FNONCT Function

Returns the value of the noncentrality parameter of an F distribution.

Category:	Mathematical

Table of Contents

Syntax

FNONCT(x, ndf, ddf, probability)

Required Arguments

x

is a numeric random variable.

Range	x ≥ 0

ndf

is a numeric numerator degree of freedom parameter.

Range	ndf > 0

ddf

is a numeric denominator degree of freedom parameter.

Range	ddf > 0

probability

is a probability.

Range	0 < probability < 1

Details

The FNONCT function returns the nonnegative noncentrality parameter from a noncentral F distribution whose parameters are x, ndf, ddf, and nc. If probability is greater than the probability from the central F distribution whose parameters are x, ndf, and ddf, a root to this problem does not exist. In this case a missing value is returned. A Newton-type algorithm is used to find a nonnegative root nc of the equation

table with 1 row and 1 column , row1 column 1 , p sub f . open x vertical line n d f comma d d f comma n c close . minus p r o b equals 0 , end table. Click image for alternative formats.

The following relationship applies to the preceding equation:

$table with 1 row and 1 column , row1 column 1 , p sub f . open x vertical line n d f comma d d f comma n c close . equals . epsilon super fraction negative n c , over 2 end fraction end super . modified cap sigma with j equals 0 below and with infinity above . fraction open , fraction n c , over 2 end fraction , close to the j , over j factorial end fraction . i sub fraction open n d f close . x , over d d f plus . open n d f close . x end fraction end sub . open . fraction d d f , over 2 end fraction . plus j comma . fraction d d f , over 2 end fraction . close , end table. Click image for alternative formats.$

In the equation, I (. . .) is the probability from the beta distribution that is given by the following equation:

$table with 1 row and 1 column , row1 column 1 , i sub x . open eh comma b close . equals . fraction , over fraction cap gamma , open eh close , cap gamma , open b close , over cap gamma . open eh plus b close end fraction end fraction . integral , from , 0 , to , x , of . t super eh minus 1 end super . open 1 minus t close super b minus 1 end super . d t , end table. Click image for alternative formats.$

If the algorithm fails to converge to a fixed point, a missing value is returned.

Example

data work;
   x=2;
   df=4;
   ddf=5;
   do nc=1 to 3 by .5;
      prob=probf(x, df, ddf, nc);
      ncc=fnonct(x, df, ddf, prob);
      output;
   end;
run;
proc print;
run;

data work;
   x=2;
   df=4;
   ddf=5;
   do nc=1 to 3 by .5;
      prob=probf(x, df, ddf, nc);
      ncc=fnonct(x, df, ddf, prob);
      output;
   end;
run;
proc print;
run;

Output from the FNONCT Function

Last updated: April 14, 2026