;+ ; NAME: ; MPFTEST ; ; AUTHOR: ; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770 ; craigm@lheamail.gsfc.nasa.gov ; UPDATED VERSIONs can be found on my WEB PAGE: ; http://cow.physics.wisc.edu/~craigm/idl/idl.html ; ; PURPOSE: ; Compute the probability of a given F value ; ; MAJOR TOPICS: ; Curve and Surface Fitting, Statistics ; ; CALLING SEQUENCE: ; PROB = MPFTEST(F, DOF1, DOF2, [/SIGMA, /CLEVEL, /SLEVEL ]) ; ; DESCRIPTION: ; ; The function MPFTEST() computes the probability for a value drawn ; from the F-distribution to equal or exceed the given value of F. ; This can be used for confidence testing of a measured value obeying ; the F-distribution (i.e., for testing the ratio of variances, or ; equivalently for the addition of parameters to a fitted model). ; ; P_F(X > F; DOF1, DOF2) = PROB ; ; In specifying the returned probability level the user has three ; choices: ; ; * return the confidence level when the /CLEVEL keyword is passed; ; OR ; ; * return the significance level (i.e., 1 - confidence level) when ; the /SLEVEL keyword is passed (default); OR ; ; * return the "sigma" of the probability (i.e., compute the ; probability based on the normal distribution) when the /SIGMA ; keyword is passed. ; ; Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive. ; ; For the ratio of variance test, the two variances, VAR1 and VAR2, ; should be distributed according to the chi-squared distribution ; with degrees of freedom DOF1 and DOF2 respectively. The F-value is ; computed as: ; ; F = (VAR1/DOF1) / (VAR2/DOF2) ; ; and then the probability is computed as: ; ; PROB = MPFTEST(F, DOF1, DOF2, ... ) ; ; ; For the test of additional parameters in least squares fitting, the ; user should perform two separate fits, and have two chi-squared ; values. One fit should be the "original" fit with no additional ; parameters, and one fit should be the "new" fit with M additional ; parameters. ; ; CHI1 - chi-squared value for original fit ; ; DOF1 - number of degrees of freedom of CHI1 (number of data ; points minus number of original parameters) ; ; CHI2 - chi-squared value for new fit ; ; DOF2 - number of degrees of freedom of CHI2 ; ; Note that according to this formalism, the number of degrees of ; freedom in the "new" fit, DOF2, should be less than the number of ; degrees of freedom in the "original" fit, DOF1 (DOF2 < DOF1); and ; also CHI2 < CHI1. ; ; With the above definition, the F value is computed as: ; ; F = ( (CHI1-CHI2)/(DOF1-DOF2) ) / (CHI2/DOF2) ; ; where DOF1-DOF2 is equal to M, and then the F-test probability is ; computed as: ; ; PROB = MPFTEST(F, DOF1-DOF2, DOF2, ... ) ; ; Note that this formalism assumes that the addition of the M ; parameters is a small peturbation to the overall fit. If the ; additional parameters dramatically changes the character of the ; model, then the first "ratio of variance" test is more appropriate, ; where F = (CHI1/DOF1) / (CHI2/DOF2). ; ; INPUTS: ; ; F - ratio of variances as defined above. ; ; DOF1 - number of degrees of freedom in first variance component. ; ; DOF2 - number of degrees of freedom in second variance component. ; ; ; RETURNS: ; ; Returns a scalar or vector of probabilities, as