# Welch's t-test

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 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.

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 on 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:

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

Keeping in mind that

${\displaystyle \mathrm {B} (x;a,b)=\int _{0}^{x}r^{a-1}\,(1-r)^{b-1}\,\mathrm {d} r.\!}$

and

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

${\displaystyle p_{2-tail}}$ 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.

## 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:     This code is distributed under the GNU LGPL license.    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}; 	const double CORRECT_ANSWERS[] = {0.021378001462867,0.148841696605327,0.0359722710297968,0.090773324285671,0.0107515611497845,0.00339907162713746,0.52726574965384,0.545266866977794}; //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));	error += fabs(pvalue - CORRECT_ANSWERS[1]);	printf("Test sets 2 p-value = %g\n",pvalue); 	pvalue = Pvalue(d5,sizeof(d5)/sizeof(*d5),d6,sizeof(d6)/sizeof(*d6));	error += fabs(pvalue - CORRECT_ANSWERS[2]);	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);	error += fabs(pvalue - CORRECT_ANSWERS[3]); 	pvalue = Pvalue(x,sizeof(x)/sizeof(*x),y,sizeof(y)/sizeof(*y));	error += fabs(pvalue - CORRECT_ANSWERS[4]);	printf("Test sets 5 p-value = %g\n", pvalue); 	pvalue = Pvalue(v1,sizeof(v1)/sizeof(*v1),v2,sizeof(v2)/sizeof(*v2));	error += fabs(pvalue - CORRECT_ANSWERS[5]);	printf("Test sets 6 p-value = %g\n", pvalue); 	pvalue = Pvalue(s1,sizeof(s1)/sizeof(*s1),s2,sizeof(s2)/sizeof(*s2));	error += fabs(pvalue - CORRECT_ANSWERS[6]);	printf("Test sets 7 p-value = %g\n", pvalue); 	pvalue = Pvalue(z1, 3, z2, 3);	error += fabs(pvalue - CORRECT_ANSWERS[7]);	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, pend program

Output

  -9.55949772193266  2.00085234885628  1.075156114978449E-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=: ^[email protected][email protected]<:(^[email protected][email protected](1 | ]) + +/@:^[email protected](1 + 1&| + [email protected]<.)@<:)@.(1&<:)"0mean=: +/ % #nu=: # - 1:sampvar=: +/@((- mean) ^ 2:) % nussem=: sampvar % #welch_T=: -&mean % 2 %: +&ssemnu=: nu f. : ((+&ssem ^ 2:) % +&((ssem^2:)%nu))B=: ^@(+&lngamma - [email protected]+) p2_tail=:dyad define  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 21.4d2=: 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 24.4d3=: 17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8d4=: 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 21.8d5=: 19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22d6=: 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 13.2d7=: 30.02 29.99 30.11 29.97 30.01 29.99d8=: 29.89 29.93 29.72 29.98 30.02 29.98d9=: 3 4 1 2.1da=: 490.2 340 433.9

   d1 p2_tail d20.021378   d3 p2_tail d40.148842   d5 p2_tail d60.0359723   d7 p2_tail d80.0907733   d9 p2_tail da0.0107377

