/// <summary>
/// Return the confidence interval for proportion of SUCCESS in the population at a given confidence level given the sample proportion point estimate
/// </summary>
/// <param name="proportion">sample proportion point estimate</param>
/// <param name="sampleSize">sample size</param>
/// <param name="confidence_level"></param>
/// <returns>confidence interval for proportion of SUCCESS in the population at a given confidence level</returns>
public static double[] GetConfidenceInterval(double proportion, int sampleSize, double confidence_level, bool useSimulation = false, int simulationCount = 500)
{
double standard_error = StandardError.GetStandardErrorForProportion(proportion, sampleSize);
double p1 = (1 - confidence_level) / 2;
double p2 = 1 - p1;
int expected_success_count = (int)(proportion * sampleSize);
int expected_failure_count = (int)((1 - proportion) * sampleSize);
if (expected_failure_count < 10 || expected_success_count < 10 || useSimulation) //if np < 10 or n(1-p) < 10, then CLT for proportion no longer holds and simulation should be used in place of the normal distribution
{
double[] sampleProportions = new double[simulationCount];
int simulationSampleSize = (int)System.Math.Max(10 / proportion, 10 / (1 - proportion)) * 2;
for (int i = 0; i < simulationCount; ++i)
{
int successCount = 0;
for (int j = 0; j < simulationSampleSize; ++j)
{
if (DistributionModel.GetUniform() <= proportion)
{
successCount++;
}
}
sampleProportions[i] = (double)successCount / simulationSampleSize;
}
double proportion_mu = Mean.GetMean(sampleProportions);
double proportion_sigma = StdDev.GetStdDev(sampleProportions, proportion_mu);
return(new double[] { proportion_mu + Gaussian.GetPercentile(p1) * proportion_sigma, proportion_mu + Gaussian.GetQuantile(p2) * proportion_sigma });
}
else
{
double critical_value1 = Gaussian.GetQuantile(p1);
double critical_value2 = Gaussian.GetQuantile(p2);
double[] confidence_interval = new double[2];
confidence_interval[0] = proportion + critical_value1 * standard_error;
confidence_interval[1] = proportion + critical_value2 * standard_error;
return(confidence_interval);
}
}
/// <summary>
/// Estimate the normal distribution of a sample proportion (for a categorical variable with two values { "SUCCESS", "FAILURE" })
///
/// The Centrl Limit Theorem (CLT) for proportions:
/// The distribution of sample proportions is nearly normal, centered at the population proportion, and with a standard error inversely proportional to the sample size.
///
/// Conditions for the CLT for proportions:
/// 1. Independence: Sampled observations must be independent.
/// > random sample/assignment
/// > if sampling without replacement, n < 10% population
/// 2. Sample size / skew: There should be at least 10 successes and 10 failures in the sample: np >= 10 and n(1-p) >= 10
/// </summary>
/// <param name="p"></param>
/// <param name="sampleSize"></param>
/// <returns></returns>
public static Gaussian EstimateSampleProportionDistribution(double p, int sampleSize)
{
double SE = StandardError.GetStandardErrorForProportion(p, sampleSize);
return(new Gaussian(p, SE));
}
/// <summary>
/// Calculate the confidence interval for the proportion of SUCCESS in the population at a given confidence interval, given the point estimate proprotions are known from multiple groups
///
/// Note that this is only for categorical variable with two levels : SUCCESS, FAILURE
/// </summary>
/// <param name="proportions">The point estimate proportion of SUCESS obtained from multiple groups</param>
/// <param name="sampleSizes">The sample size of each group</param>
/// <param name="confidence_level">The given confidence interval</param>
/// <returns>The confidence interval for the proportion of SUCCESS in the population at the given confidence level</returns>
public static double[] GetConfidenceInterval(double[] proportions, int[] sampleSizes, double confidence_level, bool useSimulation = false, int simulationCount = 500)
{
double p1 = (1 - confidence_level) / 2;
double p2 = 1 - p1;
bool shouldUseSimulation = useSimulation;
if (!shouldUseSimulation)
{
for (int i = 0; i < sampleSizes.Length; ++i)
{
int n_i = sampleSizes[i];
int expected_success_count = (int)(proportions[i] * n_i);
int expected_failure_count = (int)((1 - proportions[i]) * n_i);
if (expected_failure_count < 10 || expected_success_count < 10)
{
shouldUseSimulation = true;
break;
}
}
}
if (shouldUseSimulation)
{
double sucess_count = 0;
double total_count = 0;
for (int i = 0; i < sampleSizes.Length; ++i)
{
int n_i = sampleSizes[i];
sucess_count += proportions[i] * n_i;
total_count += n_i;
}
double p_hat = sucess_count / total_count;
double[] sampleProportions = new double[simulationCount];
int simulationSampleSize = (int)System.Math.Max(10 / p_hat, 10 / (1 - p_hat)) * 2;
for (int i = 0; i < simulationCount; ++i)
{
int successCount = 0;
for (int j = 0; j < simulationSampleSize; ++j)
{
if (DistributionModel.GetUniform() <= p_hat)
{
successCount++;
}
}
sampleProportions[i] = (double)successCount / simulationSampleSize;
}
double proportion_mu = Mean.GetMean(sampleProportions);
double proportion_sigma = StdDev.GetStdDev(sampleProportions, proportion_mu);
return(new double[] { proportion_mu + Gaussian.GetPercentile(p1) * proportion_sigma, proportion_mu + Gaussian.GetQuantile(p2) * proportion_sigma });
}
else
{
double[] standardErrors = new double[proportions.Length];
for (int i = 0; i < proportions.Length; ++i)
{
standardErrors[i] = StandardError.GetStandardErrorForProportion(proportions[i], sampleSizes[i]);
}
double standardError = StandardError.GetStandardErrorForWeightAverages(sampleSizes, standardErrors);
double sampleMean = Mean.GetMeanForWeightedAverage(proportions, sampleSizes);
double critical_value1 = 0;
double critical_value2 = 0;
critical_value1 = Gaussian.GetQuantile(p1);
critical_value2 = Gaussian.GetQuantile(p2);
double[] confidence_interval = new double[2];
confidence_interval[0] = sampleMean + critical_value1 * standardError;
confidence_interval[1] = sampleMean + critical_value2 * standardError;
return(confidence_interval);
}
}