Statistics/Chi-squared distribution
Statistics/Chi-squared distribution 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.
The probability density function (pdf) of the chi-squared (or χ2) distribution as used in statistics is
- , where
Here, denotes the Gamma_function.
The use of the gamma function in the equation below reflects the chi-squared distribution's origin as a special case of the gamma distribution.
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution.
The probability density function (pdf) of the gamma distribution is given by the formula
where Γ(k) is the Gamma_function, with shape parameter k and a scale parameter θ.
The cumulative probability distribution of the gamma distribution is the area under the curve of the distribution, which indicates the increasing probability of the x value of a single random point within the gamma distribution being less than or equal to the x value of the cumulative probability distribution. The gamma cumulative probability distribution function can be calculated as
where is the lower incomplete gamma function.
The lower incomplete gamma function can be calculated as
and so, for the chi-squared cumulative probability distribution and substituting chi-square k into s as k/2 and chi-squared x into x as x / 2,
Because series formulas may be subject to accumulated errors from rounding in the frequently used region where x and k are under 10 and near one another, you may instead use a mathematics function library, if available for your programming task, to calculate gamma and incomplete gamma.
- Task
- Calculate and show the values of the χ2(x; k) for k = 1 through 5 inclusive and x integer from 0 and through 10 inclusive.
- Create a function to calculate the cumulative probability function for the χ2 distribution. This will need to be reasonably accurate (at least 6 digit accuracy) for k = 3.
- Calculate and show the p values of statistical samples which result in a χ2(k = 3) value of 1, 2, 4, 8, 16, and 32. (Statistical p values can be calculated for the purpose of this task as approximately 1 - P(x), with P(x) the cumulative probability function at x for χ2.)
- The following is a chart for 4 airports:
| Airport | On Time | Delayed | Totals |
|---|---|---|---|
| Dallas/Fort Worth | 77 | 23 | 100 |
| Honolulu | 88 | 12 | 100 |
| LaGuardia | 79 | 21 | 100 |
| Orlando | 81 | 19 | 100 |
| All Totals | 325 | 75 | 400 |
| Expected per 100 | 81.25 | 18.75 | 100 |
- χ2 on a 2D table is calculated as the sum of the squares of the differences from expected divided by the expected numbers for each entry on the table. The k for the chi-squared distribution is to be calculated as df, the degrees of freedom for the table, which is a 2D parameter, (count of airports - 1) * (count of measures per airport - 1), which here is (4 - 1 )(2 - 1) = 3.
- Calculate the Chi-squared statistic for the above table and find its p value using your function for the cumulative probability for χ2 with k = 3 from the previous task.
- Stretch task
- Show how you could make a plot of the curves for the probability distribution function χ2(x; k) for k = 0, 1, 2, and 3.
- Related Tasks
- See also
[Chi Squared Test] [NIST page on the Chi-Square Distribution]
C++
#include <iostream>
#include <cmath>
#include <numbers>
#include <iomanip>
#include <array>
// The normalised lower incomplete gamma function.
double gamma_cdf(const double aX, const double aK) {
double result = 0.0;
for ( uint32_t m = 0; m <= 99; ++m ) {
result += pow(aX, m) / tgamma(aK + m + 1);
}
result *= pow(aX, aK) * exp(-aX);
return result;
}
// The cumulative probability function of the Chi-squared distribution.
double cdf(const double aX, const double aK) {
if ( aK > 1'000 && aK < 100 ) {
return 1.0;
}
return ( aX > 0.0 && aK > 0.0 ) ? gamma_cdf(aX / 2, aK / 2) : 0.0;
}
// The probability density function of the Chi-squared distribution.
double pdf(const double aX, const double aK) {
return ( aX > 0.0 ) ? pow(aX, aK / 2 - 1) * exp(-aX / 2) / ( pow(2, aK / 2) * tgamma(aK / 2) ) : 0.0;
}
int main() {
std::cout << " Values of the Chi-squared probability distribution function" << std::endl;
std::cout << " x/k 1 2 3 4 5" << std::endl;
for ( uint32_t x = 0; x <= 10; x++ ) {
std::cout << std::setw(2) << x;
for ( uint32_t k = 1; k <= 5; ++k ) {
std::cout << std::setw(10) << std::fixed << pdf(x, k);
}
std::cout << std::endl;
}
std::cout << "\n Values for the Chi-squared distribution with 3 degrees of freedom" << std::endl;
std::cout << "Chi-squared cumulative pdf p-value" << std::endl;
for ( uint32_t x : { 1, 2, 4, 8, 16, 32 } ) {
const double cumulative_pdf = cdf(x, 3);
std::cout << std::setw(6) << x << std::setw(19) << std::fixed << cumulative_pdf
<< std::setw(14) << ( 1.0 - cumulative_pdf ) << std::endl;
}
const std::array<const std::array<int32_t, 2>, 4> observed =
{ { { 77, 23 }, { 88, 12 }, { 79, 21 }, { 81, 19 } } };
const std::array<const std::array<double, 2>, 4> expected =
{ { { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 } } };
double testStatistic = 0.0;
for ( uint64_t i = 0; i < observed.size(); ++i ) {
for ( uint64_t j = 0; j < observed[0].size(); ++j ) {
testStatistic += pow(observed[i][j] - expected[i][j], 2.0) / expected[i][j];
}
}
const uint64_t degreesFreedom = ( observed.size() - 1 ) / ( observed[0].size() - 1 );
std::cout << "\nFor the airport data:" << std::endl;
std::cout << " test statistic : " << std::fixed << testStatistic << std::endl;
std::cout << " degrees of freedom : " << degreesFreedom << std::endl;
std::cout << " Chi-squared : " << std::fixed << pdf(testStatistic, degreesFreedom) << std::endl;
std::cout << " p-value : " << std::fixed << cdf(testStatistic, degreesFreedom) << std::endl;
}
- Output:
Values of the Chi-squared probability distribution function
x/k 1 2 3 4 5
