/* chi2.c
cc -o chi2 chi2.c -lm
cl -O2 chi2.c
This program calculates the chi-square significance values for given
degrees of freedom and the tail probability (type I error rate) for
given observed chi-square statistic and degree of freedom.
Ziheng Yang, October 1993.
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
double QuantileChi2 (double prob, double v);
#define QuantileGamma(prob,alpha,beta) QuantileChi2(prob,2.0*(alpha))/(2.0*(beta))
#define CDFGamma(x,alpha,beta) IncompleteGamma((beta)*(x),alpha,LnGammaFunction(alpha))
#define CDFChi2(x,v) CDFGamma(x,(v)/2.0,0.5)
double QuantileNormal (double prob);
double CDFNormal (double x);
double LnGammaFunction (double alpha);
double IncompleteGamma (double x, double alpha, double ln_gamma_alpha);
int main(int argc, char*argv[])
{
int i,j, n=20, ndf=200, nprob=8, option=0;
double df, chi2, d=1.0/n, prob[]={.005, .025, .1, .5, .90, .95, .99, .999};
if (argc!=2 && argc!=3) {
printf ("\n\nChi-square critical values\n");
for (i=0; i<ndf; i++) {
if (i%15==0) {
printf ("\n\t\t\t\tSignificance level\n");
printf ("\n DF ");
for (j=0; j<nprob; j++) printf ("%9.4f", 1-prob[j]);
printf ("\n");
}
printf ("\n%3d ", i+1);
for (j=0; j<nprob; j++)
printf ("%9.4f", QuantileChi2(prob[j],(double)(i+1)));
if (i%5==4) printf ("\n");
if (i%15==14) {
printf ("\nENTER for more, (q+ENTER) for quit... ");
if (getchar()=='q') break;
}
}
}
else if(argc==2) {
for (; ; ) {
printf ("\nd.f. & Chi^2 value (Ctrl-c to break)? ");
scanf ("%lf%lf", &df, &chi2);
if(df<1 || chi2<0) break;
prob[0] = 1-CDFChi2(chi2,df);
printf ("\ndf = %2.0f prob = %.9f = %.3e\n", df, prob[0], prob[0]);
}
}
else if(argc==3) {
df = atoi(argv[1]);
chi2 = atof(argv[2]);
if(df<1 || chi2<0) {
printf("df = %d ch2 = %.4f invalid");
exit(-1);
}
prob[0] = 1 - CDFChi2(chi2, df);
printf ("\ndf = %2.0f prob = %.9f = %.3e\n", df, prob[0], prob[0]);
}
printf ("\n");
return (0);
}
double QuantileNormal (double prob)
{
/* returns z so that Prob{x<z}=prob where x ~ N(0,1) and (1e-12)<prob<1-(1e-12)
returns (-9999) if in error
Odeh RE & Evans JO (1974) The percentage points of the normal distribution.
Applied Statistics 22: 96-97 (AS70)
*/
double a0=-.322232431088, a1=-1, a2=-.342242088547, a3=-.0204231210245;
double a4=-.453642210148e-4, b0=.0993484626060, b1=.588581570495;
double b2=.531103462366, b3=.103537752850, b4=.0038560700634;
double y, z=0, p=prob, p1;
p1 = (p<0.5 ? p : 1-p);
if (p1<1e-20) return (-9999);
y = sqrt (log(1/(p1*p1)));
z = y + ((((y*a4+a3)*y+a2)*y+a1)*y+a0) / ((((y*b4+b3)*y+b2)*y+b1)*y+b0);
return (p<0.5 ? -z : z);
}
double CDFNormal (double x)
{
/* Hill ID (1973) The normal integral. Applied Statistics, 22:424-427.
Algorithm AS 66.
adapted by Z. Yang, March 1994. Hill's routine is quite bad, and I
haven't consulted
Adams AG (1969) Algorithm 39. Areas under the normal curve.
Computer J. 12: 197-198.
*/
int invers=0;
double p, limit=10, t=1.28, y=x*x/2;
if (x<0) { invers=1; x*=-1; }
if (x>limit) return (invers?0:1);
if (x<1.28)
p = .5 - x * ( .398942280444 - .399903438504 * y
/(y + 5.75885480458 - 29.8213557808
/(y + 2.62433121679 + 48.6959930692
/(y + 5.92885724438))));
else
p = 0.398942280385 * exp(-y) /
(x - 3.8052e-8 + 1.00000615302 /
(x + 3.98064794e-4 + 1.98615381364 /
(x - 0.151679116635 + 5.29330324926 /
(x + 4.8385912808 - 15.1508972451 /
(x + 0.742380924027 + 30.789933034 /
(x + 3.99019417011))))));
return invers ? p : 1-p;
}
double LnGammaFunction (double alpha)
{
/* returns ln(gamma(alpha)) for alpha>0, accurate to 10 decimal places.
Stirling's formula is used for the central polynomial part of the procedure.
Pike MC & Hill ID (1966) Algorithm 291: Logarithm of the gamma function.
