# Welch's t-test

Given two lists of data, calculate the p-value used for Welch's t-test. This is meant to translate R's t.test(vector1, vector2, alternative="two.sided", var.equal=FALSE) for calculation of the p-value.

Welch's t-test is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Given two sets of data, calculate the p-value:

   x = {3.0,4.0,1.0,2.1}
y = {490.2,340.0,433.9}


Your task is to discern whether or not the difference in means between the two sets is statistically significant and worth further investigation. P-values are significance tests to gauge the probability that the difference in means between two data sets is significant, or due to chance. A threshold level, alpha, is usually chosen, 0.01 or 0.05, where p-values below alpha are worth further investigation and p-values above alpha are considered not significant. The p-value is not considered a final test of significance, only whether the given variable should be given further consideration.

There is more than one way of calculating the t-statistic, and you must choose which method is appropriate for you. Here we use Welch's t-test, which assumes that the variances between the two sets x and y are not equal. Welch's t-test statistic can be computed:

${\displaystyle t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }} }$

where

${\displaystyle \overline{X}_n }$ is the mean of set ${\displaystyle n}$,

and

${\displaystyle N_n}$ is the number of observations in set ${\displaystyle n}$,

and

${\displaystyle s_n }$ is the square root of the unbiased sample variance of set ${\displaystyle n}$, i.e.

${\displaystyle s_n = \sqrt{\frac{1}{N_n-1} \sum_{i=1}^{N_n} \left(X_i - \overline{X}_n\right)^2} }$

and the degrees of freedom, ${\displaystyle \nu}$ can be approximated:

${\displaystyle \nu \quad \approx \quad {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over { \quad {s_1^4 \over N_1^2 (N_1-1)} \; + \; {s_2^4 \over N_2^2 (N_2-1) } \quad }}}$

The two-tailed p-value, ${\displaystyle p}$, can be computed as a cumulative distribution function

${\displaystyle p_{2-tail} = I_{\frac{\nu}{t^2+\nu}}\left(\frac{\nu}{2}, \frac{1}{2}\right) }$

where I is the regularized incomplete beta function. This is the same as:

${\displaystyle p_{2-tail} = \frac{\Beta(\frac{\nu}{t^2+\nu};\frac{\nu}{2}, \frac{1}{2})}{\Beta(\frac{\nu}{2}, \frac{1}{2})} }$

Keeping in mind that

${\displaystyle \Beta(x;a,b) = \int_0^x r^{a-1}\,(1-r)^{b-1}\,\mathrm{d}r. \!}$

and

${\displaystyle \Beta(x,y) = \dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} =\exp(\ln\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}) = \exp((\ln(\Gamma(x)) + \ln(\Gamma(y)) - \ln(\Gamma(x+y))) \!}$

${\displaystyle p_{2-tail} }$ can be calculated in terms of gamma functions and integrals more simply:

${\displaystyle p_{2-tail}=\frac{\int_0^\frac{\nu}{t^2+\nu} r^{\frac{\nu}{2}-1}\,(1-r)^{-0.5}\,\mathrm{d}r}{\exp((\ln(\Gamma(\frac{\nu}{2})) + \ln(\Gamma(0.5)) - \ln(\Gamma(\frac{\nu}{2}+0.5)))} }$

which simplifies to

${\displaystyle p_{2-tail} = \frac{\int_0^\frac{\nu}{t^2+\nu} \frac{r^{\frac{\nu}{2}-1}}{\sqrt{1-r}}\,\mathrm{d}r}{ \exp((\ln(\Gamma(\frac{\nu}{2})) + \ln(\Gamma(0.5)) - \ln(\Gamma(\frac{\nu}{2}+0.5))) }}$

The definite integral can be approximated with Simpson's Rule but other methods are also acceptable.

The ${\displaystyle \ln(\Gamma(x))}$, or lgammal(x) function is necessary for the program to work with large a values, as Gamma functions can often return values larger than can be handled by double or long double data types. The lgammal(x) function is standard in math.h with C99 and C11 standards.

## 11l

Translation of: Python
F betain(x, p, q)
I p <= 0 | q <= 0 | x < 0 | x > 1
X ValueError(0)

I x == 0 | x == 1
R x

V acu = 1e-15
V lnbeta = lgamma(p) + lgamma(q) - lgamma(p + q)

V xx = x
V cx = 1 - x
V pp = p
V qq = q
V indx = 0B
V psq = p + q
I p < psq * x
xx = 1 - x
cx = x
pp = q
qq = p
indx = 1B

V term = 1.0
V ai = 1.0
V value = 1.0
V ns = floor(qq + cx * psq)
V rx = xx / cx
V temp = qq - ai
I ns == 0
rx = xx

L
term *= temp * rx / (pp + ai)
value += term
temp = abs(term)

I temp <= acu & temp <= acu * value
value *= exp(pp * log(xx) + (qq - 1) * log(cx) - lnbeta) / pp
R I indx {1 - value} E value

ai++
I --ns >= 0
temp = qq - ai
I ns == 0
rx = xx
E
temp = psq
psq++

F welch_ttest(a1, a2)
V n1 = a1.len
V n2 = a2.len
I n1 <= 1 | n2 <= 1
X ValueError(0)

V mean1 = sum(a1) / n1
V mean2 = sum(a2) / n2

V var1 = sum(a1.map(x -> (x - @mean1) ^ 2)) / (n1 - 1)
V var2 = sum(a2.map(x -> (x - @mean2) ^ 2)) / (n2 - 1)

V t = (mean1 - mean2) / sqrt(var1 / n1 + var2 / n2)
V df = (var1 / n1 + var2 / n2) ^ 2 / (var1 ^ 2 / (n1 ^ 2 * (n1 - 1)) + var2 ^ 2 / (n2 ^ 2 * (n2 - 1)))
V p = betain(df / (t ^ 2 + df), df / 2, 1 / 2)

R (t, df, p)

V a1 = [Float(3), 4, 1, 2.1]
V a2 = [Float(490.2), 340, 433.9]
print(welch_ttest(a1, a2))
Output:
(-9.5595, 2.00085, 0.0107516)


## C

Works with: C99

Link with -lm

This program, for example, pvalue.c, can be compiled by

clang -o pvalue pvalue.c -Wall -pedantic -std=c11 -lm -O3

or

gcc -o pvalue pvalue.c -Wall -pedantic -std=c11 -lm -O4.

This shows how pvalue can be calculated from any two arrays, using Welch's 2-sided t-test, which doesn't assume equal variance. This is the equivalent of R'st.test(vector1,vector2, alternative="two.sided", var.equal=FALSE) and as such, it is compared against R's pvalues with the same vectors/arrays to show that the differences are very small (here 10^-14).

#include <stdio.h>
#include <math.h>
#include <stdlib.h>

double Pvalue (const double *restrict ARRAY1, const size_t ARRAY1_SIZE, const double *restrict ARRAY2, const size_t ARRAY2_SIZE) {//calculate a p-value based on an array
if (ARRAY1_SIZE <= 1) {
return 1.0;
} else if (ARRAY2_SIZE <= 1) {
return 1.0;
}
double fmean1 = 0.0, fmean2 = 0.0;
for (size_t x = 0; x < ARRAY1_SIZE; x++) {//get sum of values in ARRAY1
if (isfinite(ARRAY1[x]) == 0) {//check to make sure this is a real numbere
puts("Got a non-finite number in 1st array, can't calculate P-value.");
exit(EXIT_FAILURE);
}
fmean1 += ARRAY1[x];
}
fmean1 /= ARRAY1_SIZE;
for (size_t x = 0; x < ARRAY2_SIZE; x++) {//get sum of values in ARRAY2
if (isfinite(ARRAY2[x]) == 0) {//check to make sure this is a real number
puts("Got a non-finite number in 2nd array, can't calculate P-value.");
exit(EXIT_FAILURE);
}
fmean2 += ARRAY2[x];
}
fmean2 /= ARRAY2_SIZE;
//	printf("mean1 = %lf	mean2 = %lf\n", fmean1, fmean2);
if (fmean1 == fmean2) {
return 1.0;//if the means are equal, the p-value is 1, leave the function
}
double unbiased_sample_variance1 = 0.0, unbiased_sample_variance2 = 0.0;
for (size_t x = 0; x < ARRAY1_SIZE; x++) {//1st part of added unbiased_sample_variance
unbiased_sample_variance1 += (ARRAY1[x]-fmean1)*(ARRAY1[x]-fmean1);
}
for (size_t x = 0; x < ARRAY2_SIZE; x++) {
unbiased_sample_variance2 += (ARRAY2[x]-fmean2)*(ARRAY2[x]-fmean2);
}
//	printf("unbiased_sample_variance1 = %lf\tunbiased_sample_variance2 = %lf\n",unbiased_sample_variance1,unbiased_sample_variance2);//DEBUGGING
unbiased_sample_variance1 = unbiased_sample_variance1/(ARRAY1_SIZE-1);
unbiased_sample_variance2 = unbiased_sample_variance2/(ARRAY2_SIZE-1);
const double WELCH_T_STATISTIC = (fmean1-fmean2)/sqrt(unbiased_sample_variance1/ARRAY1_SIZE+unbiased_sample_variance2/ARRAY2_SIZE);
const double DEGREES_OF_FREEDOM = pow((unbiased_sample_variance1/ARRAY1_SIZE+unbiased_sample_variance2/ARRAY2_SIZE),2.0)//numerator
/
(
(unbiased_sample_variance1*unbiased_sample_variance1)/(ARRAY1_SIZE*ARRAY1_SIZE*(ARRAY1_SIZE-1))+
(unbiased_sample_variance2*unbiased_sample_variance2)/(ARRAY2_SIZE*ARRAY2_SIZE*(ARRAY2_SIZE-1))
);
//	printf("Welch = %lf	DOF = %lf\n", WELCH_T_STATISTIC, DEGREES_OF_FREEDOM);
const double a = DEGREES_OF_FREEDOM/2;
double value = DEGREES_OF_FREEDOM/(WELCH_T_STATISTIC*WELCH_T_STATISTIC+DEGREES_OF_FREEDOM);
if ((isinf(value) != 0) || (isnan(value) != 0)) {
return 1.0;
}
if ((isinf(value) != 0) || (isnan(value) != 0)) {
return 1.0;
}

/*  Purpose:

BETAIN computes the incomplete Beta function ratio.

