/**
* 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 prediction interval of the regression
* _________________
* /1 1 (x - x̄)²
* PI(x,p,q) = t * sy * / - + - + --------
* √ q n SSx
*
* Where:
* t is the critical t for the p value
* sy is the estimated standard deviation of y
* q is the number of replications
* n is the number of data points
* x̄ is the average of the x values
* SSx = ∑(x - x̄)²
*
* If $p = .05, then we can say we are 95% confidence that the future averages of $q trials at $x
* will be within an interval of evaluate($x) ± PI($x, .05, $q).
*
* @param number $x
* @param number $p 0 < p < 1 The P value to use
* @param int $q Number of trials
*
* @return number
*/
public function PI($x, $p, $q = 1)
{
$V = $this->regressionVariance($x) + 1 / $q;
$σ² = $this->meanSquareResidual();
// The t-value
$t = StudentT::inverse2Tails($p, $this->ν);
return $t * sqrt($σ² * $V);
}
/**
* Two-sample t-test
* Test the means of two samples.
* https://en.wikipedia.org/wiki/Student%27s_t-test
*
* μ₁ - μ₂ - Δ
* z = --------------
* _________
* /σ₁² σ₂²
* / --- + ---
* √ n₁ n₂
*
* where
* μ₁ is sample mean 1
* μ₂ is sample mean 2
* Δ is the hypothesized difference between the population means (0 if testing for equal means)
* σ₁ is standard deviation of sample mean 1
* σ₂ is standard deviation of sample mean 2
* n₁ is sample size of mean 1
* n₂ is sample size of mean 2
*
* For Student's t distribution CDF, degrees of freedom:
* ν = (n₁ - 1) + (n₂ - 1)
*
* p1 = CDF above
* p2 = CDF outside
*
* @param number $μ₁ Sample mean of population 1
* @param number $μ₂ Sample mean of population 2
* @param number $n₁ Sample size of population 1
* @param number $n₂ Sample size of population 1
* @param number $σ₁ Standard deviation of sample mean 1
* @param number $σ₂ Standard deviation of sample mean 2
* @param number $Δ (Optional) hypothesized difference between the population means (0 if testing for equal means)
*
* @return array [
* t => t score
* p1 => one-tailed p value
* p2 => two-tailed p value
* ]
*/
public static function tTestTwoSample($μ₁, $μ₂, $n₁, $n₂, $σ₁, $σ₂, $Δ = 0) : array
{
// Calculate t score (test statistic)
$t = ($μ₁ - $μ₂ - $Δ) / sqrt($σ₁ ** 2 / $n₁ + $σ₂ ** 2 / $n₂);
// Degrees of freedom
$ν = $n₁ - 1 + ($n₂ - 1);
// One- and two-tailed P values
$p1 = StudentT::above(abs($t), $ν);
$p2 = StudentT::outside(-abs($t), abs($t), $ν);
return ['t' => $t, 'p1' => $p1, 'p2' => $p2];
}
