The R Project SVN R

/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998 Ross Ihaka
 *  Copyright (C) 2000 The R Development Core Team
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA.
 *
 *  SYNOPSIS
 *
 *    #include "Rmath.h"
 *    double ptukey(q, rr, cc, df, lower_tail, log_p);
 *
 *  DESCRIPTION
 *
 *    Computes the probability that the maximum of rr studentized
 *    ranges, each based on cc means and with df degrees of freedom
 *    for the standard error, is less than q.
 *
 *    The algorithm is based on that of the reference.
 *
 *  REFERENCE
 *
 *    Copenhaver, Margaret Diponzio & Holland, Burt S.
 *    Multiple comparisons of simple effects in
 *    the two-way analysis of variance with fixed effects.
 *    Journal of Statistical Computation and Simulation,
 *    Vol.30, pp.1-15, 1988.
 */

#include "nmath.h"
#include "dpq.h"

static double wprob(double w, double rr, double cc)
{
/*  wprob() :

    This function calculates probability integral of Hartley's
    form of the range.

    w     = value of range
    rr    = no. of rows or groups
    cc    = no. of columns or treatments
    ir    = error flag = 1 if pr_w probability > 1
    pr_w = returned probability integral from (0, w)

    program will not terminate if ir is raised.

    bb = upper limit of legendre integration
    iMax = maximum acceptable value of integral
    nleg = order of legendre quadrature
    ihalf = int ((nleg + 1) / 2)
    wlar = value of range above which wincr1 intervals are used to
           calculate second part of integral,
           else wincr2 intervals are used.
    C1, C2, C3 = values which are used as cutoffs for terminating
    or modifying a calculation.

    M_1_SQRT_2PI = 1 / sqrt(2 * pi);  from abramowitz & stegun, p. 3.
    M_SQRT2 = sqrt(2)
    xleg = legendre 12-point nodes
    aleg = legendre 12-point coefficients
 */
#define nleg    12
#define ihalf   6

    /* looks like this is suboptimal for double precision.
       (see how C1-C3 are used) <MM>
    */
    /* const static double iMax  = 1.; not used if = 1*/
    const static double C1 = -30.;
    const static double C2 = -50.;
    const static double C3 = 60.;
    const static double bb   = 8.;
    const static double wlar = 3.;
    const static double wincr1 = 2.;
    const static double wincr2 = 3.;
    const static double xleg[ihalf] = {
    0.981560634246719250690549090149,
    0.904117256370474856678465866119,
    0.769902674194304687036893833213,
    0.587317954286617447296702418941,
    0.367831498998180193752691536644,
    0.125233408511468915472441369464
    };
    const static double aleg[ihalf] = {
    0.047175336386511827194615961485,
    0.106939325995318430960254718194,
    0.160078328543346226334652529543,
    0.203167426723065921749064455810,
    0.233492536538354808760849898925,
    0.249147045813402785000562436043
    };
    double a, ac, pr_w, b, binc, blb, bub, c, cc1, einsum, elsum,
    pminus, pplus, qexpo, qsqz, rinsum, wi, wincr, xx;
    int j, jj;

    qsqz = w * 0.5;

    /* if w >= 16 then the integral lower bound (occurs for c=20) */
    /* is 0.99999999999995 so return a value of 1. */

    if (qsqz >= bb)
    return 1.0;

    /* find (f(w/2) - 1) ^ cc */
    /* (first term in integral of hartley's form). */

    pr_w = 2 * pnorm(qsqz, 0.,1., 1,0) - 1.; /* erf(qsqz / M_SQRT2) */
    /* if pr_w ^ cc < 2e-22 then set pr_w = 0 */
    if (pr_w >= exp(C2 / cc))
    pr_w = pow(pr_w, cc);
    else
    pr_w = 0.0;

    /* if w is large then the second component of the */
    /* integral is small, so fewer intervals are needed. */

    if (w > wlar)
    wincr = wincr1;
    else
    wincr = wincr2;

    /* find the integral of second term of hartley's form */
    /* for the integral of the range for equal-length */
    /* intervals using legendre quadrature.  limits of */
    /* integration are from (w/2, 8).  two or three */
    /* equal-length intervals are used. */

    /* blb and bub are lower and upper limits of integration. */

    blb = qsqz;
    binc = (bb - qsqz) / wincr;
    bub = blb + binc;
    einsum = 0.0;