Licensing:

Modified:

05 November 2010

Author:

Original FORTRAN77 version by KL Majumder, GP Bhattacharjee.
C version by John Burkardt.

Reference:

KL Majumder, GP Bhattacharjee,
Algorithm AS 63:
The incomplete Beta Integral,
Applied Statistics,
Volume 22, Number 3, 1973, pages 409-411.

Parameters:
https://www.jstor.org/stable/2346797?seq=1#page_scan_tab_contents
Input, double X, the argument, between 0 and 1.

Input, double P, Q, the parameters, which
must be positive.

Input, double BETA, the logarithm of the complete
beta function.

Output, int *IFAULT, error flag.
0, no error.
nonzero, an error occurred.

Output, double BETAIN, the value of the incomplete
Beta function ratio.
*/
const double beta = lgammal(a)+0.57236494292470009-lgammal(a+0.5);
const double acu = 0.1E-14;
double ai;
double cx;
int indx;
int ns;
double pp;
double psq;
double qq;
double rx;
double temp;
double term;
double xx;

//  ifault = 0;
//Check the input arguments.
if ( (a <= 0.0)) {// || (0.5 <= 0.0 )){
//    *ifault = 1;
//    return value;
}
if ( value < 0.0 || 1.0 < value )
{
//    *ifault = 2;
return value;
}
/*
Special cases.
*/
if ( value == 0.0 || value == 1.0 )   {
return value;
}
psq = a + 0.5;
cx = 1.0 - value;

if ( a < psq * value )
{
xx = cx;
cx = value;
pp = 0.5;
qq = a;
indx = 1;
}
else
{
xx = value;
pp = a;
qq = 0.5;
indx = 0;
}

term = 1.0;
ai = 1.0;
value = 1.0;
ns = ( int ) ( qq + cx * psq );
/*
Use the Soper reduction formula.
*/
rx = xx / cx;
temp = qq - ai;
if ( ns == 0 )
{
rx = xx;
}

for ( ; ; )
{
term = term * temp * rx / ( pp + ai );
value = value + term;;
temp = fabs ( term );

if ( temp <= acu && temp <= acu * value )
{
value = value * exp ( pp * log ( xx )
+ ( qq - 1.0 ) * log ( cx ) - beta ) / pp;

if ( indx )
{
value = 1.0 - value;
}
break;
}

ai = ai + 1.0;
ns = ns - 1;

if ( 0 <= ns )
{
temp = qq - ai;
if ( ns == 0 )
{
rx = xx;
}
}
else
{
temp = psq;
psq = psq + 1.0;
}
}
return value;
}
//-------------------
int main(void) {

const double d1[] = {27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4};
const double d2[] = {27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4};
const double d3[] = {17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8};
const double d4[] = {21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8};
const double d5[] = {19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0};
const double d6[] = {28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2};
const double d7[] = {30.02,29.99,30.11,29.97,30.01,29.99};
const double d8[] = {29.89,29.93,29.72,29.98,30.02,29.98};
const double x[] = {3.0,4.0,1.0,2.1};
const double y[] = {490.2,340.0,433.9};
const double v1[] = {0.010268,0.000167,0.000167};
const double v2[] = {0.159258,0.136278,0.122389};
const double s1[] = {1.0/15,10.0/62.0};
const double s2[] = {1.0/10,2/50.0};
const double z1[] = {9/23.0,21/45.0,0/38.0};
const double z2[] = {0/44.0,42/94.0,0/22.0};

0.148841696605327,
0.0359722710297968,
0.090773324285671,
0.0107515611497845,
0.00339907162713746,
0.52726574965384,
0.545266866977794};

//calculate the pvalues and show that they're the same as the R values

double pvalue = Pvalue(d1,sizeof(d1)/sizeof(*d1),d2,sizeof(d2)/sizeof(*d2));
double error = fabs(pvalue - CORRECT_ANSWERS[0]);
printf("Test sets 1 p-value = %g\n", pvalue);

pvalue = Pvalue(d3,sizeof(d3)/sizeof(*d3),d4,sizeof(d4)/sizeof(*d4));
printf("Test sets 2 p-value = %g\n",pvalue);

pvalue = Pvalue(d5,sizeof(d5)/sizeof(*d5),d6,sizeof(d6)/sizeof(*d6));
printf("Test sets 3 p-value = %g\n", pvalue);

pvalue = Pvalue(d7,sizeof(d7)/sizeof(*d7),d8,sizeof(d8)/sizeof(*d8));
printf("Test sets 4 p-value = %g\n", pvalue);

pvalue = Pvalue(x,sizeof(x)/sizeof(*x),y,sizeof(y)/sizeof(*y));
printf("Test sets 5 p-value = %g\n", pvalue);

pvalue = Pvalue(v1,sizeof(v1)/sizeof(*v1),v2,sizeof(v2)/sizeof(*v2));
printf("Test sets 6 p-value = %g\n", pvalue);

pvalue = Pvalue(s1,sizeof(s1)/sizeof(*s1),s2,sizeof(s2)/sizeof(*s2));
printf("Test sets 7 p-value = %g\n", pvalue);

pvalue = Pvalue(z1, 3, z2, 3);
printf("Test sets z p-value = %g\n", pvalue);

printf("the cumulative error is %g\n", error);
return 0;
}

Output:
Test sets 1 p-value = 0.021378
Test sets 2 p-value = 0.148842
Test sets 3 p-value = 0.0359723
Test sets 4 p-value = 0.0907733
Test sets 5 p-value = 0.0107516
Test sets 6 p-value = 0.00339907
Test sets 7 p-value = 0.527266
Test sets z p-value = 0.545267
the cumulative error is 1.06339e-14

If your computer does not have lgammal, add the following function before main and replace lgammal with lngammal in the calculate_Pvalue function:

#include <stdio.h>
#include <math.h>

long double lngammal(const double xx) {
unsigned int j;
double x,y,tmp,ser;
const double cof[6] = {
76.18009172947146,    -86.50532032941677,
24.01409824083091,    -1.231739572450155,
0.1208650973866179e-2,-0.5395239384953e-5
};

y = x = xx;
tmp = x + 5.5 - (x + 0.5) * logl(x + 5.5);
ser = 1.000000000190015;
for (j=0;j<=5;j++)
ser += (cof[j] / ++y);
return(log(2.5066282746310005 * ser / x) - tmp);
}


## Fortran

### Using IMSL

Using IMSL TDF function. With Absoft Pro Fortran, compile with af90 %FFLAGS% %LINK_FNL% pvalue.f90. Alternatively, the program shows the p-value computed using the IMSL BETAI function.

subroutine welch_ttest(n1, x1, n2, x2, t, df, p)
use tdf_int
implicit none
integer :: n1, n2
double precision :: x1(n1), x2(n2)
double precision :: m1, m2, v1, v2, t, df, p
m1 = sum(x1) / n1
m2 = sum(x2) / n2
v1 = sum((x1 - m1)**2) / (n1 - 1)
v2 = sum((x2 - m2)**2) / (n2 - 1)
t = (m1 - m2) / sqrt(v1 / n1 + v2 / n2)
df = (v1 / n1 + v2 / n2)**2 / &
(v1**2 / (n1**2 * (n1 - 1)) + v2**2 / (n2**2 * (n2 - 1)))
p = 2d0 * tdf(-abs(t), df)
end subroutine

program pvalue
use betai_int
implicit none
double precision :: x(4) = [3d0, 4d0, 1d0, 2.1d0]
double precision :: y(3) = [490.2d0, 340.0d0, 433.9d0]
double precision :: t, df, p
call welch_ttest(4, x, 3, y, t, df, p)
print *, t, df, p
print *, betai(df / (t**2 + df), 0.5d0 * df, 0.5d0)
end program


Output

  -9.55949772193266  2.00085234885628  1.075156114978449E-002
1.075156114978449E-002

### Using SLATEC

With Absoft Pro Fortran, compile with af90 -m64 pvalue.f90 %SLATEC_LINK%.

subroutine welch_ttest(n1, x1, n2, x2, t, df, p)
implicit none
integer :: n1, n2
double precision :: x1(n1), x2(n2)
double precision :: m1, m2, v1, v2, t, df, p
double precision :: dbetai

m1 = sum(x1) / n1
m2 = sum(x2) / n2
v1 = sum((x1 - m1)**2) / (n1 - 1)
v2 = sum((x2 - m2)**2) / (n2 - 1)
t = (m1 - m2) / sqrt(v1 / n1 + v2 / n2)
df = (v1 / n1 + v2 / n2)**2 / &
(v1**2 / (n1**2 * (n1 - 1)) + v2**2 / (n2**2 * (n2 - 1)))
p = dbetai(df / (t**2 + df), 0.5d0 * df, 0.5d0)
end subroutine

program pvalue
implicit none
double precision :: x(4) = [3d0, 4d0, 1d0, 2.1d0]
double precision :: y(3) = [490.2d0, 340.0d0, 433.9d0]
double precision :: t, df, p

call welch_ttest(4, x, 3, y, t, df, p)
print *, t, df, p
end program


Output

  -9.55949772193266  2.00085234885628  1.075156114978449E-002

### Using GSL

Works with: Fortran version 95

Instead of implementing the t-distribution by ourselves, we bind to GNU Scientific Library:

module t_test_m

use, intrinsic :: iso_c_binding
use, intrinsic :: iso_fortran_env, only: wp => real64
implicit none
private

public :: t_test, wp

interface
function gsl_cdf_tdist_p(x, nu) bind(c, name='gsl_cdf_tdist_P')
import
real(c_double), value :: x
real(c_double), value :: nu
real(c_double) :: gsl_cdf_tdist_p
end function gsl_cdf_tdist_p
end interface

contains

!> Welch T test
impure subroutine t_test(x, y, p, t, df)
real(wp), intent(in) :: x(:), y(:)
real(wp), intent(out) :: p       !! p-value
real(wp), intent(out) :: t       !! T value
real(wp), intent(out) :: df      !! degrees of freedom
integer :: n1, n2
real(wp) :: m1, m2, v1, v2