0 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.241971 0.303265 0.241971 0.151633 0.080657
2 0.103777 0.183940 0.207554 0.183940 0.138369
3 0.051393 0.111565 0.154180 0.167348 0.154180
4 0.026995 0.067668 0.107982 0.135335 0.143976
5 0.014645 0.041042 0.073225 0.102606 0.122042
6 0.008109 0.024894 0.048652 0.074681 0.097304
7 0.004553 0.015099 0.031873 0.052845 0.074371
8 0.002583 0.009158 0.020667 0.036631 0.055112
9 0.001477 0.005554 0.013296 0.024995 0.039887
10 0.000850 0.003369 0.008500 0.016845 0.028335
Values for the Chi-squared distribution with 3 degrees of freedom
Chi-squared cumulative pdf p-value
1 0.198748 0.801252
2 0.427593 0.572407
4 0.738536 0.261464
8 0.953988 0.046012
16 0.998866 0.001134
32 0.999999 0.000001
For the airport data:
test statistic : 4.512821
degrees of freedom : 3
Chi-squared : 0.088754
p-value : 0.788850
FreeBASIC
#define pi 4 * Atn(1)
Function gamma (x As Double) As Double
Dim As Integer k
Dim As Double p(12)
Dim As Double accm = p(1)
If accm = 0 Then
accm = Sqr(2 * pi)
p(1) = accm
Dim As Double k1factrl = 1
For k = 2 To 12
p(k) = Exp(13 - k) * (13 - k)^(k - 1.5) / k1factrl
k1factrl *= -(k - 1)
Next k
End If
For k = 2 To 12
accm += p(k) / (x + k - 1)
Next
accm *= Exp(-(x + 12)) * (x + 12)^(x + .5)
Return accm / x
End Function
Function pdf(x As Double, k As Double) As Double
'probability density function
Return Iif(x <= 0, 0, x^(k / 2 - 1) * Exp(-x / 2) / (2^(k / 2) * gamma(k / 2)))
End Function
Function cpdf(x As Double, k As Double) As Double
'cumulative probability distribution function
Dim As Double t = Exp(-x / 2) * (x / 2)^(k / 2)
Dim As Double s, term
Dim As Uinteger m = 0
Do
term = (x / 2)^m / gamma(k / 2 + m + 1)
s += term
m += 1
Loop Until Abs(term) < 1e-15
Return t * s
End Function
Dim As Integer x, k, i, j
Print " Values of the Chi-squared probability distribution function"
Print " x k = 1 k = 2 k = 3 k = 4 k = 5"
For x = 0 To 10
Print Using "## "; x;
For k = 1 To 5
Print Using "#.############ "; pdf(x, k);
Next
Print
Next
Print !"\n Values for Chi-squared with 3 degrees of freedom"
Print "Chi-squared cum pdf P value"
Dim As Uinteger tt(5) = {1, 2, 4, 8, 16, 32}
For x = 0 To Ubound(tt)
Dim As Double cpdff = cpdf(tt(x), 3)
Print Using " ## #.############ #.############"; tt(x); cpdff; 1-cpdff
Next
Dim As Uinteger airport(3,1) = {{77, 23}, {88, 12}, {79, 21}, {81, 19}}
Dim As Double expected(1) = {81.25, 18.75}
Dim As Double dsum = 0
For i = 0 To Ubound(airport,1)
For j = 0 To Ubound(airport,2)
dsum += (airport(i, j) - expected(j))^2 / expected(j)
Next
Next
Dim As Double dof = Ubound(airport,1) / Ubound(airport,2)
Print Using !"\nFor the airport data, diff total is #.############"; dsum
Print Spc(14); "degrees of freedom is"; dof
Print Spc(21); Using "Chi-squared is #.############"; pdf(dsum, dof)
Print Spc(25); Using "P value is #.############"; cpdf(dsum, dof)
Sleep- Output:
Values of the Chi-squared probability distribution function
x k = 1 k = 2 k = 3 k = 4 k = 5
0 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000
1 0.241970724519 0.303265329856 0.241970724519 0.151632664928 0.080656908173
2 0.103776874355 0.183939720586 0.207553748710 0.183939720586 0.138369165807
3 0.051393443268 0.111565080074 0.154180329804 0.167347620111 0.154180329804
4 0.026995483257 0.067667641618 0.107981933026 0.135335283237 0.143975910702
5 0.014644982562 0.041042499312 0.073224912810 0.102606248280 0.122041521349
6 0.008108695555 0.024893534184 0.048652173330 0.074680602552 0.097304346659
7 0.004553342922 0.015098691711 0.031873400451 0.052845420989 0.074371267720
8 0.002583373169 0.009157819444 0.020666985354 0.036631277777 0.055111960944
9 0.001477282804 0.005554498269 0.013295545236 0.024995242211 0.039886635707
10 0.000850036660 0.003368973500 0.008500366603 0.016844867498 0.028334555342
Values for Chi-squared with 3 degrees of freedom
Chi-squared cum pdf P value
1 0.198748043099 0.801251956901
2 0.427593295529 0.572406704471
4 0.738535870051 0.261464129949
8 0.953988294311 0.046011705689
16 0.998866015710 0.001133984290
32 0.999999476654 0.000000523346
For the airport data, diff total is 4.512820512821
degrees of freedom is 3
Chi-squared is 0.088753925984
P value is 0.788850426319
Java
import java.util.List;
public final class StatisticsChiSquaredDistribution {
public static void main(String[] aArgs) {
System.out.println(" Values of the Chi-squared probability distribution function");
System.out.println(" x/k 1 2 3 4 5");
for ( int x = 0; x <= 10; x++ ) {
System.out.print(String.format("%2d", x));
for ( int k = 1; k <= 5; k++ ) {
System.out.print(String.format("%10.6f", pdf(x, k)));
}
System.out.println();
}
System.out.println();
System.out.println(" Values for the Chi-squared distribution with 3 degrees of freedom");
System.out.println("Chi-squared cumulative pdf p-value");
for ( int x : List.of( 1, 2, 4, 8, 16, 32 ) ) {
final double cdf = cdf(x, 3);
System.out.println(String.format("%6d%19.6f%14.6f", x, cdf, ( 1.0 - cdf )));
}
final int[][] observed = { { 77, 23 }, { 88, 12 }, { 79, 21 }, { 81, 19 } };
final double[][] expected = { { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 } };
double testStatistic = 0.0;
for ( int i = 0; i < observed.length; i++ ) {
for ( int j = 0; j < observed[0].length; j++ ) {
testStatistic += Math.pow(observed[i][j] - expected[i][j], 2.0) / expected[i][j];
}
}
final int degreesFreedom = ( observed.length - 1 ) / ( observed[0].length - 1 );
System.out.println();
System.out.println("For the airport data:");
System.out.println(" test statistic : " + String.format("%.6f", testStatistic));
System.out.println(" degrees of freedom : " + degreesFreedom);
System.out.println(" Chi-squared : " + String.format("%.6f", pdf(testStatistic, degreesFreedom)));
System.out.println(" p-value : " + String.format("%.6f", cdf(testStatistic, degreesFreedom)));
}
// The gamma function.
