Is group mean and overall mean independent?

[colstat]

In an ANOVA experiment, you have,
group mean \overline{y}_.
overall mean \overline{y}_..

Are they independent?
Some samples overlap to calculate these means.

What is var(\overline{y}_.-\overline{y}_..) ?

 

[pmsrw3]

You mean under the null hypothesis, I presume. No, they're not independent, for the reason you give.

 

[colstat]

What is var(\overline{y}_.-\overline{y}_..) ?

 

[pmsrw3]

Umm, I'd have to look it up or work it out, and I'm too lazy to do that right now. But you should be able to find it in any stats book.

 

[colstat]

Ok. Is it something like this?
var(\overline{y}_.) + var(\overline{y}_..) + something else?

 

[pmsrw3]

Ok. Is it something like this?
var(\overline{y}_.) + var(\overline{y}_..) + something else?

Well, in general Var(a-b) = Var(a) + Var(b) - 2Cov(a,b), so you could write it in the form you give. But there's also a more precise formula in this case based on partitioning the sums of squares that contribute to the distinct variances.

 

[colstat]

what is 2Cov(a,b) in this case?

 

[pmsrw3]

what is 2Cov(a,b) in this case?

Cov(a,b) is the covariance of a and b, \mathbb{E}\left[\left(a-\mu_a\right)\left(b-\mu_b\right)\right]. Is that what you're asking, or are you asking me what it actually evaluates to when a=\overline{y}_{.} and b=\overline{y}_{..}?

 

[colstat]

evaluation.

 

[pmsrw3]

evaluation.

Write \overline{y}_{..} as a linear combination of the \overline{y}_{.}. Now use Cov(a,b+c) = Cov(a,b) + Cov(a,c) to expand it. The \overline{y}_{.} are all independent (under the null hypothesis), so all the cross covariances vanish. Cov(a, a) = Var(a), and Var(a/n) = Var(a)/n^2.

 

[colstat]

The book gives standard error: s\sqrt{1/n.-1/N}, where s = \sqrt{MSE}

How do I go from standard error to variance?

 

[pmsrw3]

The book gives standard error: s\sqrt{1/n.-1/N}, where s = \sqrt{MSE}

How do I go from standard error to variance?

The square of a standard error is an estimate (usually unbiased) of the variance of whatever it's the standard error of.

 

[colstat]

The square of a standard error is an estimate (usually unbiased) of the variance of whatever it's the standard error of.

I didn't get the last part of your sentence, are you saying standard error squared is variance?

 

[pmsrw3]

I didn't get the last part of your sentence, are you saying standard error squared is variance?

Yes, standard error squared is (an estimate of) variance. But the variance of what? The last part of the sentence was concerned with that.

In general, if you do an ANOVA or a regression, you'll get a bunch of standard errors. You get an SE for the overall mean, you get an SE for each group mean, if it's a multilevel or two-way ANOVA there will be some more SEs, and if it's a regression there'll be an SE for each coefficient. If I give you a number and I tell you it's the standard error of coefficient 2, then you know that the square of that number is an estimate of the variance of coefficient 2. And so on, for all the other SEs. That's what I meant.

 

[colstat]

I thought standard deviation squared is variance. So, does standard deviation not exist, or are they just the same thing?

 

[pmsrw3]

A standard error is a kind of a standard deviation. But usually "standard deviation" is used for the sample measurements themselves, whereas "standard error" is for estimators. For instance, if you have 4 measurements, y₁ - y₄, the square of the standard deviation is an estimate of the variance of anyone of the y_i. The square of the standard error of the mean (which is half the standard deviation) is an estimate of the variance of the mean of the four y's.

 

[colstat]

so, is this correct? var(\overline{y}_i.-\overline{y}..) = s²(1/n_i-1/N)

 

[pmsrw3]

I'm really not sure without working it out or looking it up. You said your book said that was SE, but you didn't say what it's the SE of. I don't do ANOVA so often that I have all the formulas in my head (and anyway, I rely on a computer for the actual calculations, like any sensible person).

 

[colstat]

umm, I think it should be right with the following sources I looked up:

1. "Applied Linear Regression Models" 5th by Kutner and Li P738 Example
2. "A First Course in Design and Analysis of Experiments" by Oehlert P44 top box and P43 bottom, free pdf on Oehlert's page.

 

[pmsrw3]

umm, I think it should be right...

Of course it's RIGHT, but what is it? You said it's "the standard error". The standard error of WHAT? And why do you think this particular SE should be the variance of the difference between the global mean and the group means? I would have guessed it was most likely the variance of the global mean.

 

[colstat]

Here is the pdf that contains the SE. Page 44.