n1 = size(x)
n2 = size(y)
m1 = sum(x)/n1
m2 = sum(y)/n2
v1 = sum((x - m1)**2)/(n1 - 1)
v2 = sum((y - m2)**2)/(n2 - 1)

t = (m1 - m2)/sqrt(v1/n1 + v2/n2)
df = (v1/n1 + v2/n2)**2/(v1**2/(n1**2*(n1 - 1)) + v2**2/(n2**2*(n2 - 1)))
p = 2*gsl_cdf_tdist_p(-abs(t), df)

end subroutine t_test

end module t_test_m

program main
use t_test_m, only: t_test, wp
implicit none
real(wp) :: x(4) = [3.0_wp, 4.0_wp, 1.0_wp, 2.1_wp]
real(wp) :: y(3) = [490.2_wp, 340.0_wp, 433.9_wp]
real(wp) :: t, df, p

call t_test(x, y, p, t, df)
print *, t, df, p

end program main


Output

  -9.5594977219326580        2.0008523488562844        1.0751561149784494E-002

## Go

package main

import (
"fmt"
"math"
)

var (
d1 = []float64{27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6,
23.1, 19.6, 19.0, 21.7, 21.4}
d2 = []float64{27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2,
21.9, 22.1, 22.9, 20.5, 24.4}
d3 = []float64{17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8}
d4 = []float64{21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8,
20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8}
d5 = []float64{19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0}
d6 = []float64{28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7,
23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2}
d7 = []float64{30.02, 29.99, 30.11, 29.97, 30.01, 29.99}
d8 = []float64{29.89, 29.93, 29.72, 29.98, 30.02, 29.98}
x  = []float64{3.0, 4.0, 1.0, 2.1}
y  = []float64{490.2, 340.0, 433.9}
)

func main() {
fmt.Printf("%.6f\n", pValue(d1, d2))
fmt.Printf("%.6f\n", pValue(d3, d4))
fmt.Printf("%.6f\n", pValue(d5, d6))
fmt.Printf("%.6f\n", pValue(d7, d8))
fmt.Printf("%.6f\n", pValue(x, y))
}

func mean(a []float64) float64 {
sum := 0.
for _, x := range a {
sum += x
}
return sum / float64(len(a))
}

func sv(a []float64) float64 {
m := mean(a)
sum := 0.
for _, x := range a {
d := x - m
sum += d * d
}
return sum / float64(len(a)-1)
}

func welch(a, b []float64) float64 {
return (mean(a) - mean(b)) /
math.Sqrt(sv(a)/float64(len(a))+sv(b)/float64(len(b)))
}

func dof(a, b []float64) float64 {
sva := sv(a)
svb := sv(b)
n := sva/float64(len(a)) + svb/float64(len(b))
return n * n /
(sva*sva/float64(len(a)*len(a)*(len(a)-1)) +
svb*svb/float64(len(b)*len(b)*(len(b)-1)))
}

func simpson0(n int, upper float64, f func(float64) float64) float64 {
sum := 0.
nf := float64(n)
dx0 := upper / nf
sum += f(0) * dx0
sum += f(dx0*.5) * dx0 * 4
x0 := dx0
for i := 1; i < n; i++ {
x1 := float64(i+1) * upper / nf
xmid := (x0 + x1) * .5
dx := x1 - x0
sum += f(x0) * dx * 2
sum += f(xmid) * dx * 4
x0 = x1
}
return (sum + f(upper)*dx0) / 6
}

func pValue(a, b []float64) float64 {
ν := dof(a, b)
t := welch(a, b)
g1, _ := math.Lgamma(ν / 2)
g2, _ := math.Lgamma(.5)
g3, _ := math.Lgamma(ν/2 + .5)
return simpson0(2000, ν/(t*t+ν),
func(r float64) float64 { return math.Pow(r, ν/2-1) / math.Sqrt(1-r) }) /
math.Exp(g1+g2-g3)
}

Output:
0.021378
0.148842
0.035972
0.090773
0.010751


## J

Implementation:

integrate=: adverb define
'a b steps'=. 3{.y,128
size=. (b - a)%steps
size * +/ u |: 2 ]\ a + size * i.>:steps
)
simpson  =: adverb def '6 %~ +/ 1 1 4 * u y, -:+/y'

lngamma=: ^.@!@<:(^.@!@(1 | ]) + +/@:^.@(1 + 1&| + i.@<.)@<:)@.(1&<:)"0
mean=: +/ % #
nu=: # - 1:
sampvar=: +/@((- mean) ^ 2:) % nu
ssem=: sampvar % #
welch_T=: -&mean % 2 %: +&ssem
nu=: nu f. : ((+&ssem ^ 2:) % +&((ssem^2:)%nu))
B=: ^@(+&lngamma - lngamma@+)

t=. x welch_T y  NB. need numbers for numerical integration
v=. x nu y
F=. ^&(_1+v%2) % 2 %: 1&-
lo=. 0
hi=. v%(t^2)+v
(F f. simpson integrate lo,hi) % 0.5 B v%2
)


integrate and simpson are from the Numerical integration task.

lngamma is from http://www.jsoftware.com/pipermail/programming/2015-July/042174.html -- for values less than some convenient threshold (we use 1, but we could use a modestly higher threshold), we calculate it directly. For larger values we compute the fractional part directly and rebuild the log of the factorial using the sum of the logs.

mean is classic J - most J tutorials will include this

The initial definition of nu (degrees of freedom of a data set), as well as the combining form (approximating degrees of freedom for two sets of data) is from Welch's t test. (Verb definitions can be forward referenced, even in J's tacit definitions, but it seems clearer to specify these definitions so they only depend on previously declared definitions.)

sampvar is sample variance (or: standard deviation squared)

ssem is squared standard error of the mean

Also... please ignore the highlighting of v in the definition of p2_tail. In this case, it's F that's the verb, v is just another number (the degrees of freedom for our two data sets. (But this is a hint that in explicit conjunction definitions, v would be the right verb argument. Unfortunately, the wiki's highlighting implementation is not capable of distinguishing that particular context from other contexts.)

d1=: 27.5 21 19 23.6 17 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19 21.7 21.4
d2=: 27.1 22 20.8 23.4 23.4 23.5 25.8 22 24.8 20.2 21.9 22.1 22.9 20.5 24.4
d3=: 17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8
d4=: 21.5 22.8 21 23 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8
d5=: 19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22
d6=: 28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18 23.9 21.6 24.3 20.4 24 13.2
d7=: 30.02 29.99 30.11 29.97 30.01 29.99
d8=: 29.89 29.93 29.72 29.98 30.02 29.98
d9=: 3 4 1 2.1
da=: 490.2 340 433.9


   d1 p2_tail d2
0.021378
d3 p2_tail d4
0.148842
d5 p2_tail d6
0.0359723
d7 p2_tail d8
0.0907733
d9 p2_tail da
0.0107377


## Java

Using the Apache Commons Mathematics Library.

import org.apache.commons.math3.distribution.TDistribution;

public class WelchTTest {
public static double[] meanvar(double[] a) {
double m = 0.0, v = 0.0;
int n = a.length;

for (double x: a) {
m += x;
}
m /= n;

for (double x: a) {
v += (x - m) * (x - m);
}
v /= (n - 1);

return new double[] {m, v};

}

public static double[] welch_ttest(double[] x, double[] y) {
double mx, my, vx, vy, t, df, p;
double[] res;
int nx = x.length, ny = y.length;

res = meanvar(x);
mx = res[0];
vx = res[1];

res = meanvar(y);
my = res[0];
vy = res[1];

t = (mx-my)/Math.sqrt(vx/nx+vy/ny);
df = Math.pow(vx/nx+vy/ny, 2)/(vx*vx/(nx*nx*(nx-1))+vy*vy/(ny*ny*(ny-1)));
TDistribution dist = new TDistribution(df);
p = 2.0*dist.cumulativeProbability(-Math.abs(t));
return new double[] {t, df, p};
}

public static void main(String[] args) {
double x[] = {3.0, 4.0, 1.0, 2.1};
double y[] = {490.2, 340.0, 433.9};
double res[] = welch_ttest(x, y);
System.out.println("t = " + res[0]);
System.out.println("df = " + res[1]);
System.out.println("p = " + res[2]);
}
}


Result

javac -cp .;L:\java\commons-math3-3.6.1.jar WelchTTest.java
java -cp .;L:\java\commons-math3-3.6.1.jar WelchTTest
t = -9.559497721932658
df = 2.0008523488562844
p = 0.010751561149784485

## jq

1. Translation of: Wren
Works with: jq

Works with gojq, the Go implementation of jq

Notice how jq supports the closure, f, in the same way as Wren.

jq's lgamma returns the natural logarithm of the absolute value of the gamma function of x.

def mean: add / length;

# Sample variance using division by (length-1)
def variance:
mean as $m | (reduce .[] as$x (0; . + (($x -$m) | .*.))) / (length-1) ;

def welch(a; b):
((a|mean) - (b|mean)) /
(((a|variance/length) + (b|variance/length)) | sqrt) ;

def dof(a; b):
(a|variance) as $sva | (b|variance) as$svb
| (a|length) as $la | (b|length) as$lb
| ($sva/$la + $svb/$lb) as $n |$n * $n / ($sva*$sva/($la*$la*($la-1)) + $svb*$svb/($lb*$lb*($lb-1))) ; def simpson0(nf; upper; filter): (upper/nf) as$dx0
| {sum: (( (0|filter) + ($dx0 * 0.5|filter) * 4) *$dx0),
x0: $dx0 } | reduce range(1; nf) as$i (.;
( ($i + 1) * upper / nf ) as$x1
| ((.x0 + $x1) * 0.5) as$xmid
| ($x1 - .x0) as$dx
| .sum = .sum + ((.x0|filter)*2 + ($xmid|filter)*4) *$dx
| .x0 = $x1) | (.sum + (upper|filter)*$dx0) / 6 ;

def pValue(a; b):
dof(a; b) as $nu | def f: . as$r
| pow($r; ($nu/2) - 1) / ((1 - $r)|sqrt); welch(a; b) as$t
| (($nu/2)|lgamma) as$g1
| (0.5|lgamma) as $g2 | (($nu/2 + 0.5)|lgamma) as $g3 | simpson0(2000;$nu/($t*$t + $nu); f) / (($g1 + $g2 -$g3)|exp) ;

def d1: [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4];
def d2: [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4];
def d3: [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8];
def d4: [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6,
21.5, 22.5, 23.5, 21.5, 21.8];
def d5: [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0];
def d6: [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9,
21.6, 24.3, 20.4, 24.0, 13.2];
def d7: [30.02, 29.99, 30.11, 29.97, 30.01, 29.99];
def d8: [29.89, 29.93, 29.72, 29.98, 30.02, 29.98];
def x : [3.0, 4.0, 1.0, 2.1];
def y : [490.2, 340.0, 433.9];

pValue(d1; d2),
pValue(d3; d4),
pValue(d5; d6),
pValue(d7; d8),
pValue(x; y)
Output:
0.02137800146288292
0.1488416966053347
0.03597227102982764
0.09077332428566065
0.010750673736239608