private static double gamma(double aX) {
if ( aX < 0.5 ) {
return Math.PI / ( Math.sin(Math.PI * aX) * gamma(1.0 - aX) );
}
final double[] probabilities = new double[] {
0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059,
12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 };
aX -= 1.0;
double t = probabilities[0];
for ( int i = 1; i < 9; i++ ) {
t += probabilities[i] / ( aX + i );
}
final double w = aX + 7.5;
return Math.sqrt(2.0 * Math.PI) * Math.pow(w, aX + 0.5) * Math.exp(-w) * t;
}
// The probability density function of the Chi-squared distribution.
private static double pdf(double aX, double aK) {
return ( aX > 0.0 ) ?
Math.pow(aX, aK / 2 - 1) * Math.exp(-aX / 2) / ( Math.pow(2, aK / 2) * gamma(aK / 2) ) : 0.0;
}
// The cumulative probability function of the Chi-squared distribution.
private static double cdf(double aX, double aK) {
if ( aX > 1_000 && aK < 100 ) {
return 1.0;
}
return ( aX > 0.0 && aK > 0.0 ) ? gammaCDF(aX / 2, aK / 2) : 0.0;
}
// The normalised lower incomplete gamma function.
private static double gammaCDF(double aX, double aK) {
double result = 0.0;
for ( int m = 0; m <= 99; m++ ) {
result += Math.pow(aX, m) / gamma(aK + m + 1);
}
result *= Math.pow(aX, aK) * Math.exp(-aX);
return result;
}
}
- Output:
Values of the Chi-squared probability distribution function
x/k 1 2 3 4 5
0 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.241971 0.303265 0.241971 0.151633 0.080657
2 0.103777 0.183940 0.207554 0.183940 0.138369
3 0.051393 0.111565 0.154180 0.167348 0.154180
4 0.026995 0.067668 0.107982 0.135335 0.143976
5 0.014645 0.041042 0.073225 0.102606 0.122042
6 0.008109 0.024894 0.048652 0.074681 0.097304
7 0.004553 0.015099 0.031873 0.052845 0.074371
8 0.002583 0.009158 0.020667 0.036631 0.055112
9 0.001477 0.005554 0.013296 0.024995 0.039887
10 0.000850 0.003369 0.008500 0.016845 0.028335
Values for the Chi-squared distribution with 3 degrees of freedom
Chi-squared cumulative pdf p-value
1 0.198748 0.801252
2 0.427593 0.572407
4 0.738536 0.261464
8 0.953988 0.046012
16 0.998866 0.001134
32 0.999999 0.000001
For the airport data:
test statistic : 4.512821
degrees of freedom : 3
Chi-squared : 0.088754
p-value : 0.788850
jq
Also works with gojq, the Go implementation of jq.
The implementation of Chi_cdf here uses the recusive relation for the gamma function for efficiency and to simplify the avoidance of numerical overflow issues.
Formatting
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# For formatting numbers
# ... but leave non-numbers, 0, and numbers using E alone
def round($dec):
def rpad($len): tostring | ($len - length) as $l | . + ("0" * $l);
if type == "string" then .
elif . == 0 then "0"
else pow(10;$dec) as $m
| . * $m | round / $m
| tostring
| (capture("(?<n>^[^.]*[.])(?<f>[0-9]*$)")
| .n + (.f|rpad($dec)))
// if test("^[0-9]+$") then . + "." + ("" | rpad($dec)) else null end
// .
