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package breeze.stats
import scala.math.pow
import scala.math.sqrt
import breeze.stats.distributions._
import breeze.linalg.support._
import CanTraverseValues._
/**
* This package contains hypothesis tests.
*
*/
package object hypothesis {
/**
* Implements two tailed Welch's T Test (equivalent to t.test in R)
* Returns a p value
*/
def tTest[T](it1: TraversableOnce[T],it2 : Traversable[T])(implicit numeric: Numeric[T]): Double = tTest[TraversableOnce[Double]](it1.map(numeric.toDouble), it2.map(numeric.toDouble)) //explicit type annotation ensures that the compiler runs the CanTraverseValues implementation of it
def tTest[X](it1 : X, it2 : X)(implicit ct: CanTraverseValues[X,Double]): Double = {
//first sample
val MeanAndVariance(mu1, var1, n1) = meanAndVariance(it1)
//second sample
val MeanAndVariance(mu2, var2, n2) = meanAndVariance(it2)
require(var1 > 0 && var2 > 0, "Two Sample T Test requires that both"
+"samples have variance > 0")
val tScore = (mu1 - mu2) / sqrt( (var1 / n1) + (var2 / n2) ) // T statistic
val dof = pow((var1/n1) + (var2/n2), 2) / ( pow(var1,2)/( pow(n1,2)*(n1-1) ) +
pow(var2,2)/(pow(n2,2)*(n2-1)) ) //Welch–Satterthwaite equation
new StudentsT(dof)(RandBasis.mt0).unnormalizedPdf(tScore) //return p value
}
def tTest[T](it1: Traversable[T])(implicit numeric: Numeric[T]):Double = tTest[TraversableOnce[Double]](it1.map(numeric.toDouble)) //explicit type annotation ensures that the compiler runs the CanTraverseValues implementation of it
def tTest[X](it1: X)(implicit ct:CanTraverseValues[X,Double]):Double = {
val MeanAndVariance(mu1, var1, n1) = meanAndVariance(it1)
val Z = mu1 / sqrt( var1/n1 )
val dof = n1-1
new StudentsT(dof)(RandBasis.mt0).unnormalizedPdf(Z) //return p value
}
case class Chi2Result(chi2: Double, pVal: Double)
private def chiSquaredTerm(e: Double, o: Double) = (e-o)*(e-o)/e
def chi2Test(successControl: Long, trialsControl: Long, successVariant: Long, trialsVariant: Long):Chi2Result = {
/*
* Takes 2 Bernoulli trials, and determines using a chi2 test whether there is a
* statistically significant difference between the probabilities of success
* between control and a variant.
*/
val meanP = (successControl + successVariant).toDouble / (trialsControl + trialsVariant).toDouble
val chi2 = (chiSquaredTerm(meanP * trialsControl, successControl) + chiSquaredTerm((1-meanP) * trialsControl, trialsControl - successControl)
+ chiSquaredTerm(meanP * trialsVariant, successVariant) + chiSquaredTerm((1-meanP) * trialsVariant, trialsVariant - successVariant))
val pVal = 1.0 - Gamma(0.5, 2.0).cdf(chi2)
Chi2Result(chi2, pVal)
}
/**
* Takes a sequence of N Bernoulli trials, and determines using a chi2 test whether there is a
* statistically significant difference between the N variants and a control.
* I.e., the variants may differ from each other, but this only determines whether
* they differ from control.
*
* The pVal reported in the results is the probability (assuming the null hypothesis) of
* a false positive at least this large in *any* variant, not in one particular variant.
* I.e., multiple comparisons are corrected for.
*/
def chi2Test(control: (Long, Long), trials: Seq[(Long, Long)]): Seq[Chi2Result] = {
val numTrials = trials.size
trials.map( x => chi2Test(control._1, control._2, x._1, x._2) ).map(r => Chi2Result(r.chi2, sidakCorrectedPVal(r.pVal, numTrials)))
}
/**
* Takes a p-value run for a single statistical test,
* and then corrects for multiple comparisons.
*
* I.e., if you run n tests with a p-value cutoff of 5%
* yielding p-values p1, p2, ..., pn, then if
* sidakCorrectedPVal(p1,n) < 5% or sidakCorrectedPVal(p2, n) < 5%, etc,
* you can reject the null hypothesis.
*/
def sidakCorrectedPVal(p: Double, n: Int): Double = {
1.0 - pow(1.0-p, n)
}
/**
* Takes a p-value run for a single statistical test,
* and then corrects for multiple comparisons.
*
* This function is the inverse of sidakCorrectedPVal.
* If you run n tests and want a 5% chance of false positive (assuming null
* hypothesis is true) across *all* tests, then you can
* run each individual test with a p-value cutoff of
* sidakCorrectedPValCutoff(0.05, n).
*/
def sidakCorrectedPValCutoff(p: Double, n: Int): Double = {
1.0 - pow(1.0-p, 1.0/n.toDouble)
}
}
