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help for ^ncf^
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Non-central F programs
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^ncfprob^ F' lambda v1 v2 /* yields p
^ncfinv^ p lambda v1 v2 /* yields F'
^ncfncp^ F' p v1 v2 /* yields lambda
^ncfn^ F' lambda p v1 /* yields v2
where
^F'^ is the observed F-value (^F'^ > 0)
^lambda^ is the noncentrality parameter (^lambda^ > 0)
^v1^ is the numerator degrees of freedom (^v1^ > 0)
^v2^ is the denominator degrees of freedom (^v2^ > 0)
^p^ is the probability (0 < ^p^ < 1)
Generally, ^v1^ and ^v2^ are integers, but any positive real value can be
entered for these parameters.
For each program, entering the program name with no parameters displays
the command syntax.
Description
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Let Z_1,...,Z_n be n independent random variables, where
Z_j ~ N(delta_j, 1), j=1,...,n
and X = (Z_1)^^2 + ... + (Z_n)^^2,
and where W ~ N(a,b) means W is distributed Normal with mean a and variance b.
Then X is said to have a noncentral chi-square distribution with
n degrees of freedom and noncentrality
lambda = sum[(delta_j)^^2]/2.
That is, X ~ ChiSq(n,lambda).
When delta_j = 0 for all j, then X is said to have a central
chi-square distribution, i.e., X ~ ChiSq(n).
Now, let X1 and X2 be independent random variables where
X1 ~ ChiSq(v1, lambda) and
X2 ~ ChiSq(v2).
X1/v1
Then X = ------- ~ F(v1, v2, lambda)
X2/v2
That is X is said to have a noncentral F distribution with v1 and v2
degrees of freedom and noncentrality lambda.
^ncf^xxxx is a family of immediate programs, all related to the noncentral F
distribution. Each program computes a missing parameter, given the other
parameters, such that P( F <= ^F'^ | ^lambda^, ^v1^, ^v2^ ) = ^p^.
for | use
-------+-----------------------------------------------------------
^p^ | ^ncfprob^ -- Cumulative non-central F probabilities
^F'^ | ^ncfinv^ -- Inverse cumulative non-central F values
^lambda^ | ^ncfncp^ -- Noncentrality parameter of the non-central F
^v2^ | ^ncfn^ -- Denominator df for the cumulative non-central F
The core program, ^ncfprob^, computes cumulative probability from zero to ^F'^
of the non-central F distribution for noncentrality parameter, ^lambda^, and
positive degrees of freedom ^v1^ in the numerator and ^v2^ in the denominator.
That is, ^ncfprob^ computes ^p^ such that P( F <= ^F'^ | ^lambda^, ^v1^, ^v2^ ) = p.
Each program prints its computed value and returns it in global ^S_1^ and in
a result ^r()^. The value computed and the name of the returned parameter
for each program are:
^ncfprob^ -- ^r(p)^ -- the probability p
^ncfinv^ -- ^r(F)^ -- the critical F'
^ncfncp^ -- ^r(lambda)^ -- the noncentrality parameter lambda
^ncfn^ -- ^r(v2)^ -- the denominator degrees of freedom v2
Warning: For ^ncfn^, there are many combinations of input parameters that are
not admissable (i.e, do not lead to a solution for ^v2^.) In particular, ^v2^
can go to infinity when ^F'^ is even moderately close to ^lambda^ and/or ^p^
or ^1 - p^ even moderately approaches zero. The user is cautioned to carefully
consider the reasonableness of the combination requested. Regardless, ^ncfn^
stops with an error message when the calculated ^v2^ exceeds 10,000.
As an example, consider the following hypothetical combination:
. ^ncfn 6 4 .8 1^
The above requests denominator degrees of freedom, ^v2^, such that
P(F<=^6^ | lambda=^4^, v1=^1^, ^v2^) = ^.8^
Since ^ncfprob^ calculates the ^p^-value given the other parameters, a quick
check for reasonableness can be performed by running ^ncfprob^ with selected
values of ^v2^ while holding the other parameters fixed at F'=^6^, lambda=^4^
and v1=^1^. This gives the following:
^v2 p^
-----------------------
1 .28460415
10 .34823550
100 .35216989
1000 .35235044
10000 .35236548
100000 .35236695
1000000 .35236710
10000000 .35236712
For these parameters, it is apparent that the only admissable values for ^p^
are in the range of about .28 to .36, regardless of the value of ^v2^.
Clearly, there is no value for ^v2^ that will allow ^p^ to be ^.8^, thus this
is an inadmissable combination for ^ncfn^.
Note
----
Some of these programs require installation of programs ^ridder^
(see STB-24, insert ssi5.4 for ^ridder^).
Author
------
Thomas J. Steichen
Examples
--------
. ^ncfprob 3.6 3.0 4 43^ > gives p for F' <= 3.6, lambda = 3.0, v1 = 4, v2 = 43
. ^ncfinv .95 3.0 4 43^ > gives F' for p = .95, lambda = 3.0, v1 = 4, v2 = 43
. ^ncfncp 3.6 .95 4 43^ > gives lambda for F' = 3.6, p = .95, v1 = 4, v2 = 43
. ^ncfn 3.6 1 .95 4^ > gives v2 for F' = 3.6, lambda = 1, p = .95, v1 = 4
. ^ncfprob^ > displays the command syntax