    /* integrate over each interval */

    cc1 = cc - 1.0;
    for (wi = 1; wi <= wincr; wi++) {
    elsum = 0.0;
    a = 0.5 * (bub + blb);

    /* legendre quadrature with order = nleg */

    b = 0.5 * (bub - blb);

    for (jj = 1; jj <= nleg; jj++) {
        if (ihalf < jj) {
        j = (nleg - jj) + 1;
        xx = xleg[j-1];
        } else {
        j = jj;
        xx = -xleg[j-1];
        }
        c = b * xx;
        ac = a + c;

        /* if exp(-qexpo/2) < 9e-14, */
        /* then doesn't contribute to integral */

        qexpo = ac * ac;
        if (qexpo > C3)
        break;

#ifdef OLD
        if (ac > 0.)    pplus = 1.0 + erf(ac / M_SQRT2);
        else        pplus = erfc(-(ac / M_SQRT2));

        if (ac > w) pminus = 1.0 + erf((ac / M_SQRT2) - (w / M_SQRT2));
        else    pminus = erfc((w / M_SQRT2) - (ac / M_SQRT2));
#else
        pplus = 2 * pnorm(ac, 0., 1., 1,0);
        pminus= 2 * pnorm(ac, w,  1., 1,0);
#endif

        /* if rinsum ^ (cc-1) < 9e-14, */
        /* then doesn't contribute to integral */

        rinsum = (pplus * 0.5) - (pminus * 0.5);
        if (rinsum >= exp(C1 / cc1)) {
        rinsum = (aleg[j-1] * exp(-(0.5 * qexpo))) * pow(rinsum, cc1);
        elsum += rinsum;
        }
    }
    elsum *= (((2.0 * b) * cc) * M_1_SQRT_2PI);
    einsum += elsum;
    blb = bub;
    bub += binc;
    }

    /* if pr_w ^ rr < 9e-14, then return 0 */
    pr_w = einsum + pr_w;
    if (pr_w <= exp(C1 / rr))
    return 0.;

    pr_w = pow(pr_w, rr);
    if (pr_w >= 1.)/* 1 was iMax was eps */
    return 1.;
    return pr_w;
} /* wprob() */


double ptukey(double q, double rr, double cc, double df,
          int lower_tail, int log_p)
{
/*  function ptukey() [was qprob() ]:

    q = value of studentized range
    rr = no. of rows or groups
    cc = no. of columns or treatments
    df = degrees of freedom of error term
    ir[0] = error flag = 1 if wprob probability > 1
    ir[1] = error flag = 1 if qprob probability > 1

    qprob = returned probability integral over [0, q]

    The program will not terminate if ir[0] or ir[1] are raised.

    All references in wprob to Abramowitz and Stegun
    are from the following reference:

    Abramowitz, Milton and Stegun, Irene A.
    Handbook of Mathematical Functions.
    New York:  Dover publications, Inc. (1970).

    All constants taken from this text are
    given to 25 significant digits.

    nlegq = order of legendre quadrature
    ihalfq = int ((nlegq + 1) / 2)
    eps = max. allowable value of integral
    eps1 & eps2 = values below which there is
              no contribution to integral.

    d.f. <= dhaf:   integral is divided into ulen1 length intervals.  else
    d.f. <= dquar:  integral is divided into ulen2 length intervals.  else
    d.f. <= deigh:  integral is divided into ulen3 length intervals.  else
    d.f. <= dlarg:  integral is divided into ulen4 length intervals.

    d.f. > dlarg:   the range is used to calculate integral.

    M_LN2 = log(2)

    xlegq = legendre 16-point nodes
    alegq = legendre 16-point coefficients

    The coefficients and nodes for the legendre quadrature used in
    qprob and wprob were calculated using the algorithms found in:

    Stroud, A. H. and Secrest, D.
    Gaussian Quadrature Formulas.
    Englewood Cliffs,
    New Jersey:  Prentice-Hall, Inc, 1966.

