/**
* Cumulative distribution function
*
* Fᵥ,ᵤ(x) = Fᵥ,ᵤ(x), if x ≥ 0
* = 1 - Fᵥ,₋ᵤ(x) if x < 0
*
* @param number $x
* @param int $ν Degrees of freedom
* @param number $μ Noncentrality parameter
*
* @return number
*/
public static function CDF($x, int $ν, $μ)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'ν' => $ν, 'μ' => $μ]);
if ($μ == 0) {
return StudentT::CDF($x, $ν);
}
if ($x >= 0) {
return self::F($x, $ν, $μ);
}
return 1 - self::F($x, $ν, -$μ);
}
/**
* The probabilty associated with each parameter's t value
*
* t probability = Student's T CDF(t,ν)
*
* where:
* t = t value
* ν = n - p - alpha degrees of freedom
*
* alpha = 1 if the regression includes a constant term
*
* @return array [m => p, b => p]
*/
public function tProbability() : array
{
$ν = $this->ν;
$t = $this->tValues();
return ['m' => StudentT::CDF($t['m'], $ν), 'b' => StudentT::CDF($t['b'], $ν)];
}
/**
* One-sample Student's t-test
* Compares sample mean to the population mean.
* https://en.wikipedia.org/wiki/Student%27s_t-test
*
* Hₐ - H₀ M - μ M - μ M - μ
* z = ------- = ----- = ----- = -----
* σ σ SEM σ/√n
*
* p1 = CDF below if left tailed
* = CDF above if right tailed
* p2 = CDF outside
*
* @param number $Hₐ Alternate hypothesis (M Sample mean)
* @param number $s SD of sample
* @param int $n Sample size
* @param number $H₀ Null hypothesis (μ₀ Population mean)
*
* @return array [
* z => z score
* p1 => one-tailed p value (left or right tail depends on how Hₐ differs from H₀)
* p2 => two-tailed p value
* ]
*/
public static function tTestOneSample($Hₐ, $s, $n, $H₀) : array
{
// Calculate test statistic t
$t = self::tScore($Hₐ, $s, $n, $H₀);
// Degrees of freedom
$ν = $n - 1;
// One- and two-tailed P values
if ($Hₐ < $H₀) {
$p1 = StudentT::CDF($t, $ν);
} else {
$p1 = StudentT::above($t, $ν);
}
$p2 = StudentT::outside(-abs($t), abs($t), $ν);
return ['t' => $t, 'p1' => $p1, 'p2' => $p2];
}