Communications of the Association for Computing Machinery, 9:684
*/
double x=alpha, f=0, z;
if (x<7) {
f=1; z=x-1;
while (++z<7) f*=z;
x=z; f=-log(f);
}
z = 1/(x*x);
return f + (x-0.5)*log(x) - x + .918938533204673
+ (((-.000595238095238*z+.000793650793651)*z-.002777777777778)*z
+.083333333333333)/x;
}
double IncompleteGamma (double x, double alpha, double ln_gamma_alpha)
{
/* returns the incomplete gamma ratio I(x,alpha) where x is the upper
limit of the integration and alpha is the shape parameter.
returns (-1) if in error
ln_gamma_alpha = ln(Gamma(alpha)), is almost redundant.
(1) series expansion if (alpha>x || x<=1)
(2) continued fraction otherwise
RATNEST FORTRAN by
Bhattacharjee GP (1970) The incomplete gamma integral. Applied Statistics,
19: 285-287 (AS32)
*/
int i;
double p=alpha, g=ln_gamma_alpha;
double accurate=1e-8, overflow=1e30;
double factor, gin=0, rn=0, a=0,b=0,an=0,dif=0, term=0, pn[6];
if (x==0) return (0);
if (x<0 || p<=0) return (-1);
factor=exp(p*log(x)-x-g);
if (x>1 && x>=p) goto l30;
/* (1) series expansion */
gin=1; term=1; rn=p;
l20:
rn++;
term*=x/rn; gin+=term;
if (term > accurate) goto l20;
gin*=factor/p;
goto l50;
l30:
/* (2) continued fraction */
a=1-p; b=a+x+1; term=0;
pn[0]=1; pn[1]=x; pn[2]=x+1; pn[3]=x*b;
gin=pn[2]/pn[3];
l32:
a++; b+=2; term++; an=a*term;
for (i=0; i<2; i++) pn[i+4]=b*pn[i+2]-an*pn[i];
if (pn[5] == 0) goto l35;
rn=pn[4]/pn[5]; dif=fabs(gin-rn);
if (dif>accurate) goto l34;
if (dif<=accurate*rn) goto l42;
l34:
gin=rn;
l35:
for (i=0; i<4; i++) pn[i]=pn[i+2];
if (fabs(pn[4]) < overflow) goto l32;
for (i=0; i<4; i++) pn[i]/=overflow;
goto l32;
l42:
gin=1-factor*gin;
l50:
return (gin);
}
double QuantileChi2 (double prob, double v)
{
/* returns z so that Prob{x<z}=prob where x is Chi2 distributed with df=v
returns -1 if in error. 0.000002<prob<0.999998
RATNEST FORTRAN by
Best DJ & Roberts DE (1975) The percentage Quantiles of the
Chi2 distribution. Applied Statistics 24: 385-388. (AS91)
Converted into C by Ziheng Yang, Oct. 1993.
*/
double e=.5e-6, aa=.6931471805, p=prob, g;
double xx, c, ch, a=0,q=0,p1=0,p2=0,t=0,x=0,b=0,s1,s2,s3,s4,s5,s6;
if (p<.000002 || p>.999998 || v<=0) return (-1);
g = LnGammaFunction (v/2);
xx=v/2; c=xx-1;
if (v >= -1.24*log(p)) goto l1;
ch=pow((p*xx*exp(g+xx*aa)), 1/xx);
if (ch-e<0) return (ch);
goto l4;
l1:
if (v>.32) goto l3;
ch=0.4; a=log(1-p);
l2:
q=ch; p1=1+ch*(4.67+ch); p2=ch*(6.73+ch*(6.66+ch));
t=-0.5+(4.67+2*ch)/p1 - (6.73+ch*(13.32+3*ch))/p2;
ch-=(1-exp(a+g+.5*ch+c*aa)*p2/p1)/t;
if (fabs(q/ch-1)-.01 <= 0) goto l4;
else goto l2;
l3:
x = QuantileNormal (p);
p1=0.222222/v; ch=v*pow((x*sqrt(p1)+1-p1), 3.0);
if (ch>2.2*v+6) ch=-2*(log(1-p)-c*log(.5*ch)+g);
l4:
q=ch; p1=.5*ch;
if ((t=IncompleteGamma (p1, xx, g))<0) {
printf ("\nerr IncompleteGamma");
return (-1);
}
p2=p-t;
t=p2*exp(xx*aa+g+p1-c*log(ch));
b=t/ch; a=0.5*t-b*c;
s1=(210+a*(140+a*(105+a*(84+a*(70+60*a))))) / 420;
s2=(420+a*(735+a*(966+a*(1141+1278*a))))/2520;
s3=(210+a*(462+a*(707+932*a)))/2520;
s4=(252+a*(672+1182*a)+c*(294+a*(889+1740*a)))/5040;
s5=(84+264*a+c*(175+606*a))/2520;
s6=(120+c*(346+127*c))/5040;
ch+=t*(1+0.5*t*s1-b*c*(s1-b*(s2-b*(s3-b*(s4-b*(s5-b*s6))))));
if (fabs(q/ch-1) > e) goto l4;
return (ch);
}