## 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

## 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


## 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 Burkardt's betain

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 warnings;use strict; 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;} use List::Util 'sum';sub calculate_Pvalue {	my $array1 = shift; my$array2 = shift;	if (scalar @$array1 <= 1) { return 1.0; } if (scalar @$array2 <= 1) {	return 1.0;	}	my $mean1 = sum(@{$array1 });	$mean1 /= scalar @$array1;	my $mean2 = sum(@{$array2 });	$mean2 /= scalar @$array2;	if ($mean1 ==$mean2) {	return 1.0;	}	my $variance1 = 0.0; my$variance2 = 0.0;	foreach my $x (@$array1) {	$variance1 += ($x-$mean1)*($x-$mean1); } foreach my$x (@$array2) {$variance2 += ($x-$mean2)*($x-$mean2);	}	if (($variance1 == 0.0) && ($variance2 == 0.0)) {	return 1.0;	}	$variance1 =$variance1/(scalar @$array1-1);$variance2 = $variance2/(scalar @$array2-1);	my $array1_size = scalar @$array1;	my $array2_size = scalar @$array2;	my $WELCH_T_STATISTIC = ($mean1-$mean2)/sqrt($variance1/$array1_size+$variance2/$array2_size); my$DEGREES_OF_FREEDOM = (($variance1/$array1_size+$variance2/(scalar @$array2))**2)	/	(	($variance1*$variance1)/($array1_size*$array1_size*($array1_size-1))+ ($variance2*$variance2)/($array2_size*$array2_size*($array2_size-1))	);	my $A =$DEGREES_OF_FREEDOM/2;	my $value =$DEGREES_OF_FREEDOM/($WELCH_T_STATISTIC*$WELCH_T_STATISTIC+$DEGREES_OF_FREEDOM);#from here, translation of John Burkhardt's C my$beta = lgamma($A)+0.57236494292470009-lgamma($A+0.5);	my $acu = 10**(-15); my($ai,$cx,$indx,$ns,$pp,$psq,$qq,$rx,$temp,$term,$xx);# Check the input arguments.	return $value if$A <= 0.0;# || $q <= 0.0; return$value if $value < 0.0 || 1.0 <$value;# Special cases	return $value if$value == 0.0 || $value == 1.0;$psq = $A + 0.5;$cx = 1.0 - $value; if ($A < $psq *$value) {		($xx,$cx, $pp,$qq, $indx) = ($cx, $value, 0.5,$A, 1);	} else {		($xx,$pp, $qq,$indx) = ($value,$A, 0.5, 0);	}	$term = 1.0;$ai = 1.0;	$value = 1.0;$ns = int($qq +$cx * $psq);#Soper reduction formula.$rx = $xx /$cx;	$temp =$qq - $ai;$rx = $xx if$ns == 0;	while (1) {		$term =$term * $temp *$rx / ( $pp +$ai );		$value =$value + $term;$temp = abs ($term); if ($temp <= $acu &&$temp <= $acu *$value) {	   	$value =$value * exp ($pp * log($xx)	                          + ($qq - 1.0) * log($cx) - $beta) /$pp;	   	$value = 1.0 -$value if $indx; last; }$ai = $ai + 1.0;$ns = $ns - 1; if (0 <=$ns) {			$temp =$qq - $ai;$rx = $xx if$ns == 0;		} else {			$temp =$psq;			$psq =$psq + 1.0;		}	}	return $value;}my @d1 = (27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4);my @d2 = (27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4);my @d3 = (17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8);my @d4 = (21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8);my @d5 = (19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0);my @d6 = (28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2);my @d7 = (30.02,29.99,30.11,29.97,30.01,29.99);my @d8 = (29.89,29.93,29.72,29.98,30.02,29.98);my @x = (3.0,4.0,1.0,2.1);my @y = (490.2,340.0,433.9);my @s1 = (1.0/15,10.0/62.0);my @s2 = (1.0/10,2/50.0);my @v1 = (0.010268,0.000167,0.000167);my @v2 = (0.159258,0.136278,0.122389);my @z1 = (9/23.0,21/45.0,0/38.0);my @z2 = (0/44.0,42/94.0,0/22.0); my @CORRECT_ANSWERS = (0.021378001462867,0.148841696605327,0.0359722710297968,0.090773324285671,0.0107515611497845,0.00339907162713746,0.52726574965384,0.545266866977794); my$pvalue = calculate_Pvalue(\@d1,\@d2);my $error = abs($pvalue - $CORRECT_ANSWERS[0]);printf("Test sets 1 p-value = %.14g\n",$pvalue); $pvalue = calculate_Pvalue(\@d3,\@d4);$error += abs($pvalue -$CORRECT_ANSWERS[1]);printf("Test sets 2 p-value = %.14g\n",$pvalue);$pvalue = calculate_Pvalue(\@d5,\@d6);$error += abs($pvalue - $CORRECT_ANSWERS[2]);printf("Test sets 3 p-value = %.14g\n",$pvalue); $pvalue = calculate_Pvalue(\@d7,\@d8);$error += abs($pvalue -$CORRECT_ANSWERS[3]);printf("Test sets 4 p-value = %.14g\n",$pvalue);$pvalue = calculate_Pvalue(\@x,\@y);$error += abs($pvalue - $CORRECT_ANSWERS[4]);$pvalue = calculate_Pvalue(\@v1,\@v2);$error += abs($pvalue - $CORRECT_ANSWERS[5]);printf("Test sets 6 p-value = %.14g\n",$pvalue); $pvalue = calculate_Pvalue(\@s1,\@s2);$error += abs($pvalue -$CORRECT_ANSWERS[6]);printf("Test sets 7 p-value = %.14g\n",$pvalue);$pvalue = calculate_Pvalue(\@z1,\@z2);$error += abs($pvalue - $CORRECT_ANSWERS[7]);printf("Test sets z p-value = %.14g\n",$pvalue); printf("the cumulative error is %g\n", $error); Output: Test sets 1 p-value = 0.021378001462867 Test sets 2 p-value = 0.14884169660533 Test sets 3 p-value = 0.035972271029797 Test sets 4 p-value = 0.090773324285667 Test sets 5 p-value = 0.010751561149784 Test sets 6 p-value = 0.0033990716271375 Test sets 7 p-value = 0.52726574965384 Test sets z p-value = 0.54526686697779 the cumulative error is 4.87106e-15 ### Using Math::AnyNum like Sidef This is us a simpler solution, and 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


