- Central moment
- Skewness
- Kurtosis
- Standard Error of the Mean (SEM)
- Confidence interval
In probability and statistics, a random variable is a variable whose
value is subject to variations due to chance.
A random variable can take on a set of possible different values
(similarly to other mathematical variables), each with an associated
probability, in contrast to other mathematical variables.
The mathematical function describing the possible values of a random
variable and their associated probabilities is known as a probability
distribution. Random variables can be discrete, that is, taking any of a
specified finite or countable list of values, endowed with a probability
mass function, characteristic of a probability distribution; or
continuous, taking any numerical value in an interval or collection of
intervals, via a probability density function that is characteristic of
a probability distribution; or a mixture of both types.
https://en.wikipedia.org/wiki/Random_variable
/**
* SSreg - The Sum Squares of the regression (Explained sum of squares)
*
* The sum of the squares of the deviations of the predicted values from
* the mean value of a response variable, in a standard regression model.
* https://en.wikipedia.org/wiki/Explained_sum_of_squares
*
* SSreg = ∑(ŷᵢ - ȳ)²
* When a constant is fit to the regression, the average of y = average of ŷ.
*
* In the case where the constant is not fit, we use the sum of squares of the predicted value
* SSreg = ∑ŷᵢ²
*
* @return number
*/
public function sumOfSquaresRegression()
{
if ($this->fit_constant == 1) {
return RandomVariable::sumOfSquaresDeviations($this->Yhat());
}
return array_sum(Single::square($this->reg_Yhat));
}
/**
* Two-way ANOVA
* Examines the influence of two different categorical independent variables on
* one continuous dependent variable. The two-way ANOVA not only aims at assessing
* the main effect of each independent variable but also if there is any interaction
* between them (using the F distribution).
* https://en.wikipedia.org/wiki/Two-way_analysis_of_variance
*
* Produces the following analysis of the data:
*
* ANOVA hypothesis test summary data
*
* | SS | df | MS | F | P |
* Factor A | | | | | |
* Factor B | | | | | |
* Interaction | | | | | |
* Error | | | |
* Total | | |
*
* where:
* Interaction = Factor A X Factor B working together
* Error is within groups
* SS = Sum of squares
* df = Degrees of freedom
* MS = Mean squares
* F = F statistic
* P = P value
*
* Data summary tables for:
* Factor A
* Factor B
* Factor AB (Interaction)
* Total
*
* | N | Sum | Mean | SS | Variance | SD | SEM |
* 0 | | | | | | | |
* 1 | | | | | | | |
* ... | | | | | | | |
* Total | | | | | | | |
*
* where:
* Each row is the summary for a sample, numbered from 0 to m - 1
* m = Number of samples
* N = Sample size
* SS = Sum of squares
* SD = Standard deviation
* SEM = Standard error of the mean
*
* Calculations
*
* Sum of Squares
* SST (sum of squares total)
* ∑⟮xᵢ − μ⟯²
* where:
* xᵢ = each element of all samples
* μ = mean total of all elements of all samples
*
* SSA, SSB (sum of squares for each factor A and B)
* ∑n(x - μ)²
* where:
* n = sample size
* x = sample mean
* μ = mean total of all elements of all samples
*
* SSW (sum of squares within - error)
* ∑∑⟮x − μ⟯² Sum of sum of squared deviations of each sample
* where:
* x = mean of each AB
* μ = mean of the sample
*
* SSAB (sum of squares AB - interaction)
* SSAB = SST - SSA - SSB - SSW;
*
* Degrees of Freedom
* dfT (degrees of freedom for the total)
* n - 1
*
* dfA (degrees of freedom factor A)
* r - 1
*
* dfB (degrees of freedom factor B)
* c - 1
*
* dfAB (degrees of freedom factor AB - interaction)
* (r - 1)(c - 1)
*
* dfW (degrees of freedom within - error)
* n - rc
*
* where:
* n = number of samples
* r = number of rows (number of factor As)
* c = number of columns (number of factor Bs)
*
* Mean Squares
* MSA (Mean squares factor A)
* SSA / dfA
*
* MSB (Mean squares factor B)
* SSB / dfB
*
* MSAB (Mean squares factor AB - interaction)
* SSAB / dfAB
*
* MSW (Mean squares within - error)
* SSW / dfW
*
* F Test Statistics
* FA = MSA / MSW
* FB = MSB / MSW
* FAB = MSAB / MSW
*
* P values
* PA = F distribution CDF above FA with degrees of freedom dfA and dfW
* PB = F distribution CDF above FB with degrees of freedom dfA and dfW
* PAB = F distribution CDF above FAB with degrees of freedom dfAB and dfW
*
* Example input data for ...$data parameter:
* | Factor B₁ | Factor B₂ | ⋯
* Factor A₁ | 4, 6, 8 | 6, 6, 9 | ⋯
* Factor A₂ | 4, 8, 9 | 7, 10, 13 | ⋯
* ⋮ ⋮ ⋮ ⋮
* @param array ...$data Samples to analyze [
* // Factor A₁
* [
* [4, 6, 8] // Factor B₁
* [6, 6, 9] // Factor B₂
* ⋮
* ],
* // Factor A₂
* [
* [4, 8, 9] // Factor B₁
* [7, 10, 13] // Factor B₂
* ⋮
* ],
* ...
