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.
Task Description
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:
where
is the mean of set ,
and
is the number of observations in set ,
and
is the square root of the unbiased sample variance of set , i.e.
and the degrees of freedom, can be approximated:
The two-tailed p-value, , can be computed as a cumulative distribution function
where I is the regularized incomplete beta function. This is the same as:
Keeping in mind that
and
can be calculated in terms of gamma functions and integrals more simply:
which simplifies to
The definite integral can be approximated with Simpson's Rule but other methods are also acceptable.
The , 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
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).
<lang C>#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; } </lang>
- 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:
<lang C>#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);
}
</lang>
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.
<lang fortran>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</lang>
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%.
<lang fortran>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</lang>
Output
-9.55949772193266 2.00085234885628 1.075156114978449E-002
Go
<lang 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)
}</lang>
- Output:
0.021378 0.148842 0.035972 0.090773 0.010751
J
Implementation:
<lang J>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@+)
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
)</lang>
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.)
Data for task examples: <lang J>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</lang>
Task examples: <lang J> 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</lang>
Java
Using the Apache Commons Mathematics Library. <lang java>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]);
}
}</lang>
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
<lang julia>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</lang>
- 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: <lang scala>// 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)))
}</lang>
- Output:
0.021378 0.148842 0.035972 0.090773 0.010751
Octave
<lang octave>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</lang>
PARI/GP
<lang parigp>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])</lang>
- 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.
<lang perl>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";
}</lang>
- Output:
0.0213780014628667 0.148841696605327 0.0359722710297968 0.0907733242856612 0.0107515340333929
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.
<lang perl> 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 @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 $pvalue = calculate_Pvalue($left,$right);
$error += abs($pvalue - shift @answers);
printf("p-value = %.14g\n",$pvalue);
} printf("cumulative error is %g\n", $error);</lang>
- 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
Perl 6
Integration using Simpson's Rule
Perhaps "inspired by C example" may be more accurate. Gamma subroutine from Gamma function task.
<lang perl6>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) );
}
- 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 }</lang>
- Output:
0.0213780014628669 0.148841696605328 0.0359722710297969 0.0907733242856673 0.010751534033393
Using Burkardt's betain
This uses the Soper reduction formula to evaluate the integral, which converges much more quickly than Simpson's formula.
<lang perl6>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 $error = 0; my @answers = ( 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 $pvalue = p-value @left, @right;
printf("p-value = %.14g\n",$pvalue);
$error += abs($pvalue - shift @answers);
} printf("cumulative error is %g\n", $error);</lang>
- Output:
p-value = 0.021378001462867 p-value = 0.14884169660533 p-value = 0.035972271029797 p-value = 0.090773324285667 p-value = 0.010751561149784 p-value = 0.0033990716271375 p-value = 0.52726574965384 p-value = 0.54526686697779 cumulative error is 5.30131e-15
Phix
<lang Phix>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</lang>
- Output:
0.02137800146 0.1488416966 0.03597227103 0.09077332429 0.01075067374
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)
<lang Phix>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}</lang>
- 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
<lang python>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)</lang>
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.