## Julia

Works with: Julia version 0.6
using HypothesisTests

d1 = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4]
d2 = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4]

d3 = [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8]
d4 = [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8]

d5 = [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0]
d6 = [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2]

d7 = [30.02, 29.99, 30.11, 29.97, 30.01, 29.99]
d8 = [29.89, 29.93, 29.72, 29.98, 30.02, 29.98]

x = [  3.0,   4.0,   1.0, 2.1]
y = [490.2, 340.0, 433.9]

for (y1, y2) in ((d1, d2), (d3, d4), (d5, d6), (d7, d8), (x, y))
ttest = UnequalVarianceTTest(y1, y2)
println("\nData:\n  y1 = $y1\n y2 =$y2\nP-value for unequal variance TTest: ", round(pvalue(ttest), 4))
end

Output:
Data:
y1 = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4]
y2 = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4]
P-value for unequal variance TTest: 0.0214

Data:
y1 = [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8]
y2 = [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8]
P-value for unequal variance TTest: 0.1488

Data:
y1 = [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0]
y2 = [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2]
P-value for unequal variance TTest: 0.036

Data:
y1 = [30.02, 29.99, 30.11, 29.97, 30.01, 29.99]
y2 = [29.89, 29.93, 29.72, 29.98, 30.02, 29.98]
P-value for unequal variance TTest: 0.0908

Data:
y1 = [3.0, 4.0, 1.0, 2.1]
y2 = [490.2, 340.0, 433.9]
P-value for unequal variance TTest: 0.0108


## Kotlin

This program brings in code from other tasks for gamma functions and integration by Simpson's rule as Kotlin doesn't have these built-in:

// version 1.1.4-3

typealias Func = (Double) -> Double

fun square(d: Double) = d * d

fun sampleVar(da: DoubleArray): Double {
if (da.size < 2) throw IllegalArgumentException("Array must have at least 2 elements")
val m = da.average()
return da.map { square(it - m) }.sum() / (da.size - 1)
}

fun welch(da1: DoubleArray, da2: DoubleArray): Double {
val temp = sampleVar(da1) / da1.size + sampleVar(da2) / da2.size
return (da1.average() - da2.average()) / Math.sqrt(temp)
}

fun degreesFreedom(da1: DoubleArray, da2: DoubleArray): Double {
val s1 = sampleVar(da1)
val s2 = sampleVar(da2)
val n1 = da1.size
val n2 = da2.size
val temp1 = square(s1 / n1 + s2 / n2)
val temp2 = square(s1) / (n1 * n1 * (n1 - 1)) + square(s2) / (n2 * n2 * (n2 - 1))
return temp1 / temp2
}

fun gamma(d: Double): Double {
var dd = d
val p = doubleArrayOf(
0.99999999999980993,
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
)
val g = 7
if (dd < 0.5) return Math.PI / (Math.sin(Math.PI * dd) * gamma(1.0 - dd))
dd--
var a = p[0]
val t = dd + g + 0.5
for (i in 1 until p.size) a += p[i] / (dd + i)
return Math.sqrt(2.0 * Math.PI) * Math.pow(t, dd + 0.5) * Math.exp(-t) * a
}

fun lGamma(d: Double) = Math.log(gamma(d))

fun simpson(a: Double, b: Double, n: Int, f: Func): Double {
val h = (b - a) / n
var sum = 0.0
for (i in 0 until n) {
val x = a + i * h
sum += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0
}
return sum * h
}

fun p2Tail(da1: DoubleArray, da2: DoubleArray): Double {
val nu = degreesFreedom(da1, da2)
val t = welch(da1, da2)
val g = Math.exp(lGamma(nu / 2.0) + lGamma(0.5) - lGamma(nu / 2.0 + 0.5))
val b = nu / (t * t + nu)
val f: Func = { r ->  Math.pow(r, nu / 2.0 - 1.0) / Math.sqrt(1.0 - r) }
return simpson(0.0, b, 10000, f) / g   // n = 10000 seems more than enough here
}

fun main(args: Array<String>) {
val da1 = doubleArrayOf(
27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6,
23.1, 19.6, 19.0, 21.7, 21.4
)
val da2 = doubleArrayOf(
27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2,
21.9, 22.1, 22.9, 20.5, 24.4
)
val da3 = doubleArrayOf(
17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8
)
val da4 = doubleArrayOf(
21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8,
20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8
)
val da5 = doubleArrayOf(
19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0
)
val da6 = doubleArrayOf(
28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7,
23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2
)
val da7 = doubleArrayOf(30.02, 29.99, 30.11, 29.97, 30.01, 29.99)
val da8 = doubleArrayOf(29.89, 29.93, 29.72, 29.98, 30.02, 29.98)

val x = doubleArrayOf(3.0, 4.0, 1.0, 2.1)
val y = doubleArrayOf(490.2, 340.0, 433.9)

val f = "%.6f"
println(f.format(p2Tail(da1, da2)))
println(f.format(p2Tail(da3, da4)))
println(f.format(p2Tail(da5, da6)))
println(f.format(p2Tail(da7, da8)))
println(f.format(p2Tail(x, y)))
}

Output:
0.021378
0.148842
0.035972
0.090773
0.010751


## Nim

Translation of: Kotlin
import math, stats, strutils, sugar

func sqr(f: float): float = f * f

func degreesFreedom(da1, da2: openArray[float]): float =
let s1 = varianceS(da1)
let s2 = varianceS(da2)
let n1 = da1.len.toFloat
let n2 = da2.len.toFloat
let n = sqr(s1 / n1 + s2 / n2)
let d = sqr(s1) / (n1 * n1 * (n1 - 1)) + sqr(s2) / (n2 * n2 * (n2 - 1))
result = n / d

func welch(da1, da2: openArray[float]): float =
let f = varianceS(da1) / da1.len.toFloat + varianceS(da2) / da2.len.toFloat
result = (mean(da1) - mean(da2)) / sqrt(f)

func simpson(a, b: float; n: int; f: float -> float): float =
let h = (b - a) / n.toFloat
var sum = 0.0
for i in 0..<n:
let x = a + i.toFloat * h
sum += (f(x) + 4 * f(x + h / 2) + f(x + h)) / 6
result = sum * h

func p2Tail(da1, da2: openArray[float]): float =
let ν = degreesFreedom(da1, da2)
let t = welch(da1, da2)
let g = exp(lGamma(ν / 2) + lGamma(0.5) - lGamma(ν / 2 + 0.5))
let b = ν / (t * t + ν)
proc f(r: float): float = pow(r, ν / 2 - 1) / sqrt(1 - r)
result = simpson(0, b, 10000, f) / g    # n = 10000 seems more than enough here.

when isMainModule:

const
Da1 = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6,
23.1, 19.6, 19.0, 21.7, 21.4]
Da2 = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2,
21.9, 22.1, 22.9, 20.5, 24.4]
Da3 = [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8]
Da4 = [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8,
20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8]
Da5 = [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0]
Da6 = [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7,
23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2]
Da7 = [30.02, 29.99, 30.11, 29.97, 30.01, 29.99]
Da8 = [29.89, 29.93, 29.72, 29.98, 30.02, 29.98]

X = [3.0, 4.0, 1.0, 2.1]
Y = [490.2, 340.0, 433.9]

echo p2Tail(Da1, Da2).formatFloat(ffDecimal, 6)
echo p2Tail(Da3, Da4).formatFloat(ffDecimal, 6)
echo p2Tail(Da5, Da6).formatFloat(ffDecimal, 6)
echo p2Tail(Da7, Da8).formatFloat(ffDecimal, 6)
echo p2Tail(X, Y).formatFloat(ffDecimal, 6)

Output:
0.021378
0.148842
0.035972
0.090773
0.010751

## Maple

WelschTTest:=proc(x::list(numeric),y::list(numeric))
uses Statistics;
local n1:=nops(x),n2:=nops(y),
m1:=Mean(x),m2:=Mean(y),
v1:=Variance(x),v2:=Variance(y),
t,nu,p;
t:=(m1-m2)/sqrt(v1/n1+v2/n2);
nu:=(v1/n1+v2/n2)^2/(v1^2/(n1^2*(n1-1))+v2^2/(n2^2*(n2-1)));
p:=2*CDF(StudentTDistribution(nu),-abs(t));
t,nu,p
end proc:

x:=[3,4,1,2.1]:
y:=[490.2,340,433.9]:
WelschTTest(x,y);
# -9.55949772193266, 2.00085234885628, 0.0107515611497845

## Octave

Translation of: Stata
x = [3.0,4.0,1.0,2.1];
y = [490.2,340.0,433.9];
n1 = length(x);
n2 = length(y);
v1 = var(x);
v2 = var(y);
t = (mean(x)-mean(y))/(sqrt(v1/n1+v2/n2));
df = (v1/n1+v2/n2)^2/(v1^2/(n1^2*(n1-1))+v2^2/(n2^2*(n2-1)));
p = betainc(df/(t^2+df),df/2,1/2);
[t df p]

ans =

-9.559498   2.000852   0.010752


## PARI/GP

B2(x,y)=exp(lngamma(x)+lngamma(y)-lngamma(x+y))
B3(x,a,b)=a--;b--;intnum(r=0,x,r^a*(1-r)^b)
Welch2(u,v)=my(m1=vecsum(u)/#u, m2=vecsum(v)/#v, v1=var(u,m1), v2=var(v,m2), s=v1/#u+v2/#v, t=(m1-m2)/sqrt(s), nu=s^2/(v1^2/#u^2/(#u-1)+v2^2/#v^2/(#v-1))); B3(nu/(t^2+nu),nu/2,1/2)/B2(nu/2,1/2);
Welch2([3,4,1,2.1], [490.2,340,433.9])
Output:
%1 = 0.010751561149784496723954539777213062928

## Perl

### Using Math::AnyNum

Uses Math::AnyNum for gamma and pi. It is possible to use some other modules (e.g. Math::Cephes) if Math::AnyNum has problematic dependencies.