end;Chi-squared pdf and cdf
def Chi2_pdf($x; $k):
if $x <= 0 then 0
else # ((-$x/2)|exp) * pow($x; $k/2 -1) / (pow(2;$k/2) * (($k/2)|gamma))
((-$x/2) + ($k/2 -1) * ($x|log) - ( ($k/2)*(2|log) + (($k/2)|lgamma))) | exp
end;
# $k is the degrees of freedom
# Use recursive relation for gamma: G(x+1) = x * G(x)
def Chi2_cdf($x; $k):
if $x == 0 then 0
elif $x > (1e3 * $k) then 1
else 1e-15 as $tol # for example
| { s: 0, m: 0, term: (1 / ((($k/2)+1)|gamma)) }
| until (.term|length < $tol;
.s += .term
| .m += 1
| .term *= (($x/2) / (($k/2) + .m )) )
| .s * ( ((-$x/2) + ($k/2)*(($x/2)|log)) | exp)
end ;The Tasks
def tables:
def r: round(6) | lpad(10);
" Values of the χ2 probability distribution function",
" x/k 1 2 3 4 5",
(range(0;11) as $x
| "\($x|lpad(2)) \(reduce range(1; 6) as $k (""; . + (Chi2_pdf($x; $k) | r) ))" ),
"\n Values for χ2 with 3 degrees of freedom",
"χ2 cum pdf p-value",
( (1, 2, 4, 8, 16, 32) as $x
| Chi2_cdf($x; 3) as $cdf
| "\($x|lpad(2)) \($cdf|r)\(1-$cdf|r)"
);
def airport: [[77, 23], [88, 12], [79, 21], [81, 19]];
def expected: [81.25, 18.75];
def airport($airport; $expected):
def dof: ($airport|length - 1) / ($airport[0]|length - 1);
reduce range(0; $airport|length) as $i (0;
reduce range(0; $airport[0]|length) as $j (.;
. + (($airport[$i][$j] - $expected[$j]) | .*.) / $expected[$j] ) )
| ("\nFor airport data table:",
" diff sum : \(.)",
" d.o.f. : \(dof)",
" χ2 value : \(Chi2_pdf(.; dof))",
" p-value : \(Chi2_cdf(.; dof)|round(4))" ) ;
tables,
airport(airport; expected)- Output:
Values of the χ2 probability distribution function
x/k 1 2 3 4 5
0 0 0 0 0 0
1 0.241971 0.303265 0.241971 0.151633 0.080657
2 0.103777 0.183940 0.207554 0.183940 0.138369
3 0.051393 0.111565 0.154180 0.167348 0.154180
4 0.026995 0.067668 0.107982 0.135335 0.143976
5 0.014645 0.041042 0.073225 0.102606 0.122042
6 0.008109 0.024894 0.048652 0.074681 0.097304
7 0.004553 0.015099 0.031873 0.052845 0.074371
8 0.002583 0.009158 0.020667 0.036631 0.055112
9 0.001477 0.005554 0.013296 0.024995 0.039887
10 0.000850 0.003369 0.008500 0.016845 0.028335
Values for χ2 with 3 degrees of freedom
χ2 cum pdf p-value
1 0.198748 0.801252
2 0.427593 0.572407
4 0.738536 0.261464
8 0.953988 0.046012
16 0.998866 0.001134
32 0.999999 1e-06
For airport data table:
diff sum : 4.512820512820513
d.o.f. : 3
χ2 value : 0.088753925984435
p-value : 0.7889
Julia
""" Rosetta Code task rosettacode.org/wiki/Statistics/Chi-squared_distribution """
""" gamma function to 12 decimal places """
function gamma(x)
p = [ 0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 ]
if x < 0.5
return π / (sinpi(x) * gamma(1.0 - x))
else
x -= 1.0
t = p[1]
for i in 1:8
t += p[i+1] / (x + i)
end
end
w = x + 7.5
return sqrt(2.0 * π) * w^(x+0.5) * exp(-w) * t
end
""" Chi-squared function, the probability distribution function (pdf) for chi-squared """
function χ2(x, k)
return x > 0 ? x^(k/2 - 1) * exp(-x/2) / (2^(k/2) * gamma(k / 2)) : 0
end
""" lower incomplete gamma by series formula with gamma """
function gamma_cdf(k, x)
return x^k * exp(-x) * sum(x^m / gamma(k + m + 1) for m in 0:100)
end
""" Cumulative probability function (cdf) for chi-squared """
function cdf_χ2(x, k)
return x <= 0 || k <= 0 ? 0.0 : gamma_cdf(k / 2, x / 2)
end
println("x χ2 k = 1 k = 2 k = 3 k = 4 k = 5")
println("-"^93)
for x in 0:10
print(lpad(x, 2))
for k in 1:5
s = string(χ2(x, k))
print(lpad(s[1:min(end, 13)], 18), k % 5 == 0 ? "\n" : "")
end
end
println("\nχ2 x cdf for χ2 P value (df=3)\n", "-"^36)
for p in [1, 2, 4, 8, 16, 32]
cdf = round(cdf_χ2(p, 3), digits=10)
println(lpad(p, 2), " ", cdf, " ", round(1.0 - cdf, digits=10))
end
airportdata = [ 77 23 ;
88 12;
79 21;
81 19 ]
expected_data = [ 81.25 18.75 ;
81.25 18.75 ;
81.25 18.75 ;
81.25 18.75 ; ]
dtotal = sum((airportdata[i] - expected_data[i])^2/ expected_data[i] for i in 1:length(airportdata))
println("\nFor the airport data, diff total is $dtotal, χ2 is ", χ2(dtotal, 3), ", p value ", cdf_χ2(dtotal, 3))
using Plots
x = 0.0:0.01:10
y = [map(p -> χ2(p, k), x) for k in 0:3]
plot(x, y, yaxis=[-0.1, 0.5], labels=[0 1 2 3])
- Output:
Graph of Chi-Squared for k values 0 through 3
x χ2 k = 1 k = 2 k = 3 k = 4 k = 5 --------------------------------------------------------------------------------------------- 0 0 0 0 0 0 1 0.24197072451 0.30326532985 0.24197072451 0.15163266492 0.08065690817 2 0.10377687435 0.18393972058 0.20755374871 0.18393972058 0.13836916580 3 0.05139344326 0.11156508007 0.15418032980 0.16734762011 0.15418032980 4 0.02699548325 0.06766764161 0.10798193302 0.13533528323 0.14397591070 5 0.01464498256 0.04104249931 0.07322491280 0.10260624827 0.12204152134 6 0.00810869555 0.02489353418 0.04865217332 0.07468060255 0.09730434665 7 0.00455334292 0.01509869171 0.03187340045 0.05284542098 0.07437126772 8 0.00258337316 0.00915781944 0.02066698535 0.03663127777 0.05511196094 9 0.00147728280 0.00555449826 0.01329554523 0.02499524221 0.03988663570 10 0.00085003666 0.00336897349 0.00850036660 0.01684486749 0.02833455534 χ2 x cdf for χ2 P value (df=3) ------------------------------------ 1 0.1987480431 0.8012519569 2 0.4275932955 0.5724067045 4 0.7385358701 0.2614641299 8 0.9539882943 0.0460117057 16 0.9988660157 0.0011339843 32 0.9999994767 5.233e-7 For the airport data, diff total is 4.512820512820512, χ2 is 0.08875392598443503, p value 0.7888504263193064
Nim
Adapted from Python solution.
