Non-central F programs ----------------------
^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 -----------
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 <steicht@@rjrt.com>
Examples --------
. ^ncfprob 3.6 3.0 4 43^ > gives p for F' <= 3.6, lambda = 3.0, v1 = 4, v2 = 4 > 3
. ^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