## Perl 6

### Integration using Simpson's Rule

Works with: Rakudo version 2017.08
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 \Welsh-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 = $DoF / (Welsh-𝒕-statistic² +$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($sa + 1.5) );} # Testingfor ( [<27.5 21.0 19.0 23.6 17.0 17.9 16.9 20.1 21.9 22.6 23.1 19.6 19.0 21.7 21.4>], [<27.1 22.0 20.8 23.4 23.4 23.5 25.8 22.0 24.8 20.2 21.9 22.1 22.9 20.5 24.4>], [<17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8>], [<21.5 22.8 21.0 23.0 21.6 23.6 22.5 20.7 23.4 21.8 20.7 21.7 21.5 22.5 23.6 21.5 22.5 23.5 21.5 21.8>], [<19.8 20.4 19.6 17.8 18.5 18.9 18.3 18.9 19.5 22.0>], [<28.2 26.6 20.1 23.3 25.2 22.1 17.7 27.6 20.6 13.7 23.2 17.5 20.6 18.0 23.9 21.6 24.3 20.4 24.0 13.2>], [<30.02 29.99 30.11 29.97 30.01 29.99>], [<29.89 29.93 29.72 29.98 30.02 29.98>], [<3.0 4.0 1.0 2.1>], [<490.2 340.0 433.9>]) -> @left, @right { say p-value @left, @right } Output: 0.0213780014628669 0.148841696605328 0.0359722710297969 0.0907733242856673 0.010751534033393  ### Using Burkardt's betain Works with: Rakudo version 2018.10 Translation of: Perl This uses the Soper reduction formula to evaluate the integral, which converges much more quickly than Simpson's formula.  sub lgamma ( Num(Real) \n --> Num ){ use NativeCall; sub lgamma (num64 --> num64) is native {} lgamma( n )} sub pvalue (@a, @b) { if @a.elems <= 1 { return 1.0; } if @b.elems <= 1 { return 1.0; } my Rat$mean1 = @a.sum / @a.elems;  my Rat $mean2 = @b.sum / @b.elems; if$mean1 == $mean2 { return 1.0; } my Rat$variance1 = 0.0;  my Rat $variance2 = 0.0; for @a ->$i {    $variance1 += ($mean1 - $i)**2#";" unnecessary for last statement in block } for @b ->$i {    $variance2 += ($mean2 - $i)**2 } if ($variance1 == 0 && $variance2 == 0) { return 1.0; }$variance1 /= (@a.elems - 1);  $variance2 /= (@b.elems - 1); my$WELCH_T_STATISTIC = ($mean1-$mean2)/sqrt($variance1/@a.elems+$variance2/@b.elems);	my $DEGREES_OF_FREEDOM = (($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 = $DEGREES_OF_FREEDOM/2; my$value = $DEGREES_OF_FREEDOM/($WELCH_T_STATISTIC*$WELCH_T_STATISTIC+$DEGREES_OF_FREEDOM);  my $beta = lgamma($A)+0.57236494292470009-lgamma($A+0.5); my Rat$acu = 10**(-15);	my ($ai,$cx,$indx,$ns,$pp,$psq,$qq,$rx,$temp,$term,$xx);# Check the input arguments. return$value if $A <= 0.0;# ||$q <= 0.0;	return $value if$value < 0.0 || 1.0 < $value;# Special cases return$value if $value == 0.0 ||$value == 1.0;	$psq =$A + 0.5;	$cx = 1.0 -$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 = 1.0;	$ai = 1.0;$value = 1.0;	$ns =$qq + $cx *$psq;        $ns =$ns.Int;#Soper reduction formula.	