* ]
*
* @return array [
* ANOVA => [
* factorA => [SS, df, MS, F, P],
* factorB => [SS, df, MS, F, P],
* factorAB => [SS, df, MS, F, P],
* error => [SS, df, MS],
* total => [SS, df],
* ],
* total_summary => [n, sum, mean, SS, variance, sd, sem],
* summary_factorA => [
* 0 => [n, sum, mean, SS, variance, sd, sem],
* 1 => [n, sum, mean, SS, variance, sd, sem],
* ...
* ],
* summary_factorB => [
* 0 => [n, sum, mean, SS, variance, sd, sem],
* 1 => [n, sum, mean, SS, variance, sd, sem],
* ...
* ],
* summary_factorAB => [
* 0 => [n, sum, mean, SS, variance, sd, sem],
* 1 => [n, sum, mean, SS, variance, sd, sem],
* ...
* ]
* ]
* @throws BadDataException if less than two A factors, or if B factors or values have different number elements
*/
public static function twoWay(array ...$data)
{
// Must have at least two rows (two types of factor A)
$r = count($data);
if ($r < 2) {
throw new Exception\BadDataException('Must have at least two rows (two types of factor A)');
}
// All samples must have the same number the second factor B
$c = count($data[0]);
for ($i = 1; $i < $r; $i++) {
if (count($data[$i]) !== $c) {
throw new Exception\BadDataException('All samples must have the same number of the second factor B');
}
}
// Each AB factor interaction must have the same number of values
$v = count($data[0][0]);
for ($i = 0; $i < $r; $i++) {
for ($j = 0; $j < $c; $j++) {
if (count($data[$i][$j]) !== $v) {
throw new Exception\BadDataException('Each AB factor interaction must have the same number of values');
}
}
}
// Aggregates for all elements, rows (factor A), and columns (factor B)
$all_elements = [];
$A_elements = [];
$B_elements = [];
// Summaries for factor A, factor B, AB, and total
$summary_A = [];
$summary_B = [];
$summary_AB = [];
$summary_total = [];
// Summary data for each AB
// And aggregate all elements and elements for factor A
foreach ($data as $A => $Bs) {
$A_elements[$A] = [];
foreach ($Bs as $B => $values) {
// Aggregates
$all_elements = array_merge($all_elements, $values);
$A_elements[$A] = array_merge($A_elements[$A], $values);
// AB summary
$summary_AB[$A][$B] = [];
$summary_AB[$A][$B]['n'] = $c;
$summary_AB[$A][$B]['sum'] = array_sum($values);
$summary_AB[$A][$B]['mean'] = Average::mean($values);
$summary_AB[$A][$B]['SS'] = RandomVariable::sumOfSquares($values);
$summary_AB[$A][$B]['variance'] = Descriptive::sampleVariance($values);
$summary_AB[$A][$B]['sd'] = Descriptive::sd($values);
$summary_AB[$A][$B]['sem'] = RandomVariable::standardErrorOfTheMean($values);
}
}
// Aggregate elements for factor B
for ($B = 0; $B < $c; $B++) {
$B_elements[$B] = [];
foreach ($data as $factor1s) {
$B_elements[$B] = array_merge($B_elements[$B], $factor1s[$B]);
}
}
// Factor A summary
foreach ($A_elements as $A => $elements) {
$summary_A[$A] = [];
$summary_A[$A]['n'] = count($elements);
$summary_A[$A]['sum'] = array_sum($elements);
$summary_A[$A]['mean'] = Average::mean($elements);
$summary_A[$A]['SS'] = RandomVariable::sumOfSquares($elements);
$summary_A[$A]['variance'] = Descriptive::sampleVariance($elements);
$summary_A[$A]['sd'] = Descriptive::sd($elements);
$summary_A[$A]['sem'] = RandomVariable::standardErrorOfTheMean($elements);
}
// Factor B summary
foreach ($B_elements as $B => $elements) {
$summary_B[$B] = [];
$summary_B[$B]['n'] = count($elements);
$summary_B[$B]['sum'] = array_sum($elements);
$summary_B[$B]['mean'] = Average::mean($elements);
$summary_B[$B]['SS'] = RandomVariable::sumOfSquares($elements);
$summary_B[$B]['variance'] = Descriptive::sampleVariance($elements);
$summary_B[$B]['sd'] = Descriptive::sd($elements);
$summary_B[$B]['sem'] = RandomVariable::standardErrorOfTheMean($elements);
}
