/// <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);
}
}