Translation of: Sidef
use utf8;
use List::Util qw(sum);
use Math::AnyNum qw(gamma pi);

sub p_value () {
my ($A,$B) = @_;

(@$A > 1 && @$B > 1) || return 1;

my $x̄_a = sum(@$A) / @$A; my$x̄_b = sum(@$B) / @$B;

my $a_var = sum(map { ($x̄_a - $_)**2 } @$A) / (@$A - 1); my$b_var = sum(map { ($x̄_b -$_)**2 } @$B) / (@$B - 1);

($a_var &&$b_var) || return 1;

my $Welsh_𝒕_statistic = ($x̄_a - $x̄_b) / sqrt($a_var/@$A +$b_var/@$B); my$DoF = ($a_var/@$A + $b_var/@$B)**2 / (
$a_var**2 / (@$A**3 - @$A**2) +$b_var**2 / (@$B**3 - @$B**2));

my $sa =$DoF / 2 - 1;
my $x =$DoF / ($Welsh_𝒕_statistic**2 +$DoF);
my $N = 65355; my$h  = $x /$N;

my ($sum1,$sum2) = (0, 0);

foreach my $k (0 ..$N - 1) {
my $i =$h * $k;$sum1 += ($i +$h/2)**$sa / sqrt(1 - ($i + $h/2));$sum2 += $i**$sa / sqrt(1-$i); } ($h/6 * ($x**$sa / sqrt(1-$x) + 4*$sum1 + 2*$sum2) / (gamma($sa + 1) * sqrt(pi) / gamma($sa + 1.5)))->numify; } my @tests = ( [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4], [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4], [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8], [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8], [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0], [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2], [30.02, 29.99, 30.11, 29.97, 30.01, 29.99], [29.89, 29.93, 29.72, 29.98, 30.02, 29.98], [3.0, 4.0, 1.0, 2.1], [490.2, 340.0, 433.9], ); while (@tests) { my ($left, $right) = splice(@tests, 0, 2); print p_value($left, $right), "\n"; }  Output: 0.0213780014628667 0.148841696605327 0.0359722710297968 0.0907733242856612 0.0107515340333929  ### Using Burkhardt's 'incomplete beta' We use a slightly more accurate lgamma than the C code. Note that Perl can be compiled with different underlying floating point representations -- double, long double, or quad double. Translation of: C use strict; use warnings; use List::Util 'sum'; sub lgamma { my$x = shift;
my $log_sqrt_two_pi = 0.91893853320467274178; my @lanczos_coef = ( 0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 ); my$base = $x + 7.5; my$sum = 0;
$sum +=$lanczos_coef[$_] / ($x + $_) for reverse (1..8);$sum += $lanczos_coef[0];$sum = $log_sqrt_two_pi + log($sum/$x) + ( ($x+0.5)*log($base) -$base );
$sum; } sub calculate_P_value { my ($array1,$array2) = (shift, shift); return 1 if @$array1 <= 1 or @$array2 <= 1; my$mean1 = sum(@$array1); my$mean2 = sum(@$array2);$mean1 /= scalar @$array1;$mean2 /= scalar @$array2; return 1 if$mean1 == $mean2; my ($variance1,$variance2);$variance1 += ($mean1-$_)**2 for @$array1;$variance2 += ($mean2-$_)**2 for @$array2; return 1 if$variance1 == 0 and $variance2 == 0;$variance1 = $variance1/(@$array1-1);
$variance2 =$variance2/(@$array2-1); my$Welch_t_statistic = ($mean1-$mean2)/sqrt($variance1/@$array1+$variance2/@$array2);
my $DoF = (($variance1/@$array1+$variance2/@$array2)**2) / ( ($variance1*$variance1)/(@$array1*@$array1*(@$array1-1)) +
($variance2*$variance2)/(@$array2*@$array2*(@$array2-1)) ); my$A     = $DoF / 2; my$value = $DoF / ($Welch_t_statistic**2 + $DoF); return$value if $A <= 0 or$value <= 0 or 1 <= $value; # from here, translation of John Burkhardt's C code my$beta = lgamma($A) + 0.57236494292470009 - lgamma($A+0.5); # constant is lgamma(.5), but more precise than 'lgamma' routine
my $eps = 10**-15; my($ai,$cx,$indx,$ns,$pp,$psq,$qq,$qq_ai,$rx,$term,$xx);

$psq =$A + 0.5;
$cx = 1 -$value;
if ($A <$psq * $value) { ($xx, $cx,$pp, $qq,$indx) = ($cx,$value, 0.5,  $A, 1) } else { ($xx,      $pp,$qq, $indx) = ($value,         $A, 0.5, 0) }$term = $ai =$value = 1;
$ns = int$qq + $cx *$psq;