import std/[math, strformat, sugar]
func χ²(x: float; k: int): float =
## χ² function, the probability distribution function (pdf) for χ².
if x > 0: x.pow(k / 2 - 1) * exp(-x/2) / (2.pow(k / 2) * gamma(k / 2)) else: 0
func gammaCdf(k, x: float): float =
## Lower incomplete gamma by series formula with gamma.
for m in 0..99:
result += x^m / gamma(k + m.toFloat + 1)
result *= pow(x, k) * exp(-x)
func cdfχ²(x: float; k: int): float =
## Cumulative probability function (cdf) for χ².
if x > 0 and k > 0: gammaCdf(k / 2, x / 2) else: 0
echo " Values of the χ2 probability distribution function"
echo " x/k 1 2 3 4 5"
for x in 0..10:
stdout.write &"{x:2} "
for k in 1..5:
stdout.write &"{χ²(x.toFloat, k):10.6f}"
echo()
echo()
echo " Values for χ2 with 3 degrees of freedom"
echo "χ² cum pdf p-value"
for x in [1, 2, 4, 8, 16, 32]:
let cdf = cdfχ²(x.toFloat, 3)
echo &"{x:2} {cdf:10.6f} {1 - cdf:10.6f}"
const
AirportData = [[float 77, 23], [float 88, 12], [float 79, 21], [float 81, 19]]
Expected = [[81.25, 18.75], [81.25, 18.75], [81.25, 18.75], [81.25, 18.75]]
var dtotal = 0.0
for pos in [0, 1]:
for i, d in AirportData:
dtotal += (d[pos] - Expected[i][pos])^2 / Expected[i][pos]
echo()
echo &"For the airport data, diff total is {dtotal:.6f}, " &
&"χ² is {χ²(dtotal, 3):.6f}, p value is {cdfχ²(dtotal, 3):.6f}."
### Stretch task ###
import gnuplot
let x = collect(for n in 0..9999: n / 1000)
withGnuplot:
cmd "set multiplot"
cmd "set yrange [-0.02:0.5]"
for k in 0..3:
let y = collect(for p in x: χ²(p, k))
plot(x, y, &"k = {k}", "with lines")
- Output:
Values of the χ2 probability distribution function
x/k 1 2 3 4 5
0 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.241971 0.303265 0.241971 0.151633 0.080657
2 0.103777 0.183940 0.207554 0.183940 0.138369
3 0.051393 0.111565 0.154180 0.167348 0.154180
4 0.026995 0.067668 0.107982 0.135335 0.143976
5 0.014645 0.041042 0.073225 0.102606 0.122042
6 0.008109 0.024894 0.048652 0.074681 0.097304
7 0.004553 0.015099 0.031873 0.052845 0.074371
8 0.002583 0.009158 0.020667 0.036631 0.055112
9 0.001477 0.005554 0.013296 0.024995 0.039887
10 0.000850 0.003369 0.008500 0.016845 0.028335
Values for χ2 with 3 degrees of freedom
χ² cum pdf p-value
1 0.198748 0.801252
2 0.427593 0.572407
4 0.738536 0.261464
8 0.953988 0.046012
16 0.998866 0.001134
32 0.999999 0.000001
For the airport data, diff total is 4.512821, χ² is 0.088754, p value is 0.788850.
Perl
use v5.36;
use Math::MPFR;
use List::Util 'sum';
sub Gamma ($z) {
my $g = Math::MPFR->new();
Math::MPFR::Rmpfr_gamma($g, Math::MPFR->new($z), 0);
$g;
}
sub chi2($x,$k) { $x>0 && $k>0 ? ($x**($k/2 - 1) * exp(-$x/2)/(2**($k/2)*Gamma($k / 2))) : 0 }
sub gamma_cdf($k,$x) { $x**$k * exp(-$x) * sum map { $x** $_ / Gamma($k+$_+1) } 0..100 }
sub cdf_chi2($x,$k) { ($x <= 0 or $k <= 0) ? 0.0 : gamma_cdf($k / 2, $x / 2) }
print 'x χ² ';
print "k = $_" . ' 'x13 for 1..5;
say "\n" . '-' x (my $width = 93);
for my $x (0..10) {
printf '%2d', $x;
printf ' %.' . (int(($width-2)/5)-4) . 'f', chi2($x, $_) for 1..5;
say '';
}
say "\nχ² x cdf for χ² P value (df=3)\n" . '-' x 36;
for my $p (map { 2**$_ } 0..5) {
my $cdf = cdf_chi2($p, 3);
printf "%2d %-.10f %-.10f\n", $p, $cdf, 1-$cdf;
}
my @airport = <77 23 88 12 79 21 81 19>;
my @expected = split ' ', '81.25 18.75 ' x 4;
my $dtotal;
$dtotal += ($airport[$_] - $expected[$_])**2 / $expected[$_] for 0..$#airport;
printf "\nFor the airport data, diff total is %.5f, χ² is %.5f, p value %.5f\n", $dtotal, chi2($dtotal, 3), cdf_chi2($dtotal, 3);
- Output:
x χ² k = 1 k = 2 k = 3 k = 4 k = 5 --------------------------------------------------------------------------------------------- 0 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 1 0.24197072451914 0.30326532985632 0.24197072451914 0.15163266492816 0.08065690817305 2 0.10377687435515 0.18393972058572 0.20755374871030 0.18393972058572 0.13836916580686 3 0.05139344326792 0.11156508007421 0.15418032980377 0.16734762011132 0.15418032980377 4 0.02699548325659 0.06766764161831 0.10798193302638 0.13533528323661 0.14397591070183 5 0.01464498256193 0.04104249931195 0.07322491280963 0.10260624827987 0.12204152134939 6 0.00810869555494 0.02489353418393 0.04865217332964 0.07468060255180 0.09730434665928 7 0.00455334292164 0.01509869171116 0.03187340045148 0.05284542098906 0.07437126772012 8 0.00258337316926 0.00915781944437 0.02066698535409 0.03663127777747 0.05511196094425 9 0.00147728280398 0.00555449826912 0.01329554523581 0.02499524221105 0.03988663570744 10 0.00085003666025 0.00336897349954 0.00850036660252 0.01684486749771 0.02833455534173 χ² x cdf for χ² P value (df=3) ------------------------------------ 1 0.1987480431 0.8012519569 2 0.4275932955 0.5724067045 4 0.7385358701 0.2614641299 8 0.9539882943 0.0460117057 16 0.9988660157 0.0011339843 32 0.9999994767 0.0000005233 For the airport data, diff total is 4.51282, χ² is 0.08875, p value 0.78885
Phix
You can run this online here.