$rx =$xx / $cx;$temp = $qq -$ai;	$rx =$xx if $ns == 0; while (True) {$term = $term *$temp * $rx / ($pp + $ai );$value = $value +$term;		$temp =$term.abs;		if $temp <=$acu && $temp <=$acu * $value {$value = $value * ($pp * $xx.log + ($qq - 1.0) * $cx.log -$beta).exp / $pp;$value = 1.0 - $value if$indx;	   	last;		}	 	$ai =$ai + 1.0;		$ns--; if 0 <=$ns {			$temp =$qq - $ai;$rx = $xx if$ns == 0;		} else {			$temp =$psq;			$psq =$psq + 1.0;		}	}	return $value;}my @d1 = 27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4;my @d2 = 27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4;my @d3 = 17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8;my @d4 = 21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6,21.5,22.5,23.5,21.5,21.8;my @d5 = 19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0;my @d6 = 28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9,21.6,24.3,20.4,24.0,13.2;my @d7 = 30.02,29.99,30.11,29.97,30.01,29.99;my @d8 = 29.89,29.93,29.72,29.98,30.02,29.98;my @x = 3.0,4.0,1.0,2.1;my @y = 490.2,340.0,433.9;my @s1 = 1.0/15,10.0/62.0;my @s2 = 1.0/10,2/50.0;my @v1 = 0.010268,0.000167,0.000167;my @v2 = 0.159258,0.136278,0.122389;my @z1 = 9/23.0,21/45.0,0/38.0;my @z2 = 0/44.0,42/94.0,0/22.0; my @CORRECT_ANSWERS = (0.021378001462867,0.148841696605327,0.0359722710297968,0.090773324285671,0.0107515611497845,0.00339907162713746,0.52726574965384,0.545266866977794); my$pvalue = pvalue(@d1, @d2);my $error = abs($pvalue - @CORRECT_ANSWERS[0]);printf("Test sets 1 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@d3, @d4);$error += abs($pvalue - @CORRECT_ANSWERS[1]);printf("Test sets 2 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@d5, @d6);$error += abs($pvalue - @CORRECT_ANSWERS[2]);printf("Test sets 3 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@d7, @d8);$error += abs($pvalue - @CORRECT_ANSWERS[3]);printf("Test sets 4 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@x, @y);$error += abs($pvalue - @CORRECT_ANSWERS[4]);printf("Test sets 5 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@v1, @v2);$error += abs($pvalue - @CORRECT_ANSWERS[5]);printf("Test sets 6 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@s1, @s2);$error += abs($pvalue - @CORRECT_ANSWERS[6]);printf("Test sets 7 p-value = %.14g\n",$pvalue);$pvalue = pvalue(@z1, @z2);$error += abs($pvalue - @CORRECT_ANSWERS[7]);printf("Test sets 8 p-value = %.14g\n",$pvalue); printf("the cumulative error is %g\n",$error);  
Output:
Test sets 1 p-value = 0.021378001462867
Test sets 2 p-value = 0.14884169660533
Test sets 3 p-value = 0.035972271029797
Test sets 4 p-value = 0.090773324285667
Test sets 5 p-value = 0.010751561149784
Test sets 6 p-value = 0.0033990716271375
Test sets 7 p-value = 0.52726574965384
Test sets 8 p-value = 0.54526686697779
the cumulative error is 5.50254e-15