# Soper reduction formula
$qq_ai =$qq - $ai;$rx = $ns == 0 ?$xx : $xx /$cx;
while (1) {
$term =$term * $qq_ai *$rx / ( $pp +$ai );
$value =$value + $term;$qq_ai = abs($term); if ($qq_ai <= $eps &&$qq_ai <= $eps *$value) {
$value =$value * exp ($pp * log($xx) + ($qq - 1) * log($cx) - $beta) /$pp;
$value = 1 -$value if $indx; last; }$ai++;
$ns--; if ($ns >= 0) {
$qq_ai =$qq - $ai;$rx = $xx if$ns == 0;
} else {
$qq_ai =$psq;
$psq =$psq + 1;
}
}
$value } my @answers = ( 0.021378001462867, 0.148841696605327, 0.0359722710297968, 0.090773324285671, 0.0107515611497845, 0.00339907162713746, 0.52726574965384, 0.545266866977794, ); my @tests = ( [27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4], [27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4], [17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8], [21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8], [19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0], [28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2], [30.02,29.99,30.11,29.97,30.01,29.99], [29.89,29.93,29.72,29.98,30.02,29.98], [3.0,4.0,1.0,2.1], [490.2,340.0,433.9], [0.010268,0.000167,0.000167], [0.159258,0.136278,0.122389], [1.0/15,10.0/62.0], [1.0/10,2/50.0], [9/23.0,21/45.0,0/38.0], [0/44.0,42/94.0,0/22.0], ); my$error = 0;
while (@tests) {
my ($left,$right) = splice(@tests, 0, 2);
my $pvalue = calculate_P_value($left,$right);$error += abs($pvalue - shift @answers); printf("p-value = %.14g\n",$pvalue);
}
printf("cumulative error is %g\n", $error);  Output: p-value = 0.021378001462867 p-value = 0.14884169660533 p-value = 0.035972271029797 p-value = 0.090773324285661 p-value = 0.010751561149784 p-value = 0.0033990716271375 p-value = 0.52726574965384 p-value = 0.54526686697779 cumulative error is 1.11139e-14 ## Phix Translation of: Go Translation of: Kotlin with javascript_semantics function mean(sequence a) return sum(a) / length(a) end function function sv(sequence a) integer la = length(a) atom m := mean(a), tot := 0 for i=1 to la do atom d = a[i] - m tot += d * d end for return tot / (la-1) end function function welch(sequence a, b) integer la = length(a), lb = length(b) return (mean(a) - mean(b)) / sqrt(sv(a)/la+sv(b)/lb) end function function dof(sequence a, b) integer la = length(a), lb = length(b) atom sva := sv(a), svb := sv(b), n := sva/la + svb/lb return n * n / (sva*sva/(la*la*(la-1)) + svb*svb/(lb*lb*(lb-1))) end function function f(atom r, v) return power(r, v/2-1) / sqrt(1-r) end function function simpson0(integer n, atom high, v) atom tot := 0, dx0 := high / n, x0 := dx0, x1, xmid, dx tot += f(0,v) * dx0 tot += f(dx0*.5,v) * dx0 * 4 for i=1 to n-1 do x1 := (i+1) * high / n xmid := (x0 + x1) * .5 dx := x1 - x0 tot += f(x0,v) * dx * 2 tot += f(xmid,v) * dx * 4 x0 = x1 end for return (tot + f(high,v)*dx0) / 6 end function constant p = { 0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 } function gamma(atom d) atom dd = d, g = 7 if dd<0.5 then return PI / (sin(PI*dd) * gamma(1-dd)) end if dd -= 1 atom a = p[1], t = dd + g + 0.5 for i=2 to length(p) do a += p[i] / (dd + i - 1) end for return sqrt(2*PI) * power(t, dd + 0.5) * exp(-t) * a end function function lGamma(atom d) return log(gamma(d)) end function function pValue(sequence ab) sequence {a, b} = ab atom v := dof(a, b), t := welch(a, b), g1 := lGamma(v / 2), g2 := lGamma(.5), g3 := lGamma(v/2 + .5) return simpson0(2000, v/(t*t+v), v) / exp(g1+g2-g3) end function constant tests = {{{27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4}, {27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4}}, {{17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8}, {21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8}}, {{19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0}, {28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2}}, {{30.02, 29.99, 30.11, 29.97, 30.01, 29.99}, {29.89, 29.93, 29.72, 29.98, 30.02, 29.98}}, {{3.0, 4.0, 1.0, 2.1}, {490.2, 340.0, 433.9}} } for i=1 to length(tests) do ?pValue(tests[i]) end for  Output: 0.0213780015 0.1488416966 0.035972271 0.0907733243 0.0107506737  Translation of: Python The above was a bit off on the fifth test, so I also tried this. using gamma() from Gamma_function#Phix (the one from above is probably also fine, but I didn't test that) with javascript_semantics --<copy of gamma from Gamma_function#Phix> sequence c = repeat(0,12) function gamma(atom z) atom accm = c[1] if accm=0 then accm = sqrt(2*PI) c[1] = accm atom k1_factrl = 1 -- (k - 1)!*(-1)^k with 0!==1 for k=2 to 12 do c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl k1_factrl *= -(k-1) end for end if for k=2 to 12 do accm += c[k]/(z+k-1) end for accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1) return accm/z end function --</copy of gamma> function lgamma(atom d) return log(gamma(d)) end function function betain(atom x, p, q) if p<=0 or q<=0 or x<0 or x>1 then ?9/0 end if if x == 0 or x == 1 then return x end if atom acu = 1e-15, lnbeta = lgamma(p) + lgamma(q) - lgamma(p + q), psq = p + q, cx = 1-x bool indx = (p<psq*x) if indx then {cx,x,p,q} = {x,1-x,q,p} end if atom term = 1, ai = 1, val = 1, ns = floor(q + cx*psq), rx = iff(ns=0?x:x/cx), temp = q - ai while true do term *= temp * rx / (p + ai) val += term temp = abs(term) if temp<=acu and temp<=acu*val then val *= exp(p*log(x) + (q-1)*log(cx) - lnbeta) / p return iff(indx?1-val:val) end if ai += 1 ns -= 1 if ns>=0 then temp = q - ai if ns == 0 then rx = x end if else temp = psq psq += 1 end if end while end function function welch_ttest(sequence ab) sequence {a, b} = ab integer la = length(a), lb = length(b) atom ma = sum(a)/la, mb = sum(b)/lb, va = sum(sq_power(sq_sub(a,ma),2))/(la-1), vb = sum(sq_power(sq_sub(b,mb),2))/(lb-1), n = va/la + vb/lb, t = (ma-mb)/sqrt(n), df = (n*n) / (va*va/(la*la*(la-1)) + vb*vb/(lb*lb*(lb-1))) return betain(df/(t*t+df), df/2, 1/2) end function constant tests = {{{27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4}, {27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4}}, {{17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8}, {21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8}}, {{19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0}, {28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2}}, {{30.02, 29.99, 30.11, 29.97, 30.01, 29.99}, {29.89, 29.93, 29.72, 29.98, 30.02, 29.98}}, {{3.0, 4.0, 1.0, 2.1}, {490.2, 340.0, 433.9}}, {{0.010268,0.000167,0.000167}, {0.159258,0.136278,0.122389}}, {{1.0/15,10.0/62.0}, {1.0/10,2/50.0}}, {{9/23.0,21/45.0,0/38.0}, {0/44.0,42/94.0,0/22.0}}}, correct = {0.021378001462867, 0.148841696605327, 0.0359722710297968, 0.090773324285671, 0.0107515611497845, 0.00339907162713746, 0.52726574965384, 0.545266866977794} atom cerr = 0 for i=1 to length(tests) do atom r = welch_ttest(tests[i]) ?r cerr += abs(r-correct[i]) end for ?{"cumulative error",cerr}  Output: 0.02137800146 0.1488416966 0.03597227103 0.09077332429 0.01075156115 0.003399071627 0.5272657497 0.545266867 {"cumulative error",1.989380882e-14} -- (32 bit/p2js) {"cumulative error",4.915115776e-15} -- (64-bit)  ## Python ### Using NumPy & SciPy import numpy as np import scipy as sp import scipy.stats def welch_ttest(x1, x2): n1 = x1.size n2 = x2.size m1 = np.mean(x1) m2 = np.mean(x2) v1 = np.var(x1, ddof=1) v2 = np.var(x2, ddof=1) t = (m1 - m2) / np.sqrt(v1 / n1 + v2 / n2) df = (v1 / n1 + v2 / n2)**2 / (v1**2 / (n1**2 * (n1 - 1)) + v2**2 / (n2**2 * (n2 - 1))) p = 2 * sp.stats.t.cdf(-abs(t), df) return t, df, p welch_ttest(np.array([3.0, 4.0, 1.0, 2.1]), np.array([490.2, 340.0, 433.9])) (-9.559497721932658, 2.0008523488562844, 0.01075156114978449)  ### Using betain from AS 63 First, the implementation of betain (translated from the Stata program in the discussion page). The original Fortran code is under copyrighted by the Royal Statistical Society. The C translation is under GPL, written by John Burkardt. The exact statement of the RSS license is unclear. import math def betain(x, p, q): if p <= 0 or q <= 0 or x < 0 or x > 1: raise ValueError if x == 0 or x == 1: return x acu = 1e-15 lnbeta = math.lgamma(p) + math.lgamma(q) - math.lgamma(p + q) psq = p + q if p < psq * x: xx = 1 - x cx = x pp = q qq = p indx = True else: xx = x cx = 1 - x pp = p qq = q indx = False term = ai = value = 1 ns = math.floor(qq + cx * psq) rx = xx / cx temp = qq - ai if ns == 0: rx = xx while True: term *= temp * rx / (pp + ai) value += term temp = abs(term) if temp <= acu and temp <= acu * value: value *= math.exp(pp * math.log(xx) + (qq - 1) * math.log(cx) - lnbeta) / pp return 1 - value if indx else value ai += 1 ns -= 1 if ns >= 0: temp = qq - ai if ns == 0: rx = xx else: temp = psq psq += 1  The Python code is then straightforward: import math def welch_ttest(a1, a2): n1 = len(a1) n2 = len(a2) if n1 <= 1 or n2 <= 1: raise ValueError mean1 = sum(a1) / n1 mean2 = sum(a2) / n2 var1 = sum((x - mean1)**2 for x in a1) / (n1 - 1) var2 = sum((x - mean2)**2 for x in a2) / (n2 - 1) t = (mean1 - mean2) / math.sqrt(var1 / n1 + var2 / n2) df = (var1 / n1 + var2 / n2)**2 / (var1**2 / (n1**2 * (n1 - 1)) + var2**2 / (n2**2 * (n2 - 1))) p = betain(df / (t**2 + df), df / 2, 1 / 2) return t, df, p  Example a1 = [3, 4, 1, 2.1] a2 = [490.2, 340, 433.9] print(welch_ttest(a1, a2))  Output (-9.559497721932658, 2.0008523488562844, 0.01075156114978449) ## R #!/usr/bin/R printf <- function(...) cat(sprintf(...)) #allows printing to greater number of digits #https://stackoverflow.com/questions/13023274/how-to-do-printf-in-r#13023329 d1 <- c(27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4) d2 <- c(27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4) d3 <- c(17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8) d4 <- c(21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8) d5 <- c(19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0) d6 <- c(28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2) d7 <- c(30.02,29.99,30.11,29.97,30.01,29.99) d8 <- c(29.89,29.93,29.72,29.98,30.02,29.98) x <- c(3.0,4.0,1.0,2.1) y <- c(490.2,340.0,433.9) v1 <- c(0.010268,0.000167,0.000167); v2<- c(0.159258,0.136278,0.122389); s1<- c(1.0/15,10.0/62.0); s2<- c(1.0/10,2/50.0); z1<- c(9/23.0,21/45.0,0/38.0); z2<- c(0/44.0,42/94.0,0/22.0); results <- t.test(d1,d2, alternative="two.sided", var.equal=FALSE) printf("%.15g\n", results$p.value);
results <- t.test(d3,d4, alternative="two.sided", var.equal=FALSE)
printf("%.15g\n", results$p.value); results <- t.test(d5,d6, alternative="two.sided", var.equal=FALSE) printf("%.15g\n", results$p.value);
results <- t.test(d7,d8, alternative="two.sided", var.equal=FALSE)
printf("%.15g\n", results$p.value); results <- t.test(x,y, alternative="two.sided", var.equal=FALSE) printf("%.15g\n", results$p.value);
results <- t.test(v1,v2, alternative="two.sided", var.equal=FALSE)
printf("%.15g\n", results$p.value); results <- t.test(s1,s2, alternative="two.sided", var.equal=FALSE) printf("%.15g\n", results$p.value);
results <- t.test(z1,z2, alternative="two.sided", var.equal=FALSE)
printf("%.15g\n", results$p.value);  The output here is used to compare against C's output above. Output: 0.021378001462867 0.148841696605327 0.0359722710297968 0.090773324285671 0.0107515611497845 0.00339907162713746 0.52726574965384 0.545266866977794  ## Racket Translation of: C #lang racket (require math/statistics math/special-functions) (define (p-value S1 S2 #:n (n 11000)) (define σ²1 (variance S1 #:bias #t)) (define σ²2 (variance S2 #:bias #t)) (define N1 (sequence-length S1)) (define N2 (sequence-length S2)) (define σ²/sz1 (/ σ²1 N1)) (define σ²/sz2 (/ σ²2 N2)) (define degrees-of-freedom (/ (sqr (+ σ²/sz1 σ²/sz2)) (+ (/ (sqr σ²1) (* (sqr N1) (sub1 N1))) (/ (sqr σ²2) (* (sqr N2) (sub1 N2)))))) (define a (/ degrees-of-freedom 2)) (define a-1 (sub1 a)) (define x (let ((welch-t-statistic (/ (- (mean S1) (mean S2)) (sqrt (+ σ²/sz1 σ²/sz2))))) (/ degrees-of-freedom (+ (sqr welch-t-statistic) degrees-of-freedom)))) (define h (/ x n)) (/ (* (/ h 6) (+ (* (expt x a-1) (expt (- 1 x) -1/2)) (* 4 (for/sum ((i (in-range 0 n))) (* (expt (+ (* h i) (/ h 2)) a-1) (expt (- 1 (+ (* h i) (/ h 2))) -1/2)))) (* 2 (for/sum ((i (in-range 0 n))) (* (expt (* h i) a-1) (expt (- 1 (* h i)) -1/2)))))) (* (gamma a) 1.77245385090551610 (/ (gamma (+ a 1/2)))))) (module+ test (list (p-value (list 27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4) (list 27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4)) (p-value (list 17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8) (list 21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8)) (p-value (list 19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0) (list 28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2)) (p-value (list 30.02 29.99 30.11 29.97 30.01 29.99) (list 29.89 29.93 29.72 29.98 30.02 29.98)) (p-value (list 3.0 4.0 1.0 2.1) (list 490.2 340.0 433.9))))  Output: (0.021378001462867013 0.14884169660532798 0.035972271029796624 0.09077332428567102 0.01075139991904718) ## Raku (formerly Perl 6) ### Integration using Simpson's Rule Works with: Rakudo version 2019.11 Translation of: C Perhaps "inspired by C example" may be more accurate. Gamma subroutine from Gamma function task. sub Γ(\z) { constant g = 9; z < .5 ?? π / sin(π × z) / Γ(1 - z) !! τ.sqrt × (z + g - 1/2)**(z - 1/2) × exp(-(z + g - 1/2)) × [+] < 1.000000000000000174663 5716.400188274341379136 -14815.30426768413909044 14291.49277657478554025 -6348.160217641458813289 1301.608286058321874105 -108.1767053514369634679 2.605696505611755827729 -0.7423452510201416151527e-2 0.5384136432509564062961e-7 -0.4023533141268236372067e-8 > Z× 1, |map 1/(z + *), 0..* } sub p-value (@A, @B) { return 1 if @A <= 1 or @B <= 1; my$a-mean = @A.sum / @A;
my $b-mean = @B.sum / @B; my$a-variance = @A.map( { ($a-mean -$_)² } ).sum / (@A - 1);
my $b-variance = @B.map( { ($b-mean - $_)² } ).sum / (@B - 1); return 1 unless$a-variance && $b-variance; my \Welchs-𝒕-statistic = ($a-mean - $b-mean)/($a-variance/@A + $b-variance/@B).sqrt; my$DoF = ($a-variance / @A +$b-variance / @B)² /
(($a-variance² / (@A³ - @A²)) + ($b-variance² / (@B³ - @B²)));