described above, ; and according to the /SLEVEL, /CLEVEL and /SIGMA keywords. ; ; KEYWORD PARAMETERS: ; ; SLEVEL - if set, then PROB describes the significance level ; (default). ; ; CLEVEL - if set, then PROB describes the confidence level. ; ; SIGMA - if set, then PROB is the number of "sigma" away from the ; mean in the normal distribution. ; ; EXAMPLE: ; ; chi1 = 62.3D & dof1 = 42d ; chi2 = 54.6D & dof2 = 40d ; ; f = ((chi1-chi2)/(dof1-dof2)) / (chi2/dof2) ; print, mpftest(f, dof1-dof2, dof2) ; ; This is a test for addition of parameters. The "original" ; chi-squared value was 62.3 with 42 degrees of freedom, and the ; "new" chi-squared value was 54.6 with 40 degrees of freedom. ; These values reflect the addition of 2 parameters and the ; reduction of the chi-squared value by 7.7. The significance of ; this set of circumstances is 0.071464757. ; ; REFERENCES: ; ; Algorithms taken from CEPHES special function library, by Stephen ; Moshier. (http://www.netlib.org/cephes/) ; ; MODIFICATION HISTORY: ; Completed, 1999, CM ; Documented, 16 Nov 2001, CM ; Reduced obtrusiveness of common block and math error handling, 18 ; Nov 2001, CM ; Added documentation, 30 Dec 2001, CM ; Documentation corrections (thanks W. Landsman), 17 Jan 2002, CM ; Example docs were corrected (Thanks M. Perez-Torres), 17 Feb 2002, ; CM ; Example corrected again (sigh...), 13 Feb 2003, CM ; Convert to IDL 5 array syntax (!), 16 Jul 2006, CM ; Move STRICTARR compile option inside each function/procedure, 9 Oct 2006 ; Usage message with /CONTINUE, 23 Sep 2009, CM ; ; \$Id: mpftest.pro,v 1.10 2009/09/23 20:12:46 craigm Exp \$ ;- ; Copyright (C) 1999,2001,2002,2003,2009, Craig Markwardt ; This software is provided as is without any warranty whatsoever. ; Permission to use, copy, modify, and distribute modified or ; unmodified copies is granted, provided this copyright and disclaimer ; are included unchanged. ;- forward_function cephes_incbet, cephes_incbcf, cephes_incbd, cephes_pseries ;; Set machine constants, once for this session. Double precision ;; only. pro cephes_setmachar COMPILE_OPT strictarr common cephes_machar, cephes_machar_vals if n_elements(cephes_machar_vals) GT 0 then return if (!version.release) LT 5 then dummy = check_math(1, 1) mch = machar(/double) machep = mch.eps maxnum = mch.xmax minnum = mch.xmin maxlog = alog(mch.xmax) minlog = alog(mch.xmin) maxgam = 171.624376956302725D cephes_machar_vals = {machep: machep, maxnum: maxnum, minnum: minnum, \$ maxlog: maxlog, minlog: minlog, maxgam: maxgam} if (!version.release) LT 5 then dummy = check_math(0, 0) return end ; incbet.c ; ; Incomplete beta integral ; ; ; SYNOPSIS: ; ; double a, b, x, y, incbet(); ; ; y = incbet( a, b, x ); ; ; ; DESCRIPTION: ; ; Returns incomplete beta integral of the arguments, evaluated ; from zero to x. The function is defined as ; ; x ; - - ; | (a+b) | | a-1 b-1 ; ----------- | t (1-t) dt. ; - - | | ; | (a) | (b) - ; 0 ; ; The domain of definition is 0 <= x <= 1. In this ; implementation a and b are restricted to positive values. ; The integral from x to 1 may be obtained by the symmetry ; relation ; ; 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). ; ; The integral is