Jeffrey Simons([email protected]) asks about a limit on the degrees of freedom
for the functions related to noncentral chi-square:
> From Stata manual:
>
> invnchi2(n,L,p) returns the inverse cumulative noncentral chi-squared
> distribution. If nchi2(n,L,x) = p, then invnchi2(n,L,p) = x (n must be
> an integer,1<=n<=200, 0<=L<=1000).
>
> Why does n (the degrees of freedom) need to be <=200?
>
> I am trying to utilize MacCallum et al., (1996) formulas for determining
> sample size for covariance structural models. He provides equations and
> SAS code at the end of the article and it involves the above chi2
> distributions.
>
> I wrote a program to do this in Stata but naturally, the model I want to
> test has 246 degrees of freedom. Why is there this limit?
This limit comes from the algorithm used to implement the -nchi2()- function;
from Haynam, Govindarajulu, and Leone (1970) [ref below].
The key issue here is that the algorithm involves progressive products
(factorials) and powers of values, both of which are taken directly from the
degrees of freedom. Beyond a certain point, these calculations will break
down given that they are performed on finite precision machines, thus we chose
to cut things off at n=200.
There are several approximations to this distribution, most (if not all) of
which are discussed in Johnson, Kotz, and Balakrishnan (1995). One
approximation is based on a Central Limit Argument.
Assume X2 is distributed as a non-central chi-square random variable
with n degrees of freedome and non-centrality parameter L, then
Z = (X2 - (n+L))/sqrt(2*(n+2*L))
will tend to follow the standard normal distribution for fixed n and
large L, or for fixed L and large n.
I compared the results between -invnchi2()- and this approximation fixing
n=200, using alpha = 0.05, and L ranging between 5 and 99. The relative
difference between the two ranged between 0.0040 and 0.0047. The normal
approximation was consistently less than the result from -invnchi2()- (minimum
difference was 1.12, max was 1.38).
For n=246, the normal approximation will be better, but I cannot say by how
much.
Reference:
Haynam, G.E., Z. Govindarajulu, and F.C. Leone. 1970. Tables of the cumulative
non-central chi-square distribution. In Selected Tables in
Mathematical Statistics, vol. 1, eds. H.L Harter and D.B. Owen,
Providence, RI: American Mathematical Society, pp. 1--78.
Johnson, N.L., S. Kotz, and N. Balarishnan. 1995. Continuous Univariate
Distributions, vol 2. 2d ed. New York: Wiley.
--Jeff
[email protected]
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