my $sa =$DoF / 2 - 1;
my $x =$DoF / (Welchs-𝒕-statistic² + $DoF); my$N = 65355;
my $h =$x / $N; my ($sum1, $sum2 ); for ^$N »×» $h ->$i {
$sum1 += (($i + $h / 2) **$sa) / (1 - ($i +$h / 2)).sqrt;
$sum2 +=$i           ** $sa / (1 -$i).sqrt;
}

(($h / 6) × ($x ** $sa / (1 -$x).sqrt + 4 × $sum1 + 2 ×$sum2)) /
( Γ($sa + 1) × π.sqrt / Γ($sa + 1.5) );
}

# Testing
for (
[<27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4>],
[<27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4>],

[<17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8>],
[<21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8>],

[<19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0>],
[<28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2>],

[<30.02 29.99 30.11 29.97 30.01 29.99>],
[<29.89 29.93 29.72 29.98 30.02 29.98>],

[<3.0 4.0 1.0 2.1>],
[<490.2 340.0 433.9>]
) -> @left, @right { say p-value @left, @right }

Output:
0.0213780014628669
0.148841696605328
0.0359722710297969
0.0907733242856673
0.010751534033393


### Using Burkhardt's 'incomplete beta'

Works with: Rakudo version 2019.11
Translation of: Perl

This uses the Soper reduction formula to evaluate the integral, which converges much more quickly than Simpson's formula.

sub lgamma ( Num(Real) \n --> Num ){
use NativeCall;
sub lgamma (num64 --> num64) is native {}
lgamma( n )
}

sub p-value (@a, @b) {
return 1 if @a.elems | @b.elems ≤ 1;
my $mean1 = @a.sum / @a.elems; my$mean2 = @b.sum / @b.elems;
return 1 if $mean1 ==$mean2;

my $variance1 = sum (@a «-»$mean1) X**2;
my $variance2 = sum (@b «-»$mean2) X**2;
return 1 if $variance1 |$variance2 == 0;

$variance1 /= @a.elems - 1;$variance2 /= @b.elems - 1;
my $Welchs-𝒕-statistic = ($mean1-$mean2)/sqrt($variance1/@a.elems+$variance2/@b.elems); my$DoF = ($variance1/@a.elems +$variance2/@b.elems)² /
(($variance1 ×$variance1)/(@a.elems × @a.elems × (@a.elems-1)) +
($variance2 ×$variance2)/(@b.elems × @b.elems × (@b.elems-1))
);
my $A =$DoF / 2;
my $value =$DoF / ($Welchs-𝒕-statistic² +$DoF);
return $value if$A | $value ≤ 0 or$value ≥ 1;

# from here, translation of John Burkhardt's C
my $beta = lgamma($A) + 0.57236494292470009 - lgamma($A+0.5); # constant is logΓ(.5), more precise than 'lgamma' routine my$eps   = 10**-15;
my $psq =$A + 0.5;
my $cx = 1 -$value;
my ($xx,$pp,$qq,$indx);
if $A <$psq × $value { ($xx, $cx,$pp, $qq,$indx) = $cx,$value, 0.5,  $A, 1 } else { ($xx,      $pp,$qq, $indx) =$value,  $A, 0.5, 0 } my$term = my $ai =$value = 1;
my $ns = floor$qq + $cx ×$psq;

# Soper reduction formula
my $qq-ai =$qq - $ai; my$rx = $ns == 0 ??$xx !! $xx /$cx;
loop {
$term ×=$qq-ai × $rx / ($pp + $ai);$value += $term;$qq-ai  = $term.abs; if$qq-ai ≤ $eps &$eps×$value {$value = $value × ($pp × $xx.log + ($qq - 1) × $cx.log -$beta).exp / $pp;$value = 1 - $value if$indx;
last
}
$ai++;$ns--;
if $ns ≥ 0 {$qq-ai = $qq -$ai;
$rx =$xx if $ns == 0; } else {$qq-ai = $psq;$psq  += 1;
}
}
$value } my$error = 0;
0.021378001462867,
0.148841696605327,
0.0359722710297968,
0.090773324285671,
0.0107515611497845,
0.00339907162713746,
0.52726574965384,
0.545266866977794,
);

for (
[<27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4>],
[<27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4>],

[<17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8>],
[<21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8>],

[<19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0>],
[<28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2>],

[<30.02 29.99 30.11 29.97 30.01 29.99>],
[<29.89 29.93 29.72 29.98 30.02 29.98>],

[<3.0 4.0 1.0 2.1>],
[<490.2 340.0 433.9>],

[<0.010268 0.000167 0.000167>],
[<0.159258 0.136278 0.122389>],

[<1.0/15 10.0/62.0>],
[<1.0/10 2/50.0>],

[<9/23.0 21/45.0 0/38.0>],
[<0/44.0 42/94.0 0/22.0>],
) -> @left, @right {
my $p-value = p-value @left, @right; printf("p-value = %.14g\n",$p-value);
$error += abs($p-value - shift @answers);
}
println(s"\nSuccessfully completed without errors. [total ${scala.compat.Platform.currentTime - executionStart} ms]") }  ## Scilab Translation of: Stata Scilab will print a warning because the number of degrees of freedom is not an integer. However, the underlying implementation makes use of the dcdflib Fortran library, which happily accepts a noninteger df. x = [3.0,4.0,1.0,2.1]; y = [490.2,340.0,433.9]; n1 = length(x); n2 = length(y); v1 = variance(x); v2 = variance(y); t = (mean(x)-mean(y))/(sqrt(v1/n1+v2/n2)); df = (v1/n1+v2/n2)^2/(v1^2/(n1^2*(n1-1))+v2^2/(n2^2*(n2-1))); [p, q] = cdft("PQ", -abs(t), df); [t df 2*p]  Output  ans = - 9.5594977 2.0008523 0.0107516 ## Sidef Translation of: Raku func p_value (A, B) { [A.len, B.len].all { _ > 1 } || return 1 var x̄_a = Math.avg(A...) var x̄_b = Math.avg(B...) var a_var = (A.map {|n| (x̄_a - n)**2 }.sum / A.end) var b_var = (B.map {|n| (x̄_b - n)**2 }.sum / B.end) (a_var && b_var) || return 1 var Welsh_𝒕_statistic = ((x̄_a - x̄_b) / √(a_var/A.len + b_var/B.len)) var DoF = ((a_var/A.len + b_var/B.len)**2 / ((a_var**2 / (A.len**3 - A.len**2)) + (b_var**2 / (B.len**3 - B.len**2)))) var sa = (DoF/2 - 1) var x = (DoF/(Welsh_𝒕_statistic**2 + DoF)) var N = 65355 var h = x/N var (sum1=0, sum2=0) ^N -> lazy.map { _ * h }.each { |i| sum1 += (((i + h/2) ** sa) / √(1 - (i + h/2))) sum2 += (( i ** sa) / √(1 - (i ))) } (h/6 * (x**sa / √(1-x) + 4*sum1 + 2*sum2)) / (gamma(sa + 1) * √(Num.pi) / gamma(sa + 1.5)) } # Testing var tests = [ %n<27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4>, %n<27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4>, %n<17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8>, %n<21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8>, %n<19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0>, %n<28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2>, %n<30.02 29.99 30.11 29.97 30.01 29.99>, %n<29.89 29.93 29.72 29.98 30.02 29.98>, %n<3.0 4.0 1.0 2.1>, %n<490.2 340.0 433.9> ] tests.each_slice(2, {|left, right| say p_value(left, right) })  Output: 0.0213780014628670325061113281387220205111519317756 0.148841696605327985083613019511085971435711697961 0.0359722710297967180871367618538977446933248150651 0.0907733242856668878840956275523536083406692525656 0.0107515340333929755465323718028856669932912031012  ## Stata Here is a straightforward solution using the ttest command. If one does not want the output but only the p-value, prepend the command with qui and use the result r(p) as shown below. The t statistic is r(t). Notice the data are stored in a single variable, using a group variable to distinguish the two series. Notice that here we use the option unequal of the ttest command, and not welch, so that Stata uses the Welch-Satterthwaite approximation. mat a=(3,4,1,2.1,490.2,340,433.9\1,1,1,1,2,2,2)' clear svmat double a rename (a1 a2) (x group) ttest x, by(group) unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 1 | 4 2.525 .6394985 1.278997 .4898304 4.56017 2 | 3 421.3667 43.80952 75.88032 232.8695 609.8638 ---------+-------------------------------------------------------------------- combined | 7 182.0286 86.22435 228.1282 -28.95482 393.012 ---------+-------------------------------------------------------------------- diff | -418.8417 43.81419 -607.282 -230.4014 ------------------------------------------------------------------------------ diff = mean(1) - mean(2) t = -9.5595 Ho: diff = 0 Satterthwaite's degrees of freedom = 2.00085 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.0054 Pr(|T| > |t|) = 0.0108 Pr(T > t) = 0.9946 di r(t) -9.5594977 di r(p) .01075156  The computation can easily be implemented in Mata. Here is how to compute the t statistic (t), the approximate degrees of freedom (df) and the p-value (p). st_view(a=., ., .) x = select(a[., 1], a[., 2] :== 1) y = select(a[., 1], a[., 2] :== 2) n1 = length(x) n2 = length(y) v1 = variance(x) v2 = variance(y) t = (mean(x)-mean(y))/sqrt(v1/n1+v2/n2) df = (v1/n1+v2/n2)^2/(v1^2/(n1^2*(n1-1))+v2^2/(n2^2*(n2-1))) p = 2*t(df, -abs(t)) t,df,p 1 2 3 +----------------------------------------------+ 1 | -9.559497722 2.000852349 .0107515611 | +----------------------------------------------+  ## Tcl Translation of: Racket Works with: Tcl version 8.6 Library: Tcllib (Package: math::statistics) Library: Tcllib (Package: math::special) This is not particularly idiomatic Tcl, but perhaps illustrates some of the language's relationship with the Lisp family. #!/usr/bin/tclsh package require math::statistics package require math::special namespace path {::math::statistics ::math::special ::tcl::mathfunc ::tcl::mathop} proc incf {_var {inc 1.0}} { upvar 1$_var var
if {![info exists var]} {
set var 0.0
}
set var [expr {$inc +$var}]
}