evaluated by a continued fraction expansion ; or, when b*x is small, by a power series. ; ; ACCURACY: ; ; Tested at uniformly distributed random points (a,b,x) with a and b ; in "domain" and x between 0 and 1. ; Relative error ; arithmetic domain # trials peak rms ; IEEE 0,5 10000 6.9e-15 4.5e-16 ; IEEE 0,85 250000 2.2e-13 1.7e-14 ; IEEE 0,1000 30000 5.3e-12 6.3e-13 ; IEEE 0,10000 250000 9.3e-11 7.1e-12 ; IEEE 0,100000 10000 8.7e-10 4.8e-11 ; Outputs smaller than the IEEE gradual underflow threshold ; were excluded from these statistics. ; ; ERROR MESSAGES: ; message condition value returned ; incbet domain x<0, x>1 0.0 ; incbet underflow 0.0 function cephes_incbet, aa, bb, xx COMPILE_OPT strictarr forward_function cephes_incbcf, cephes_incbd, cephes_pseries common cephes_machar, machvals MINLOG = machvals.minlog MAXLOG = machvals.maxlog MAXGAM = machvals.maxgam MACHEP = machvals.machep if aa LE 0. OR bb LE 0. then goto, DOMERR if xx LE 0. OR xx GE 1. then begin if xx EQ 0 then return, 0.D if xx EQ 1. then return, 1.D DOMERR: message, 'ERROR: domain', /info return, 0.D endif flag = 0 if bb * xx LE 1. AND xx LE 0.95 then begin t = cephes_pseries(aa, bb, xx) goto, DONE endif w = 1.D - xx if xx GT aa/(aa+bb) then begin flag = 1 a = bb b = aa xc = xx x = w endif else begin a = aa b = bb xc = w x = xx endelse if flag EQ 1 AND b*x LE 1. AND x LE 0.95 then begin t = cephes_pseries(a, b, x) goto, DONE endif ;; Choose expansion for better convergence y = x * (a+b-2.) - (a-1.) if y LT 0. then w = cephes_incbcf(a, b, x) \$ else w = cephes_incbd(a, b, x) / xc ;; Multiply w by the factor ;; a b _ _ _ ;; x (1-x) | (a+b) / ( a | (a) | (b) ) . */ y = a * alog(x) t = b * alog(xc) if (a+b) LT MAXGAM AND abs(y) LT MAXLOG AND abs(t) LT MAXLOG then begin t = ((xc^b) * (x^a)) * w * gamma(a+b) / ( a * gamma(a) * gamma(b) ) goto, DONE endif ;; Resort to logarithms y = y + t + lngamma(a+b) - lngamma(a) - lngamma(b) y = y + alog(w/a) if y LT MINLOG then t = 0.D \$ else t = exp(y) DONE: if flag EQ 1 then begin if t LE MACHEP then t = 1.D - MACHEP \$ else t = 1.D - t endif return, t end ;; Continued fraction expasion #1 for incomplete beta integral function cephes_incbcf, a, b, x COMPILE_OPT strictarr common cephes_machar, machvals MACHEP = machvals.machep big = 4.503599627370496D15 biginv = 2.22044604925031308085D-16 k1 = a k2 = a + b k3 = a k4 = a + 1. k5 = 1. k6 = b - 1. k7 = k4 k8 = a + 2. pkm2 = 0.D qkm2 = 1.D pkm1 = 1.D qkm1 = 1.D ans = 1.D r = 1.D n = 0L thresh = 3.D * MACHEP repeat begin xk = - (x * k1 * k2 ) / (k3 * k4) pk = pkm1 + pkm2 * xk qk = qkm1 + qkm2 * xk pkm2 = pkm1 pkm1 = pk qkm2 = qkm1 qkm1 = qk xk = ( x * k5 * k6 ) / ( k7 * k8) pk = pkm1 + pkm2 * xk qk = qkm1 + qkm2 * xk pkm2 = pkm1 pkm1 = pk qkm2 = qkm1 qkm1 = qk if qk NE 0 then r = pk/qk if r NE 0 then begin t = abs( (ans-r)/r ) ans = r endif else begin t = 1.D endelse if t LT thresh then goto, CDONE k1 = k1 + 1. k2 = k2 + 1. k3 = k3 + 2. k4 = k4 + 2. k5 = k5 + 1. k6 = k6 - 1. k7 = k7 + 2. k8 = k8 + 2. if abs(qk) + abs(pk) GT big then begin pkm2 = pkm2 * biginv pkm1 = pkm1 * biginv qkm2 = qkm2 * biginv qkm1 = qkm1 * biginv endif if abs(qk) LT biginv OR abs(pk) LT biginv then begin pkm2 = pkm2 * big pkm1 = pkm1 * big qkm2 = qkm2 * big qkm1 = qkm1 * big endif n = n + 1 endrep until n GE 300 CDONE: return, ans end ;; Continued fraction expansion #2 for incomplete beta integral function cephes_incbd, a, b, x COMPILE_OPT strictarr common cephes_machar, machvals MACHEP = machvals.machep big = 4.503599627370496D15 biginv = 2.22044604925031308085D-16 k1 = a k2 = b - 1. k3 = a k4 = a + 1. k5 = 1. k6 = a + b k7 = a + 1. k8 = a + 2. pkm2 = 0.D qkm2 = 1.D pkm1 = 1.D qkm1 = 1.D z = x / (1.D - x) ans = 1.D r = 1.D n = 0L thresh = 3.D * MACHEP repeat begin xk = -(z * k1 * k2) / (k3 * k4) pk = pkm1 + pkm2 * xk qk = qkm1 + qkm2 * xk pkm2 = pkm1 pkm1 = pk qkm2 = qkm1 qkm1 = qk xk = (z * k5 * k6) / (k7 * k8) pk = pkm1 + pkm2 * xk qk = qkm1 + qkm2 * xk pkm2 = pkm1 pkm1 = pk qkm2 = qkm1 qkm1 = qk if qk NE 0 then r = pk/qk if r NE 0 then begin t = abs( (ans-r)/r ) ans = r endif else begin t = 1.D endelse if t LT thresh then goto, CDONE k1 = k1 + 1. k2 = k2 - 1. k3 = k3 + 2. k4 = k4 + 2. k5 = k5 + 1. k6 = k6 + 1. k7 = k7 + 2. k8 = k8 + 2. if abs(qk) + abs(pk) GT big then begin pkm2 = pkm2 * biginv pkm1 = pkm1 * biginv qkm2 = qkm2 * biginv qkm1 = qkm1 * biginv endif if abs(qk) LT biginv OR abs(pk) LT biginv then begin pkm2 = pkm2 * big pkm1 = pkm1 * big qkm2 = qkm2 * big qkm1 = qkm1 * big endif n = n + 1 endrep until n GE 300 CDONE: return, ans end ;; Power series for incomplete beta integral. ;; Use when b*x is small and x not too close to 1 function cephes_pseries, a, b, x COMPILE_OPT strictarr common cephes_machar, machvals MINLOG = machvals.minlog MAXLOG = machvals.maxlog MAXGAM = machvals.maxgam MACHEP = machvals.machep ai = 1.D/a u = (1.D - b) * x v = u / (a + 1.D) t1 = v t = u n = 2.D s = 0.D z = MACHEP * ai while abs(v) GT z do begin u = (n-b) * x / n t = t * u v = t / (a+n) s = s + v n = n + 1.D endwhile s = s + t1 + ai u = a * alog(x) if (a+b) LT MAXGAM AND abs(u) LT MAXLOG then begin t = gamma(a+b)/(gamma(a)*gamma(b)) s = s * t * x^a endif else begin t = lngamma(a+b) - lngamma(a) - lngamma(b) + u + alog(s) if t LT MINLOG then s = 0.D else s = exp(t) endelse return, s end ; MPFTEST ; Returns the significance level of a particular F-statistic. ; P(x; nu1, nu2) is probability for F to exceed x ; x - the F-ratio ; For ratio of variance test: ; x = (chi1sq/nu1) / (chi2sq/nu2) ; p = mpftest(x, nu1, nu2) ; For additional parameter test: ; x = [ (chi1sq-chi2sq)/(nu1-nu2) ] / (chi2sq/nu2) ; p = mpftest(x, nu1-nu2, nu2) ; ; nu1 - number of DOF in chi1sq ; nu2 - number of DOF in chi2sq nu2 < nu1 function mpftest, x, nu1, nu2, slevel=slevel, clevel=clevel, sigma=sigma COMPILE_OPT strictarr if n_params() LT 3 then begin message, 'USAGE: PROB = MPFTEST(F, DOF1, DOF2, [/SIGMA, /CLEVEL, /SLEVEL ])', /cont return, !values.d_nan endif cephes_setmachar ;; Set machine constants if nu1 LT 1 OR nu2 LT 1 OR x LT 0. then begin message, 'ERROR: domain', /info return, 0.D endif w = double(nu2) / (double(nu2) + double(nu1)*double(x)) s = cephes_incbet(0.5D * nu2, 0.5D * nu1, w) ;; Return confidence level if requested if keyword_set(clevel) then return, 1D - s if keyword_set(sigma) then return, mpnormlim(s, /slevel) ;; Return significance level otherwise. return, s end