// Totals summary
$μ = Average::mean($all_elements);
$summary_total = ['n' => count($all_elements), 'sum' => array_sum($all_elements), 'mean' => $μ, 'SS' => RandomVariable::sumOfSquares($all_elements), 'variance' => Descriptive::sampleVariance($all_elements), 'sd' => Descriptive::sd($all_elements), 'sem' => RandomVariable::standardErrorOfTheMean($all_elements)];
// Sum of squares factor A
$SSA = array_sum(array_map(function ($f1) use($μ) {
return $f1['n'] * ($f1['mean'] - $μ) ** 2;
}, $summary_A));
// Sum of squares factor B
$SSB = array_sum(array_map(function ($B) use($μ) {
return $B['n'] * ($B['mean'] - $μ) ** 2;
}, $summary_B));
// Sum of squares within (error)
$SSW = 0;
foreach ($data as $A => $Bs) {
foreach ($Bs as $B => $values) {
foreach ($values as $value) {
$SSW += ($value - $summary_AB[$A][$B]['mean']) ** 2;
}
}
}
// Sum of squares total
$SST = 0;
foreach ($data as $A => $Bs) {
foreach ($Bs as $B => $values) {
foreach ($values as $value) {
$SST += ($value - $μ) ** 2;
}
}
}
// Sum of squares AB interaction
$SSAB = $SST - $SSA - $SSB - $SSW;
// Degrees of freedom
$dfA = $r - 1;
$dfB = $c - 1;
$dfAB = ($r - 1) * ($c - 1);
$dfW = $summary_total['n'] - $r * $c;
$dfT = $summary_total['n'] - 1;
// Mean squares
$MSA = $SSA / $dfA;
$MSB = $SSB / $dfB;
$MSAB = $SSAB / $dfAB;
$MSW = $SSW / $dfW;
// F test statistics
$FA = $MSA / $MSW;
$FB = $MSB / $MSW;
$FAB = $MSAB / $MSW;
// P values
$PA = F::above($FA, $dfA, $dfW);
$PB = F::above($FB, $dfB, $dfW);
$PAB = F::above($FAB, $dfAB, $dfW);
// Return ANOVA report
return ['ANOVA' => ['factorA' => ['SS' => $SSA, 'df' => $dfA, 'MS' => $MSA, 'F' => $FA, 'P' => $PA], 'factorB' => ['SS' => $SSB, 'df' => $dfB, 'MS' => $MSB, 'F' => $FB, 'P' => $PB], 'interaction' => ['SS' => $SSAB, 'df' => $dfAB, 'MS' => $MSAB, 'F' => $FAB, 'P' => $PAB], 'error' => ['SS' => $SSW, 'df' => $dfW, 'MS' => $MSW], 'total' => ['SS' => $SST, 'df' => $dfT]], 'total_summary' => $summary_total, 'summary_factorA' => $summary_A, 'summary_factorB' => $summary_B, 'summary_interaction' => $summary_AB];
}
/**
* Get a report of all the descriptive statistics over a list of numbers
* Includes mean, median, mode, range, midrange, variance, standard deviation, quartiles, etc.
*
* @param array $numbers
* @param bool $population: true means all possible observations of the system are present;
* false means a sample is used.
* @return array [ n, mean, median, mode, range, midrange, variance, sd, CV, mean_mad,
* median_mad, quartiles, skewness, kurtosis, sem, ci_95, ci_99 ]
*/
public static function describe(array $numbers, bool $population = false) : array
{
$n = count($numbers);
$μ = Average::mean($numbers);
$σ = self::standardDeviation($numbers, $population);
return ['n' => $n, 'min' => min($numbers), 'max' => max($numbers), 'mean' => $μ, 'median' => Average::median($numbers), 'mode' => Average::mode($numbers), 'range' => self::range($numbers), 'midrange' => self::midrange($numbers), 'variance' => $population ? self::populationVariance($numbers) : self::sampleVariance($numbers), 'sd' => $σ, 'cv' => $σ / $μ, 'mean_mad' => self::meanAbsoluteDeviation($numbers), 'median_mad' => self::medianAbsoluteDeviation($numbers), 'quartiles' => self::quartiles($numbers), 'midhinge' => self::midhinge($numbers), 'skewness' => $population ? RandomVariable::populationSkewness($numbers) : RandomVariable::skewness($numbers), 'ses' => RandomVariable::SES($n), 'kurtosis' => RandomVariable::kurtosis($numbers), 'sek' => RandomVariable::SEK($n), 'sem' => RandomVariable::standardErrorOfTheMean($numbers), 'ci_95' => RandomVariable::confidenceInterval($μ, $n, $σ, 95), 'ci_99' => RandomVariable::confidenceInterval($μ, $n, $σ, 99)];
}