proc sumfor {_var A B body} {
upvar 1 $_var var set var$A
set res 0
while {$var <$B} {
incf res [uplevel 1 $body] incr var } return$res
}

proc sqr {x} {expr {$x*$x}}

proc pValue {S1 S2 {n 11000}} {
set σ²1 [var $S1] set σ²2 [var$S2]
set N1  [llength $S1] set N2 [llength$S2]
set σ²/sz1 [/ ${σ²1}$N1]
set σ²/sz2 [/ ${σ²2}$N2]

set d1 [/ [sqr ${σ²1}] [* [sqr$N1] [- $N1 1]]] set d2 [/ [sqr${σ²2}] [* [sqr $N2] [-$N2 1]]]
set DoF [/ [sqr [+ ${σ²/sz1}${σ²/sz2}]] [+ $d1$d2]]

set a [/ $DoF 2.0] set welchTstat [/ [- [mean$S1] [mean $S2]] [sqrt [+${σ²/sz1} ${σ²/sz2}]]] set x [/$DoF [+ [sqr $welchTstat]$DoF]]
set h [/ $x$n]

/ [* [/ $h 6] \ [+ [* [**$x [- $a 1]] \ [** [- 1$x] -0.5]] \
[* 4 [sumfor i 0 $n { * [** [+ [*$h $i] [/$h 2]] [- $a 1]] \ [** [- 1 [*$h $i] [/$h 2]] -0.5]}]] \
[* 2 [sumfor i 0 $n { * [** [*$h $i] [-$a 1]] [** [- 1 [* $h$i]] -0.5]}]]]] \
[* [Gamma $a] 1.77245385090551610 [/ 1.0 [Gamma [+$a 0.5]]]]
}

foreach {left right} {
{ 27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4 }
{ 27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4 }

{ 17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8 }
{ 21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8 }

{ 19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0 }
{ 28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2 }

{ 30.02 29.99 30.11 29.97 30.01 29.99 }
{ 29.89 29.93 29.72 29.98 30.02 29.98 }

{ 3.0 4.0 1.0 2.1 }
{ 490.2 340.0 433.9 }
} {
puts [pValue $left$right]
}

Output:
0.021378001462853034
0.148841696604164
0.035972271029770915
0.09077332428458083
0.010751399918798182


## Wren

Translation of: Go
Library: Wren-math
Library: Wren-fmt
import "/math" for Math, Nums
import "/fmt" for Fmt

var welch = Fn.new { |a, b|
return (Nums.mean(a) - Nums.mean(b)) /
(Nums.variance(a)/a.count + Nums.variance(b)/b.count).sqrt
}

var dof = Fn.new { |a, b|
var sva = Nums.variance(a)
var svb = Nums.variance(b)
var la = a.count
var lb = b.count
var n = sva/la + svb/lb
return n * n / (sva*sva/(la*la*(la-1)) + svb*svb/(lb*lb*(lb-1)))
}

var simpson0 = Fn.new { |nf, upper, f|
var dx0 = upper/nf
var sum = (f.call(0) + f.call(dx0*0.5)*4) * dx0
var x0 = dx0
for (i in 1...nf) {
var x1 = (i + 1) * upper / nf
var xmid = (x0 + x1) * 0.5
var dx = x1 - x0
sum = sum + (f.call(x0)*2 + f.call(xmid)*4) * dx
x0 = x1
}
return (sum + f.call(upper)*dx0) / 6
}

var pValue = Fn.new { |a, b|
var nu = dof.call(a, b)
var t = welch.call(a, b)
var g1 = Math.gamma(nu/2).log
var g2 = Math.gamma(0.5).log
var g3 = Math.gamma(nu/2 + 0.5).log
var f = Fn.new { |r| r.pow(nu/2-1) / (1 - r).sqrt }
return simpson0.call(2000, nu/(t*t + nu), f) / Math.exp(g1 + g2 - g3)
}

var d1 = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4]
var d2 = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4]
var d3 = [17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8]
var d4 = [21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6,
21.5, 22.5, 23.5, 21.5, 21.8]
var d5 = [19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0]
var d6 = [28.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9,
21.6, 24.3, 20.4, 24.0, 13.2]
var d7 = [30.02, 29.99, 30.11, 29.97, 30.01, 29.99]
var d8 = [29.89, 29.93, 29.72, 29.98, 30.02, 29.98]
var x  = [3.0, 4.0, 1.0, 2.1]
var y  = [490.2, 340.0, 433.9]
Fmt.print("$0.6f", pValue.call(d1, d2)) Fmt.print("$0.6f", pValue.call(d3, d4))
Fmt.print("$0.6f", pValue.call(d5, d6)) Fmt.print("$0.6f", pValue.call(d7, d8))
Fmt.print("\$0.6f", pValue.call(x, y))

Output:
0.021378
0.148842
0.035972
0.090773
0.010751


## zkl

Translation of: C
fcn calculate_Pvalue(array1,array2){
if (array1.len()<=1 or array2.len()<=1) return(1.0);

mean1,mean2 := array1.sum(0.0),array2.sum(0.0);
if(mean1==mean2) return(1.0);
mean1/=array1.len();
mean2/=array2.len();

variance1:=array1.reduce('wrap(sum,x){ sum + (x-mean1).pow(2) },0.0);
variance2:=array2.reduce('wrap(sum,x){ sum + (x-mean2).pow(2) },0.0);

variance1/=(array1.len() - 1);
variance2/=(array2.len() - 1);

WELCH_T_STATISTIC:=(mean1-mean2)/
(variance1/array1.len() + variance2/array2.len()).sqrt();
DEGREES_OF_FREEDOM:=
( variance1/array1.len() + variance2/array2.len() ).pow(2) // numerator
/ (
(variance1*variance1)/(array1.len().pow(2)*(array1.len() - 1)) +
(variance2*variance2)/(array2.len().pow(2)*(array2.len() - 1))
);
a:=DEGREES_OF_FREEDOM/2;
x:=DEGREES_OF_FREEDOM/( WELCH_T_STATISTIC.pow(2) + DEGREES_OF_FREEDOM );
N,h := 65535, x/N;

sum1,sum2 := 0.0, 0.0;
foreach i in (N){
sum1+=((h*i + h/2.0).pow(a - 1))/(1.0 - (h*i + h/2.0)).sqrt();
sum2+=((h*i).pow(a - 1))/(1.0 - h*i).sqrt();
}
return_value:=((h/6.0)*( x.pow(a - 1)/(1.0 - x).sqrt() +
4.0*sum1 + 2.0*sum2) ) /
((0.0).e.pow(lngammal(a) + 0.57236494292470009 - lngammal(a + 0.5)));

if(return_value > 1.0) return(1.0);	// or return_value is infinite, throws
return_value;
}
fcn lngammal(xx){
var [const] cof=List(	// static
76.18009172947146,    -86.50532032941677,
24.01409824083091,    -1.231739572450155,
0.1208650973866179e-2,-0.5395239384953e-5
);

y:=x:=xx;
tmp:=x + 5.5 - (x + 0.5) * (x + 5.5).log();
ser:=1.000000000190015;
foreach x in (cof){ ser+=(x/(y+=1)); }
return((2.5066282746310005 * ser / x).log() - tmp);
}
testSets:=T(
T(T(27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4),
T(27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4)),
T(T(17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8),
T(21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8)),
T(T(19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0),
T(28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2)),
T(T(30.02,29.99,30.11,29.97,30.01,29.99),
T(29.89,29.93,29.72,29.98,30.02,29.98)),
T(T(3.0,4.0,1.0,2.1),T(490.2,340.0,433.9)) );

foreach x,y in (testSets)
{ println("Test set 1 p-value = %f".fmt(calculate_Pvalue(x,y))); }
Output:
Test set 1 p-value = 0.021378
Test set 1 p-value = 0.148842
Test set 1 p-value = 0.035972
Test set 1 p-value = 0.090773
Test set 1 p-value = 0.010752
`