-- -- demo\rosetta\Chi-squared_distribution.exw -- with javascript_semantics function gamma(atom z) constant p = { 0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 } if z<0.5 then return PI / (sin(PI*z)*gamma(1-z)) end if z -= 1; atom x := p[1]; for i=1 to length(p)-1 do x += p[i+1]/(z+i) end for atom t = z + length(p) - 1.5; return sqrt(2*PI) * power(t,z+0.5) * exp(-t) * x end function function chi_squared(atom x, k) -- Chi-squared function, the probability distribution function (pdf) for chi-squared return iff(x > 0 ? power(x,k/2-1) * exp(-x/2) / (power(2,k/2) * gamma(k / 2)) : 0) end function function gamma_cdf(atom k, x) -- lower incomplete gamma by series formula with gamma atom tot = 0 for m=0 to 100 do tot += power(x,m) / gamma(k + m + 1) end for return power(x,k) * exp(-x) * tot end function function cdf_chi_squared(atom x, k) -- Cumulative probability function (cdf) for chi-squared return iff(x<=0 or k<=0 ? 0.0 : gamma_cdf(k/2, x/2)) end function printf(1," ------------------------------------ Chi-squared ------------------------------------\n") printf(1," x k = 1 k = 2 k = 3 k = 4 k = 5\n") printf(1,repeat('-',92)&"\n") for x=0 to 10 do printf(1,"%2d",x) for k=1 to 5 do printf(1,"%18.11f%n",{chi_squared(x, k),k=5}) end for end for printf(1,"\nChi_squared x P value (df=3)\n------------------------------------\n") for p in {1, 2, 4, 8, 16, 32} do printf(1," %2d %.16g\n",{p, 1-cdf_chi_squared(p, 3)}) end for constant airportdata = { 77, 23, 88, 12, 79, 21, 81, 19 }, expected_data = { 81.25, 18.75, 81.25, 18.75, 81.25, 18.75, 81.25, 18.75 }, fmt = "\n"&""" For the airport data, diff total is %.15f, degrees of freedom is %d, ch-squared is %.15f, p value is %.15f """ integer df = length(airportdata)/2-1 atom dtotal = sum(sq_div(sq_power(sq_sub(airportdata,expected_data),2),expected_data)) printf(1,fmt,{dtotal, df, chi_squared(dtotal,df), cdf_chi_squared(dtotal, df)}) include IupGraph.e function get_data(Ihandle /*graph*/) constant colours = {CD_BLUE,CD_ORANGE,CD_GREEN,CD_RED,CD_PURPLE} sequence x = sq_div(tagset(999,0),100), xy = {{"NAMES",{"0","1","2","3","4"}}} for k=0 to 4 do xy = append(xy,{x,apply(true,chi_squared,{x,k}),colours[k+1]}) end for return xy end function IupOpen() Ihandle graph = IupGraph(get_data,`RASTERSIZE=340x180,GRID=NO`) IupSetAttributes(graph,`XMAX=10,XTICK=2,XMARGIN=10,YMAX=0.5,YTICK=0.1`) IupShow(IupDialog(graph,`TITLE="Chi-squared distribution",MINSIZE=260x200`)) if platform()!=JS then IupMainLoop() IupClose() end if
- Output:
------------------------------------ Chi-squared ------------------------------------
x k = 1 k = 2 k = 3 k = 4 k = 5
--------------------------------------------------------------------------------------------
0 0.00000000000 0.00000000000 0.00000000000 0.00000000000 0.00000000000
1 0.24197072452 0.30326532986 0.24197072452 0.15163266493 0.08065690817
2 0.10377687436 0.18393972059 0.20755374871 0.18393972059 0.13836916581
3 0.05139344327 0.11156508007 0.15418032980 0.16734762011 0.15418032980
4 0.02699548326 0.06766764162 0.10798193303 0.13533528324 0.14397591070
5 0.01464498256 0.04104249931 0.07322491281 0.10260624828 0.12204152135
6 0.00810869555 0.02489353418 0.04865217333 0.07468060255 0.09730434666
7 0.00455334292 0.01509869171 0.03187340045 0.05284542099 0.07437126772
8 0.00258337317 0.00915781944 0.02066698535 0.03663127778 0.05511196094
9 0.00147728280 0.00555449827 0.01329554524 0.02499524221 0.03988663571
10 0.00085003666 0.00336897350 0.00850036660 0.01684486750 0.02833455534
Chi_squared x P value (df=3)
------------------------------------
1 0.8012519569012007
2 0.5724067044708797
4 0.2614641299491101
8 0.0460117056892316
16 0.0011339842897863
32 5.233466446874501e-7
For the airport data, diff total is 4.512820512820513,
degrees of freedom is 3,
ch-squared is 0.088753925984435,
p value is 0.788850426319307
Python
''' rosettacode.org/wiki/Statistics/Chi-squared_distribution#Python '''
from math import exp, pi, sin, sqrt
from matplotlib.pyplot import plot, legend, ylim
def gamma(x):
''' gamma function, accurate to about 12 decimal places '''
p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]
if x < 0.5:
return pi / (sin(pi * x) * gamma(1.0 - x))
x -= 1.0
t = p[0]
for i in range(1, 9):
t += p[i] / (x + i)
w = x + 7.5
return sqrt(2.0 * pi) * w**(x+0.5) * exp(-w) * t
