/// <summary>
/// GOF test for one categorical variable with more than two levels.
///
/// Hypotheses are:
/// H_0 : actual distribution of each level = expected distribution of each level
/// H_1 : actual distribution of each level != expected distribution of each level
///
/// p-value = P(observed or more mismatch of expected and actual level distribution | H_0 is true)
///
/// Reject H_0 if p-value < alpha
/// </summary>
/// <param name="countOfEachLevel">The count of each level in the sample data for the categorical variable</param>
/// <param name="expectedPercentageOfEachLevel">The expected distribution / percentage of each level in the population for the categorical variable</param>
/// <param name="pValue">p-value which is P(observed or more extreme mismatch of expected and actual level distribution | H_0 is true</param>
/// <param name="significance_level">alpha</param>
/// <returns>True if H_0 is rejected; False if H_0 is failed to be rejected</returns>
public bool RejectH0(int[] observedCountInEachLevel, double[] expectedPercentageOfEachLevel, out double pValue, double significance_level = 0.05)
{
int sampleSize = 0;
int countOfLevels = observedCountInEachLevel.Length;
for (int i = 0; i < countOfLevels; ++i)
{
sampleSize += observedCountInEachLevel[i];
}
int[] expectedCountInEachLevel = new int[countOfLevels];
int r = sampleSize;
for (int i = 0; i < countOfLevels; ++i)
{
expectedCountInEachLevel[i] = (int)(expectedPercentageOfEachLevel[i] * sampleSize);
r -= expectedCountInEachLevel[i];
}
if (r > 0)
{
expectedCountInEachLevel[0] += r;
}
double ChiSq = 0;
for (int i = 0; i < countOfLevels; ++i)
{
ChiSq += System.Math.Pow(observedCountInEachLevel[i] - expectedCountInEachLevel[i], 2) / expectedCountInEachLevel[i];
}
pValue = 1 - ChiSquare.GetPercentile(ChiSq, countOfLevels - 1);
return(pValue < significance_level);
}
/// <summary>
/// Chi^2 independence test for categorical variables, var1 and var2
///
/// The hypotheses are:
/// H_0 : variable 1 is independent of variable 2
/// H_A : variable 1 and variable 2 are dependent
///
/// p-value = P(observed or more extreme events that favors H_A | H_0)
///
/// Now assuming H_0 is true, that is, the var1 and var2 are independent,
/// This implies the distribution of each level of var1 in each level of var2 should be the same
/// In other words, the expected distribution of each level of var1 in each level of var2 is given by distributionInEachLevel_var1
/// Now we can build a new contingency table containing the expected count corresponding to each level of both var1 and var2
///
/// Reject H_0 if p-value < alpha
/// </summary>
/// <param name="contingency_table">The contingency table in which each cell contains the counts of records in the sample data that matches the row (i.e. a var1 level) and col (i.e. a var2 level)</param>
/// <param name="pValue">p-value = P(observed or more extreme events that favors H_A | H_0)</param>
/// <param name="signficance_level">alpha</param>
/// <returns>True if H_0 is rejected; False if H_0 is failed to be rejected</returns>
public bool RejectH0(int[][] contingency_table, out double pValue, double signficance_level = 0.05)
{
int countOfLevels_var1 = contingency_table.Length;
int countOfLevels_var2 = contingency_table[0].Length;
int sampleSize = 0;
int[] countInEachLevel_var1 = new int[countOfLevels_var1];
for (int row = 0; row < countOfLevels_var1; ++row)
{
int countInLevel = 0;
for (int col = 0; col < countOfLevels_var2; ++col)
{
countInLevel += contingency_table[row][col];
}
countInEachLevel_var1[row] = countInLevel;
sampleSize += countInLevel;
}
double[] distributionInEachLevel_var1 = new double[countOfLevels_var1];
for (int row = 0; row < countOfLevels_var1; ++row)
{
distributionInEachLevel_var1[row] = (double)countInEachLevel_var1[row] / sampleSize;
}
int[] countInEachLevel_var2 = new int[countOfLevels_var2];
for (int col = 0; col < countOfLevels_var2; ++col)
{
int countInLevel = 0;
for (int row = 0; row < countOfLevels_var1; ++row)
{
countInLevel += contingency_table[row][col];
}
countInEachLevel_var2[col] = countInLevel;
}
//Now assuming H_0 is true, that is, the var1 and var2 are independent,
//This implies the distribution of each level of var1 in each level of var2 should be the same
//In other words, the expected distribution of each level of var1 in each level of var2 is given by distributionInEachLevel_var1
//Now we can build a new contingency table containing the expected count corresponding to each level of both var1 and var2
double[][] expected_contingency_table = new double[countOfLevels_var1][];
for (int row = 0; row < countOfLevels_var1; ++row)
{
expected_contingency_table[row] = new double[countOfLevels_var2];
for (int col = 0; col < countOfLevels_var2; ++col)
{
expected_contingency_table[row][col] = countInEachLevel_var2[col] * distributionInEachLevel_var1[row];
}
}
double ChiSq = 0;
for (int row = 0; row < countOfLevels_var1; ++row)
{
for (int col = 0; col < countOfLevels_var2; ++col)
{
ChiSq += System.Math.Pow(contingency_table[row][col] - expected_contingency_table[row][col], 2) / expected_contingency_table[row][col];
}
}
int df = (countOfLevels_var1 - 1) * (countOfLevels_var2 - 1);
pValue = 1 - ChiSquare.GetPercentile(ChiSq, df);
return(pValue < signficance_level);
}
