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## Why is the total variance fron a random effects ANOVA not equal to the total variance of the data?

Hi

I am hoping someone can explain this to me in an intuitive way.

Why is the total variance of all data (last column in attached table) not equal to the total variance from the random effects ANOVA (script in the attached table)?

What does this difference mean?

Jesper

BR
Jesper
6 REPLIES 6

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

Variance (and standard deviation) of all data assume all results are independent, random results from one population. If you have a random effects model, results in the same SIRINDEX group may be more alike than results in another group. Or, to put it another way, each SIRINDEX group may be a different population with a different mean. The total variance in the random effects ANOVA takes into account group mean differences and effectively "adjusts" the variance. In particular, the random effects ANOVA calculates separate within group and between group variances, then adds those up. (The random effects ANOVA does assume each group has the same / similar within population variances.)

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

The way I understand what you write is that the assumptions of the ”pure” overall standard deviation and the total standard deviation from the random effects ANOVA are different. However, I still don’t quite understand why they yield different results.

I get that the ANOVA takes into account the fact that certain groups of data may come from different populations. But my understanding is that the ANOVA tries to decompose the total variation into components coming from e.g. within-run and between-run. If those are the only factors in you ANOVA (as in my example) why does the within- and between-run variances not sum up to the total variance?

I understand the math behind both types of total variance. What I am trying to understand is the real world meanings of those - or the difference in meaning explaining the different values.

I have searched quite bit for the answer, but all can find is how to calculate and interpret the different variance components – not the total variance.

It seems to me that if you break the total variance on two and put the pieces back together, you end up with a total variance larger than what you started with. Why?

Jesper

BR
Jesper

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

Haven't had a chance to look in detail at you analyses so don't know for
sure. But if your data are unbalanced (different numbers in each group) and
the default REML method is used the ANOVA model components do not always
add to the ANOVA model total. Don't recall off-hand the theory on why.
Perhaps someone from SAS/JMP can add to this?

Highlighted

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

PS., As for why the ANOVA model total does not equal the overall sample standard deviation (say from the Distribution platform), my best thought on that is that the overall sample standard deviation is not accounting for the grouping as is thus estimating something different.

Again, anyone else have another thought? Thanks!

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

Thank you for your reply. I am still not sure I quite understand, but at least I am affirmed that it is not basic knowledge for everyone but me 😊
BR
Jesper

## Re: Why is the total variance fron a random effects ANOVA not equal to the total variance of the dat

Glad to help, somewhat at least. I actually find it hard to explain with
just words. Visualizing some data live would help, so perhaps you can find
someone nearby to help out as well. Regards!