/// <summary>
/// Two-sided or one-sided test for whether statitics of two variables are equal in the true population, var1 and var2 are paired and dependent
///
/// Hypotheses are:
/// H_0: mu_var1 = mu_var2
/// H_1: mu_var1 != mu_var2
///
/// The hypotheses can be written as
/// H_0: mu_var1 - mu_var2 = 0
/// H_1: mu_var1 - mu_var2 != 0
///
/// By Central Limt Theorem:
/// sample_mean_var1 - sample_mean_var2 ~ N(0, SE), where null_value = 0 and SE is the standard error of the sampling distribution
///
/// p-value = (sample_mean is at least ||null_value-point_estimate|| away from the null_value) | mu = null_value)
/// </summary>
/// <param name="sample_for_paired_data">a random sample consisting data paired together, var1 and var2, var1 and var2 are not independent</param>
/// <param name="one_sided">True if the test is one-sided</param>
/// <param name="significance_level"></param>
/// <returns></returns>
public bool RejectH0_PairedData(Tuple <double, double>[] sample_for_paired_data, out double pValue, double significance_level = 0.05, bool one_sided = false, bool useStudentT = false)
{
int sample_size = sample_for_paired_data.Length;
double[] diff = new double[sample_size];
for (int i = 0; i < sample_size; ++i)
{
diff[i] = sample_for_paired_data[i].Item1 - sample_for_paired_data[i].Item2;
}
double point_estimate = Mean.GetMean(diff);
double null_value = 0;
double SE = StandardError.GetStandardError(diff);
double test_statistic = System.Math.Abs(point_estimate - null_value) / SE;
double percentile = 0;
if (sample_for_paired_data.Length < 30 || useStudentT) //if sample size is smaller than 30, then CLT for population statistics such as sample mean no longer holds and Student's t distribution should be used in place of the normal distribution
{
percentile = StudentT.GetPercentile(test_statistic, sample_for_paired_data.Length - 1);
}
else
{
percentile = Gaussian.GetPercentile(test_statistic);
}
pValue = (1 - percentile) * (one_sided ? 1 : 2);
return(pValue < significance_level);
}
/// <summary>
/// Two-sided or one-sided test for a single statistic
///
/// Given that:
/// H_0 : mu = null_value
/// H_A : mu != null_value
///
/// By Central Limit Theorem:
/// sample_mean ~ N(mu, SE)
///
/// p-value = (sample_mean is at least ||null_value-point_estimate|| away from the null_value) | mu = null_value)
/// if(p-value < significance_level) reject H_0
/// </summary>
/// <param name="point_estimate">point estimate of the population statistics (e.g., sample mean, sample median, etc.)</param>
/// <param name="null_value"></param>
/// <param name="SE">standard error of the population statistics</param>
/// <param name="significance_level"></param>
/// <param name="one_sided"></param>
/// <returns></returns>
public static bool RejectH0(double point_estimate, double null_value, double SE, int sampleSize, out double pValue, double significance_level = 0.05, bool one_sided = false, bool useStudentT = false)
{
double test_statistic = System.Math.Abs(point_estimate - null_value) / SE; //This assumes that H_0 is true, that is, the true population mean, mu = null_value
double percentile = 0;
if (sampleSize < 30 || useStudentT) //if sample size is smaller than 30, then CLT for population statistics such as sample mean no longer holds and Student's t distribution should be used in place of the normal distribution
{
percentile = StudentT.GetPercentile(test_statistic, sampleSize - 1);
}
else
{
percentile = Gaussian.GetPercentile(test_statistic);
}
pValue = (1 - percentile) * (one_sided ? 1 : 2);
return(pValue < significance_level);
}
/// <summary>
/// The p-values are P(observed or more extreme coefficients != 0 | true coefficient mean is 0)
/// </summary>
/// <param name="CoeffPointEstimates">point estimates of the predictor coefficients</param>
/// <param name="CoeffSEs">standard errors of the predicator coefficients</param>
/// <param name="n">number of training records</param>
/// <param name="one_sided">whether the t distribution is one-sided</param>
/// <returns>p-values</returns>
public static double[] CalcPValues(double[] CoeffPointEstimates, double[] CoeffSEs, int n, bool one_sided = false)
{
double null_value = 0;
int k = CoeffPointEstimates.Length;
double[] pValues = new double[k];
int df = n - 1;
for (int i = 0; i < k; ++i)
{
double t = (CoeffPointEstimates[i] - null_value) / CoeffSEs[i];
double pValue = (1 - StudentT.GetPercentile(System.Math.Abs(t), df)) * (one_sided ? 1 : 2);
pValues[i] = pValue;
}
return(pValues);
}
/// <summary>
/// Two-sided or one-sided test for whether statitics of two variables are equal in the true population, var1 and var2 are independent
///
/// Hypotheses are:
/// H_0: mu_var1 = mu_var2
/// H_1: mu_var1 != mu_var2
///
/// The hypotheses can be written as
/// H_0: mu_var1 - mu_var2 = 0
/// H_1: mu_var1 - mu_var2 != 0
///
/// By Central Limt Theorem:
/// sample_mean_var1 - sample_mean_var2 ~ N(0, SE), where null_value = 0 and SE is the standard error of the sampling distribution
///
/// p-value = (sample_mean is at least ||null_value-point_estimate|| away from the null_value) | mu = null_value)
/// </summary>
/// <param name="sample_for_var1">value sample for variable 1</param>
/// <param name="sample_for_var2">value sample for variable 2</param>
/// <param name="one_sided">True if the test is one-sided</param>
/// <param name="significance_level"></param>
/// <returns></returns>
public bool RejectH0(double[] sample_for_var1, double[] sample_for_var2, out double pValue, double significance_level = 0.05, bool one_sided = false, bool useStudentT = false)
{
double pointEstimate = Mean.GetMean(sample_for_var1) - Mean.GetMean(sample_for_var2);
double null_value = 0;
double SE = StandardError.GetStandardError(sample_for_var1, sample_for_var2);
double test_statistic = System.Math.Abs(pointEstimate - null_value) / SE;
double percentile = 0;
if (sample_for_var1.Length < 30 || sample_for_var2.Length < 30 || useStudentT) //if sample size is smaller than 30, then CLT for population statistics such as sample mean no longer holds and Student's t distribution should be used in place of the normal distribution
{
int df = System.Math.Min(sample_for_var1.Length - 1, sample_for_var2.Length - 1);
percentile = StudentT.GetPercentile(test_statistic, df);
}
else
{
percentile = Gaussian.GetPercentile(test_statistic);
}
pValue = (1 - percentile) * (one_sided ? 1 : 2);
return(pValue < significance_level);
}
/// <summary>
/// Return the p-value from the Student's distribution
/// </summary>
/// <param name="t"></param>
/// <param name="dfE">degrees of freedom error obtained after ANOVA</param>
/// <returns>p-value = P(observed or more extreme values | H_0 is true)</returns>
private static double GetPValue(double t, int dfE)
{
return(StudentT.GetPercentile(System.Math.Abs(t), dfE));
}