def χ2(x, k):
''' Chi-squared function, the probability distribution function (pdf) for chi-squared '''
return x**(k/2 - 1) * exp(-x/2) / (2**(k/2) * gamma(k / 2)) if x > 0 and k > 0 else 0.0
def gamma_cdf(k, x):
''' lower incomplete gamma by series formula with gamma '''
return x**k * exp(-x) * sum(x**m / gamma(k + m + 1) for m in range(100))
def cdf_χ2(x, k):
''' Cumulative probability function (cdf) for chi-squared '''
return gamma_cdf(k / 2, x / 2) if x > 0 and k > 0 else 0.0
print('x χ2 k = 1 k = 2 k = 3 k = 4 k = 5')
print('-' * 93)
for x in range(11):
print(f'{x:2}', end='')
for k in range(1, 6):
print(f'{χ2(x, k):16.8}', end='\n' if k % 5 == 0 else '')
print('\nχ2 x P value (df=3)\n----------------------')
for p in [1, 2, 4, 8, 16, 32]:
print(f'{p:2}', ' ', 1.0 - cdf_χ2(p, 3))
AIRPORT_DATA = [[77, 23], [88, 12], [79, 21], [81, 19]]
EXPECTED = [[81.25, 18.75],
[81.25, 18.75],
[81.25, 18.75],
[81.25, 18.75]]
DTOTAL = sum((d[pos] - EXPECTED[i][pos])**2 / EXPECTED[i][pos]
for i, d in enumerate(AIRPORT_DATA) for pos in [0, 1])
print(
f'\nFor the airport data, diff total is {DTOTAL}, χ2 is {χ2(DTOTAL, 3)}, p value {cdf_χ2(DTOTAL, 3)}')
X = [x * 0.001 for x in range(10000)]
for k in range(5):
plot(X, [χ2(p, k) for p in X])
legend([0, 1, 2, 3, 4])
ylim(-0.02, 0.5)
- Output:
x χ2 k = 1 k = 2 k = 3 k = 4 k = 5 --------------------------------------------------------------------------------------------- 0 0.0 0.0 0.0 0.0 0.0 1 0.24197072 0.30326533 0.24197072 0.15163266 0.080656908 2 0.10377687 0.18393972 0.20755375 0.18393972 0.13836917 3 0.051393443 0.11156508 0.15418033 0.16734762 0.15418033 4 0.026995483 0.067667642 0.10798193 0.13533528 0.14397591 5 0.014644983 0.041042499 0.073224913 0.10260625 0.12204152 6 0.0081086956 0.024893534 0.048652173 0.074680603 0.097304347 7 0.0045533429 0.015098692 0.0318734 0.052845421 0.074371268 8 0.0025833732 0.0091578194 0.020666985 0.036631278 0.055111961 9 0.0014772828 0.0055544983 0.013295545 0.024995242 0.039886636 10 0.00085003666 0.0033689735 0.0085003666 0.016844867 0.028334555 χ2 x P value (df=3) ---------------------- 1 0.8012519569012009 2 0.5724067044708798 4 0.26146412994911117 8 0.04601170568923141 16 0.0011339842897852837 32 5.233466447984725e-07 For the airport data, diff total is 4.512820512820513, χ2 is 0.088753925984435, p value 0.7888504263193064
Raku
# 20221101 Raku programming solution
use Graphics::PLplot;
sub Γ(\z) { # https://rosettacode.org/wiki/Gamma_function#Raku
constant g = 9;
z < .5 ?? π/ sin(π * z) / Γ(1 - z)
!! sqrt(2*π) * (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 χ2(\x,\k) {x>0 && k>0 ?? (x**(k/2 - 1)*exp(-x/2)/(2**(k/2)*Γ(k / 2))) !! 0}
sub Γ_cdf(\k,\x) { x**k * exp(-x) * sum( ^101 .map: { x** $_ / Γ(k+$_+1) } ) }
sub cdf_χ2(\x,\k) { (x <= 0 or k <= 0) ?? 0.0 !! Γ_cdf(k / 2, x / 2) }
say ' 𝒙 χ² ', [~] (1..5)».&{ "𝒌 = $_" ~ ' ' x 13 };
say '-' x my \width = 93;
for 0..10 -> \x {
say x.fmt('%2d'), [~] (1…5)».&{χ2(x, $_).fmt: " %-.{((width-2) div 5)-4}f"}
}
say "\nχ² 𝒙 cdf for χ² P value (df=3)\n", '-' x 36;
for 2 «**« ^6 -> \p {
my $cdf = cdf_χ2(p, 3).fmt: '%-.10f';
say p.fmt('%2d'), " $cdf ", (1-$cdf).fmt: '%-.10f'
}
my \airport = [ <77 23>, <88 12>, <79 21>, <81 19> ];
my \expected = [ <81.25 18.75> xx 4 ];
my \dtotal = ( (airport »-« expected)»² »/» expected )».List.flat.sum;
say "\nFor the airport data, diff total is ",dtotal,", χ² is ", χ2(dtotal, 3), ", p value ", cdf_χ2(dtotal, 3);
given Graphics::PLplot.new( device => 'png', file-name => 'output.png' ) {
.begin;
.pen-width: 3 ;
.environment: x-range => [-1.0, 10.0], y-range => [-0.1, 0.5], just => 0 ;
.label: x-axis => '', y-axis => '', title => 'Chi-squared distribution' ;
for 0..3 -> \𝒌 {
.color-index0: 1+2*𝒌;
.line: (0, .1 … 10).map: -> \𝒙 { ( 𝒙, χ2( 𝒙, 𝒌 ) )».Num };
.text-viewport: side=>'t', disp=>-𝒌-2, pos=>.5, just=>.5, text=>'k = '~𝒌
} # plplot.sourceforge.net/docbook-manual/plplot-html-5.15.0/plmtex.html
.end
}
- Output:
