double> group1, List <double> group2, ANOVA anova)
{
double x_bar1 = Mean.GetMean(group1);
double x_bar2 = Mean.GetMean(group2);
int n1 = group1.Count;
int n2 = group2.Count;
int null_value = 0;
double t = GetTStatistic(x_bar1, x_bar2, n1, n2, null_value, anova.MSE);
double pValue = GetPValue(t, anova.dfE);
return(pValue);
}
