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Science Forum Index » Statistics - Math Forum » The Non-central Ch-square Distribution
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| David A. Heiser |
 Posted: Wed Jan 03, 2007 8:53 pm |
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Guest
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There were some statements made on another list about the use of the
non-central chi-square distribution. They did not appear to make sense, so I
need some enlightenment.
One statement the other party said was, "For both central and noncentral
chi-square, we assume that the vector of sample covariances is
asymptotically "normally distributed about a stable parameter." But for
noncentral chi-square, the population vector has a structure different from
the one we hypothesized."
Steiger in chapter 9 of "What If There Were No Significance Tests" is not
clear on its application. I have H&C, and their discussion (p290) is too
brief.
I want to start for the beginning, since what I said may be wrong.
The non-central parameter is it a stochastic parameter? I had viewed it as a
location parameter.
Were is it used?
What are the asymptotic properties? (as N increases)
David Heiser |
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| Guest |
| Posted: Thu Jan 04, 2007 11:15 am |
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David,
The central Chi-square distribution is derived as the sum of
independent squared standard normal random variables. Each of the
normal random variables has a mean of 0 and standard deviation of 1.
The non-central chi-square distribution allows the mean of the normal
random variables to be different from 0. The mean squared is the
noncentrality parameter, so it is a location parameter for the
non-central chi-square.
The most frequent use of the non-central chisquare is in power
calculations for hypothesis tests. You look at the size of the critical
region of the test as the mean departs from zero. As it departs from
zero, the power (size of the critical region) increases.
You will find it used most often in categorical hypothesis tests such
as chi-square tests of independence. The non-centrality paramter is a
little more complicated than just a squared mean, but it is essentially
that. Anyway, a good reference for the derivation of the noncentral
chi-square distribution is Graybill's book on Linear Models.
Mark
On Jan 3, 6:53 pm, "David A. Heiser" <dah_b...@innercite.com> wrote:
Quote: There were some statements made on another list about the use of the
non-central chi-square distribution. They did not appear to make sense, so I
need some enlightenment.
One statement the other party said was, "For both central and noncentral
chi-square, we assume that the vector of sample covariances is
asymptotically "normally distributed about a stable parameter." But for
noncentral chi-square, the population vector has a structure different from
the one we hypothesized."
Steiger in chapter 9 of "What If There Were No Significance Tests" is not
clear on its application. I have H&C, and their discussion (p290) is too
brief.
I want to start for the beginning, since what I said may be wrong.
The non-central parameter is it a stochastic parameter? I had viewed it as a
location parameter.
Were is it used?
What are the asymptotic properties? (as N increases)
David Heiser |
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