    All values matched the tables (provided in same reference)
    to 30 significant digits.

    f(x) = .5 + erf(x / sqrt(2)) / 2      for x > 0

    f(x) = erfc( -x / sqrt(2)) / 2        for x < 0

    where f(x) is standard normal c. d. f.

    if degrees of freedom large, approximate integral
    with range distribution.
 */
#define nlegq   16
#define ihalfq  8

/*  const static double eps = 1.0; not used if = 1 */
    const static double eps1 = -30.0;
    const static double eps2 = 1.0e-14;
    const static double dhaf  = 100.0;
    const static double dquar = 800.0;
    const static double deigh = 5000.0;
    const static double dlarg = 25000.0;
    const static double ulen1 = 1.0;
    const static double ulen2 = 0.5;
    const static double ulen3 = 0.25;
    const static double ulen4 = 0.125;
    const static double xlegq[ihalfq] = {
    0.989400934991649932596154173450,
    0.944575023073232576077988415535,
    0.865631202387831743880467897712,
    0.755404408355003033895101194847,
    0.617876244402643748446671764049,
    0.458016777657227386342419442984,
    0.281603550779258913230460501460,
    0.950125098376374401853193354250e-1
    };
    const static double alegq[ihalfq] = {
    0.271524594117540948517805724560e-1,
    0.622535239386478928628438369944e-1,
    0.951585116824927848099251076022e-1,
    0.124628971255533872052476282192,
    0.149595988816576732081501730547,
    0.169156519395002538189312079030,
    0.182603415044923588866763667969,
    0.189450610455068496285396723208
    };
    double ans, f2, f21, f2lf, ff4, otsum, qsqz, rotsum, t1, twa1, ulen, wprb;
    int i, j, jj;

#ifdef IEEE_754
    if (ISNAN(q) || ISNAN(rr) || ISNAN(cc) || ISNAN(df))
    ML_ERR_return_NAN;
#endif

    if (q <= 0)
    return R_DT_0;

    /* df must be > 1 */
    /* there must be at least two values */

    if (df < 2 || rr < 1 || cc < 2) ML_ERR_return_NAN;

    if(!R_FINITE(q))
    return R_DT_1;

    if (df > dlarg)
    return R_DT_val(wprob(q, rr, cc));

    /* calculate leading constant */

    f2 = df * 0.5;
    /* lgammafn(u) = log(gamma(u)) */
    f2lf = ((f2 * log(df)) - (df * M_LN2)) - lgammafn(f2);
    f21 = f2 - 1.0;

    /* integral is divided into unit, half-unit, quarter-unit, or */
    /* eighth-unit length intervals depending on the value of the */
    /* degrees of freedom. */

    ff4 = df * 0.25;
    if      (df <= dhaf)    ulen = ulen1;
    else if (df <= dquar)   ulen = ulen2;
    else if (df <= deigh)   ulen = ulen3;
    else            ulen = ulen4;

    f2lf += log(ulen);

    /* integrate over each subinterval */

    ans = 0.0;

    for (i = 1; i <= 50; i++) {
    otsum = 0.0;

    /* legendre quadrature with order = nlegq */
    /* nodes (stored in xlegq) are symmetric around zero. */

    twa1 = (2 * i - 1) * ulen;

    for (jj = 1; jj <= nlegq; jj++) {
        if (ihalfq < jj) {
        j = jj - ihalfq - 1;
        t1 = (f2lf + (f21 * log(twa1 + (xlegq[j] * ulen))))
            - (((xlegq[j] * ulen) + twa1) * ff4);
        } else {
        j = jj - 1;
        t1 = (f2lf + (f21 * log(twa1 - (xlegq[j] * ulen))))
            + (((xlegq[j] * ulen) - twa1) * ff4);

        }

        /* if exp(t1) < 9e-14, then doesn't contribute to integral */
        if (t1 >= eps1) {
        if (ihalfq < jj) {
            qsqz = q * sqrt(((xlegq[j] * ulen) + twa1) * 0.5);
        } else {
            qsqz = q * sqrt(((-(xlegq[j] * ulen)) + twa1) * 0.5);
        }

        /* call wprob to find integral of range portion */

        wprb = wprob(qsqz, rr, cc);
        rotsum = (wprb * alegq[j]) * exp(t1);
        otsum += rotsum;
        }
        /* end legendre integral for interval i */
        /* L200: */
    }

    /* if integral for interval i < 1e-14, then stop.
     * However, in order to avoid small area under left tail,
     * at least  1 / ulen  intervals are calculated.
     */
    if (i * ulen >= 1.0 && otsum <= eps2)
        break;

    /* end of interval i */
    /* L330: */

    ans += otsum;
    }

    if(otsum > eps2) { /* not converged */
    ML_ERROR(ME_PRECISION);
    }
    if (ans > 1.)
    ans = 1.;
    return R_DT_val(ans);
}