𝒙 χ² 𝒌 = 1 𝒌 = 2 𝒌 = 3 𝒌 = 4 𝒌 = 5 --------------------------------------------------------------------------------------------- 0 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 1 0.24197072451914 0.30326532985632 0.24197072451914 0.15163266492816 0.08065690817305 2 0.10377687435515 0.18393972058572 0.20755374871030 0.18393972058572 0.13836916580686 3 0.05139344326792 0.11156508007421 0.15418032980377 0.16734762011132 0.15418032980377 4 0.02699548325659 0.06766764161831 0.10798193302638 0.13533528323661 0.14397591070183 5 0.01464498256193 0.04104249931195 0.07322491280963 0.10260624827987 0.12204152134939 6 0.00810869555494 0.02489353418393 0.04865217332964 0.07468060255180 0.09730434665928 7 0.00455334292164 0.01509869171116 0.03187340045148 0.05284542098906 0.07437126772012 8 0.00258337316926 0.00915781944437 0.02066698535409 0.03663127777747 0.05511196094425 9 0.00147728280398 0.00555449826912 0.01329554523581 0.02499524221105 0.03988663570744 10 0.00085003666025 0.00336897349954 0.00850036660252 0.01684486749771 0.02833455534173 χ² 𝒙 cdf for χ² P value (df=3) ------------------------------------ 1 0.1987480431 0.8012519569 2 0.4275932955 0.5724067045 4 0.7385358701 0.2614641299 8 0.9539882943 0.0460117057 16 0.9988660157 0.0011339843 32 0.9999994767 0.0000005233 For the airport data, diff total is 4.512821, χ² is 0.08875392598443506, p value 0.7888504263193072
Wren
import "dome" for Window
import "graphics" for Canvas, Color
import "./math2" for Math
import "./iterate" for Stepped
import "./fmt" for Fmt
import "./plot" for Axes
class Chi2 {
static pdf(x, k) {
if (x <= 0) return 0
return (-x/2).exp * x.pow(k/2-1) / (2.pow(k/2) * Math.gamma(k/2))
}
static cpdf(x, k) {
var t = (-x/2).exp * (x/2).pow(k/2)
var s = 0
var m = 0
var tol = 1e-15 // say
while (true) {
var term = (x/2).pow(m) / Math.gamma(k/2 + m + 1)
s = s + term
if (term.abs < tol) break
m = m + 1
}
return t * s
}
}
System.print(" Values of the χ2 probability distribution function")
System.print(" x/k 1 2 3 4 5")
for (x in 0..10) {
Fmt.write("$2d ", x)
for (k in 1..5) {
Fmt.write("$f ", Chi2.pdf(x, k))
}
System.print()
}
System.print("\n Values for χ2 with 3 degrees of freedom")
System.print("χ2 cum pdf p-value")
for (x in [1, 2, 4, 8, 16, 32]) {
var cpdf = Chi2.cpdf(x, 3)
Fmt.print("$2d $f $f", x, cpdf, 1-cpdf)
}
var airport = [[77, 23], [88, 12], [79, 21], [81, 19]]
var expected = [81.25, 18.75]
var dsum = 0
for (i in 0...airport.count) {
for (j in 0...airport[0].count) {
dsum = dsum + (airport[i][j] - expected[j]).pow(2) / expected[j]
}
}
var dof = (airport.count - 1) / (airport[0].count - 1)
System.print("\nFor airport data table: ")
Fmt.print(" diff sum : $f", dsum)
Fmt.print(" d.o.f. : $d", dof)
Fmt.print(" χ2 value : $f", Chi2.pdf(dsum, dof))
Fmt.print(" p-value : $f", Chi2.cpdf(dsum, dof))
// generate points for plot
var Pts = List.filled(5, null)
for (k in 0..4) {
Pts[k] = []
var x = 0
while (x < 10) {
Pts[k].add([x, 10 * Chi2.pdf(x, k)])
x = x + 0.01
}
}
class Main {
construct new() {
Window.title = "Chi-squared distribution for k in [0, 4]"
Canvas.resize(1000, 600)
Window.resize(1000, 600)
Canvas.cls(Color.white)
var axes = Axes.new(100, 500, 800, 400, -0.5..10, -0.5..5)
axes.draw(Color.black, 2)
var xMarks = 0..10
var yMarks = 0..5
axes.mark(xMarks, yMarks, Color.black, 2)
var xMarks2 = Stepped.new(0..10, 2)
var yMarks2 = 0..5
axes.label(xMarks2, yMarks2, Color.black, 2, Color.black, 1, 10)
var colors = [Color.blue, Color.yellow, Color.green, Color.red, Color.purple]
for (k in 0..4) {
axes.lineGraph(Pts[k], colors[k], 2)
}
axes.rect(8, 5, 120, 110, Color.black)
for (k in 0..4) {
var y = 4.75 - k * 0.25
axes.line(8.2, y, 9, y, colors[k], 2)
y = 385 - k * 18
axes.print(750, y, k.toString, Color.black)
}
}
init() {}
update() {}
draw(alpha) {}
}
var Game = Main.new()
- Output:
Terminal output:
Values of the χ2 probability distribution function
x/k 1 2 3 4 5
0 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.241971 0.303265 0.241971 0.151633 0.080657
2 0.103777 0.183940 0.207554 0.183940 0.138369
3 0.051393 0.111565 0.154180 0.167348 0.154180
4 0.026995 0.067668 0.107982 0.135335 0.143976
5 0.014645 0.041042 0.073225 0.102606 0.122042
6 0.008109 0.024894 0.048652 0.074681 0.097304
7 0.004553 0.015099 0.031873 0.052845 0.074371
8 0.002583 0.009158 0.020667 0.036631 0.055112
9 0.001477 0.005554 0.013296 0.024995 0.039887
10 0.000850 0.003369 0.008500 0.016845 0.028335
Values for χ2 with 3 degrees of freedom
χ2 cum pdf p-value
1 0.198748 0.801252
2 0.427593 0.572407
4 0.738536 0.261464
8 0.953988 0.046012
16 0.998866 0.001134
32 0.999999 0.000001
For airport data table:
diff sum : 4.512821
d.o.f. : 3
χ2 value : 0.088754
p-value : 0.788850
