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| Zootal... |
 Posted: Tue Oct 27, 2009 10:18 pm |
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I hope this is the appropriate place to post this question. I am attempting
to show that an F-statistic is equal to a t-statistic squared, or IOW F =
t^2. I am working with linear regression and extra sum of squares tests. How
would one go about showing this? Or, can some kind soul nudge me in the
right direction? |
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| Jack Tomsky... |
| Posted: Wed Oct 28, 2009 9:28 am |
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[quote]I hope this is the appropriate place to post this
question. I am attempting
to show that an F-statistic is equal to a t-statistic
squared, or IOW F =
t^2. I am working with linear regression and extra
sum of squares tests. How
would one go about showing this? Or, can some kind
soul nudge me in the
right direction?
[/quote]
The easiest way is to work with the canonical forms.
F(m,n) = [X2(m)/m]/[X2(n)/n],
where F(m,n) is an F with m and n degrees of freedom, X2(m) is a chi-square with m degrees of freedom, X2(n) is a chi-square with n degrees of freedom, and X2(m) and X2(n) are independent.
In particular, for m = 1,
F(1,n) = X2(1)/[X2(n)/n] = = z^2/[X2(n)/n] =
[z/sqrt(X2(n)/n)]^2 = [t(n)]^2,
where z ~ N(0,1) and t(n) is Student's t with n degrees of freedom.
Thus an F with 1 and n degrees of freedom is the square of a t with n degrees of freedom.
Jack
www.tomskystatistics.com |
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| Adole... |
| Posted: Sat Nov 07, 2009 11:19 am |
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On Oct 28, 1:28 pm, Jack Tomsky <jtom... at (no spam) ix.netcom.com> wrote:
[quote]I hope this is the appropriate place to post this
question. I am attempting
to show that an F-statistic is equal to a t-statistic
squared, or IOW F > > t^2. I am working with linear regression and extra
sum of squares tests. How
would one go about showing this? Or, can some kind
soul nudge me in the
right direction?
The easiest way is to work with the canonical forms.
F(m,n) = [X2(m)/m]/[X2(n)/n],
where F(m,n) is an F with m and n degrees of freedom, X2(m) is a chi-square with m degrees of freedom, X2(n) is a chi-square with n degrees of freedom, and X2(m) and X2(n) are independent.
In particular, for m = 1,
F(1,n) = X2(1)/[X2(n)/n] = = z^2/[X2(n)/n] > [z/sqrt(X2(n)/n)]^2 = [t(n)]^2,
where z ~ N(0,1) and t(n) is Student's t with n degrees of freedom.
Thus an F with 1 and n degrees of freedom is the square of a t with n degrees of freedom.
Jackwww.tomskystatistics.com
[/quote]
The reason this is not a great explanation is because you don't
explain (or show) how X2 and Z2 are approximated to be the same.
Therefore, this equations carries assumptions which from the student's
perspective aren't demonstrated to be valid. |
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