In article <[EMAIL PROTECTED]>,
Paige Miller <[EMAIL PROTECTED]> wrote:
>Suppose I have three (or more) samples, from three (or more) different
>populations. According to my subject matter expert, he wants to estimate
>a linear combination of the means, say for example
> 0.5*mu1 + 0.5*mu2 - mu3
>where mu1, mu2 and mu3 are the population means. I know how to compute
>this estimate, it is done by simply replacing the population means with
>the sample means. If I assume the original populations are normal and
>that the population variances are equal, I can compute the variance of
>this linear combination. Pretty straightforward stuff.
>However, I want to create a t-test to test the null hypothesis that this
>linear combination of means is equal to zero, using an estimate of
>variance derived from the data, rather than a population variance, which
>is unknown. In doing so, I run into the mathematical difficulty that I
>do not know the proper degrees of freedom for this test. (And yes, I
>know that for the special case of estimating mu1 - mu2 there are
>textbook formulas for this t-test, however I really am interested in
>linear combinations of more than two means).
>I feel like I am missing something very obvious; however, if someone
>knows, or can point me to the proper formula for a t-test on a linear
>combination of means, it would be greatly appreciated.
The textbook formulas are only approximate for that case,
anyhow. However, there is a procedure which is easy to
derive, has the t distribution under normality, and which
only loses power compared to what can be done precisely.
It may even be a little more robust against non-normality.
>From your description, you have samples of size n_j with
means mu_j, and you want to do a t-test of \sum a_j*mu_j.
Let N be the smallest of the n_j, and let X_jk be the
observations from the j-th sample, m_j the sample means.
Choose, at random or otherwise, coefficients b_jkr,
r=1, ..., N-1, so that
\sum_k b_jkr = 0;
\sum_k b_jkr^2 = 1;
\sum_k b_jkr*b_jks = 0 if r != s.
The b_j's are rows of an orthogonal matrix orthogonal to
the mean vector. Then the numbers Y_jr = \sum b_jkr*X_jk
are uncorrelated (independent if normal) and have mean 0
and variance \sigma_j^2. So the sums
\sum_j a_j*Y_jr/sqrt(n_j)
are uncorrelated (independent normal) random variables
with mean 0 and variance that of \sum_j a_j*m_j. Use
this to get a t-test with N-1 degrees of freedom.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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