<lang python>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</lang>
The Python code is then straightforward:
<lang python>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</lang>
Example
<lang python>a1 = [3, 4, 1, 2.1] a2 = [490.2, 340, 433.9] print(welch_ttest(a1, a2))</lang>
Output
(-9.559497721932658, 2.0008523488562844, 0.01075156114978449)
R
<lang 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); </lang>
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
<lang racket>#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))))</lang>
- Output:
(0.021378001462867013 0.14884169660532798 0.035972271029796624 0.09077332428567102 0.01075139991904718)
Ruby
<lang Ruby>def calculate_p_value(array1, array2)
return 1.0 if array1.size <= 1
return 1.0 if array2.size <= 1
mean1 = array1.sum / array1.size
mean2 = array2.sum / array2.size
return 1.0 if mean1 == mean2
variance1 = 0.0
variance2 = 0.0
array1.each do |x|
variance1 += (mean1 - x)**2
end
array2.each do |x|
variance2 += (mean2 - x)**2
end
return 1.0 if variance1 == 0.0 && variance2 == 0.0
variance1 /= (array1.size - 1)
variance2 /= (array2.size - 1)
welch_t_statistic = (mean1 - mean2) / Math.sqrt(variance1 / array1.size + variance2 / array2.size)
degrees_of_freedom = ((variance1 / array1.size + variance2 / array2.size)**2) / (
(variance1 * variance1) / (array1.size * array1.size * (array1.size - 1)) +
(variance2 * variance2) / (array2.size * array2.size * (array2.size - 1)))
a = degrees_of_freedom / 2
value = degrees_of_freedom / (welch_t_statistic**2 + degrees_of_freedom)
beta = Math.lgamma(a)[0] + 0.57236494292470009 - Math.lgamma(a + 0.5)[0]
acu = 10**-15
return value if a <= 0
return value if value < 0.0 || value > 1.0
return value if (value == 0) || (value == 1.0)
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
end
term = 1.0
ai = 1.0
value = 1.0
ns = (qq + cx * psq).to_i
# Soper reduction formula
rx = xx / cx
temp = qq - ai
loop do
term = term * temp * rx / (pp + ai)
value += term
temp = term.abs
if temp <= acu && temp <= acu * value
value = value * Math.exp(pp * Math.log(xx) + (qq - 1.0) * Math.log(cx) - beta) / pp
value = 1.0 - value
value = 1.0 - value if indx == 0
break
end
ai += 1.0
ns -= 1
if ns >= 0
temp = qq - ai
rx = xx if ns == 0
else
temp = psq
psq += 1.0
end
end
value
end
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] s1 = [1.0 / 15, 10.0 / 62.0] s2 = [1.0 / 10, 2 / 50.0] v1 = [0.010268, 0.000167, 0.000167] v2 = [0.159258, 0.136278, 0.122389] z1 = [9 / 23.0, 21 / 45.0, 0 / 38.0] z2 = [0 / 44.0, 42 / 94.0, 0 / 22.0]
CORRECT_ANSWERS = [0.021378001462867, 0.148841696605327, 0.0359722710297968,
0.090773324285671, 0.0107515611497845, 0.00339907162713746, 0.52726574965384, 0.545266866977794].freeze
pvalue = calculate_p_value(d1, d2) error = (pvalue - CORRECT_ANSWERS[0]).abs printf("Test sets 1 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(d3, d4) error += (pvalue - CORRECT_ANSWERS[1]).abs printf("Test sets 2 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(d5, d6) error += (pvalue - CORRECT_ANSWERS[2]).abs printf("Test sets 3 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(d7, d8) error += (pvalue - CORRECT_ANSWERS[3]).abs printf("Test sets 4 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(x, y) error += (pvalue - CORRECT_ANSWERS[4]).abs printf("Test sets 5 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(v1, v2) error += (pvalue - CORRECT_ANSWERS[5]).abs printf("Test sets 6 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(s1, s2) error += (pvalue - CORRECT_ANSWERS[6]).abs printf("Test sets 7 p-value = %.14g\n", pvalue)
pvalue = calculate_p_value(z1, z2) error += (pvalue - CORRECT_ANSWERS[7]).abs printf("Test sets z p-value = %.14g\n", pvalue)
printf("the cumulative error is %g\n", error) </lang>
- 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 = 0.090773324285671 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 1.34961e-15
SAS
<lang>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;</lang>
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:
<lang sas>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;</lang>
Output
-9.559498 2.0008523 0.0107516
Scala
<lang 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]")
}</lang>
Scilab
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.
<lang>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]</lang>
Output
ans = - 9.5594977 2.0008523 0.0107516
Sidef
<lang ruby>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)
})</lang>
- 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.
<lang stata>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</lang>
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).
<lang stata>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 | +----------------------------------------------+</lang>
Tcl
This is not particularly idiomatic Tcl, but perhaps illustrates some of the language's relationship with the Lisp family.
<lang Tcl>#!/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]
} </lang>
- Output:
0.021378001462853034 0.148841696604164 0.035972271029770915 0.09077332428458083 0.010751399918798182
zkl
<lang zkl>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);
}</lang> <lang zkl>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))); }</lang>
- 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