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  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  UPDATED VERSIONs can be found on my WEB PAGE:


  Compute the probability of a given F value

Major Topics

  Curve and Surface Fitting, Statistics

Calling Sequence



  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
    * return the confidence level when the /CLEVEL keyword is passed;
    * 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
    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).


  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 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
  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.


  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.


  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,
  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

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