## Phix

Translation of: Go
Translation of: Kotlin
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) / 6end 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) * aend 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:
0.02137800146
0.1488416966
0.03597227103
0.09077332429
0.01075067374

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)

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 whileend 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 = 0for 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)
{"cumulative error",4.915115776e-15}    -- (64-bit)


## Python

### Using NumPy & SciPy

import numpy as npimport scipy as spimport 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#13023329d1 <- 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)

## SAS

Translation of: Stata
data tbl;input value group @@;cards;3 1 4 1 1 1 2.1 1 490.2 2 340 2 433.9 2;run; proc ttest data=tbl;class group;var value;run;

Output

group Method N Mean Std Dev Std Err Minimum Maximum
1   4 2.5250 1.2790 0.6395 1.0000 4.0000
2   3 421.4 75.8803 43.8095 340.0 490.2
Diff (1-2) Pooled   -418.8 48.0012 36.6615
Diff (1-2) Satterthwaite   -418.8   43.8142

group Method Mean 95% CL Mean Std Dev 95% CL Std Dev
1   2.5250 0.4898 4.5602 1.2790 0.7245 4.7688
2   421.4 232.9 609.9 75.8803 39.5077 476.9
Diff (1-2) Pooled -418.8 -513.1 -324.6 48.0012 29.9627 117.7
Diff (1-2) Satterthwaite -418.8 -607.3 -230.4

Method Variances DF t Value Pr > |t|
Pooled Equal 5 -11.42 <.0001
Satterthwaite Unequal 2.0009 -9.56 0.0108

Equality of Variances
Method Num DF Den DF F Value Pr > F
Folded F 2 3 3519.81 <.0001

Implementation in IML:

proc iml;use tbl;read all var {value} into x where(group=1);read all var {value} into y where(group=2);close tbl;n1 = nrow(x);n2 = nrow(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 = 2*probt(-abs(t), df);print t df p;quit;

Output

-9.559498 2.0008523 0.0107516

## Scala

import org.apache.commons.math3.distribution.TDistribution object WelchTTest extends App {   val res = welchTtest(Array(3.0, 4.0, 1.0, 2.1), Array(490.2, 340.0, 433.9))   def welchTtest(x: Array[Double], y: Array[Double]) = {     def square[T](x: T)(implicit num: Numeric[T]): T = {      import num._      x * x    }     def count[A](a: Seq[A])(implicit num: Fractional[A]): A =      a.foldLeft(num.zero) { case (cnt, _) => num.plus(cnt, num.one) }     def mean[A](a: Seq[A])(implicit num: Fractional[A]): A = num.div(a.sum, count(a))     def variance[A](a: Seq[A])(implicit num: Fractional[A]) =      num.div(a.map(xs => square(num.minus(xs, mean(a)))).sum, num.minus(count(a), num.one))     val (nx, ny) = (x.length, y.length)    val (vx, vy) = (variance(x), variance(y))    val qt = vx / nx + vy / ny    val t = (mean(x) - mean(y)) / math.sqrt(qt)    val df = square(qt) / (square(vx) / (square(nx) * (nx - 1)) + square(vy) / (square(ny) * (ny - 1)))    val p = 2.0 * new TDistribution(df).cumulativeProbability(-math.abs(t))    (t, df, p)  }   println(s"t  = ${res._1}\ndf =${res._2}\np  = ${res._3}") println(s"\nSuccessfully completed without errors. [total${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: Perl 6
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))} # Testingvar 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)'clearsvmat double arename (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::statisticspackage require math::specialnamespace 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


## 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
`