st: intraclass correlations

[David Airey <david.airey@vanderbilt.edu>]

Previously May Boggess (mboggess@stata.com) posted the following answer 
to how to calculate the intraclass correlations with 2 random factors. 
I had two questions about this code. The first in interspersed and the 
second follows.


> OK, now for two random factors. For these formulas I must turn to the
> comprehensive experimental design text by Winer et. al. "Statistical
> Principles of Experimental Design" (page 413). The formula is what you
> would expect it to be: ICC=variance of A/(total variance). So again, we
> count the number of repetitions, and this time the number of levels of
> each factor, compute the variance components, and then ICC.
>
> This time I am going to generate my own balanced dataset. Again, if 
> your
> design is not balanced, you just have to work harder to get the 
> variance
> components (the formulas for this are in Winer).
>
> *----two random factors--------------*
> *--- generate balanced data
>
> clear
> set obs 36
> generate A=int(_n/6+1)
> replace A=1 if A==7
> table A
> bysort A: generate B=int(_n/2+1)
> replace B=1 if B==4
> generate response=uniform()
>
> *----------------
> local depvar="response"
> local A="A"
> local B="B"
>
> table  `A' `B'
> local N=r(N)
> display `N'
> egen tag=tag(`A')
> egen sum=sum(tag)
> local p= sum[1]
> display `p'
>
> drop sum tag
> egen tag=tag(`B')
> egen sum=sum(tag)
> local q= sum[1]
> display `q'
> drop sum tag
>
> local n=`N'/(`p'*`q')
> display `n'
>
> anova `depvar'  `A' `B' `A'*`B'
> test `A'
> local MSA=r(ss)/r(df)
> display `MSA'
> local MSE=r(rss)/r(df_r)
> display `MSE'
Here May has picked up MSE from the test of A. Should she not have 
picked up MSE immediately after the anova command via the ereturn list?



> test `B'
> local MSB=r(ss)/r(df)
> display `MSB'
> test `A'*`B'
> local MSAB=r(ss)/r(df)
> display `MSAB'
>
> local sigA=(`MSA'-`MSAB')/(`n'*`q')
> local sigB=(`MSB'-`MSAB')/(`n'*`p')
> local sigAB=(`MSAB'-`MSE')/`n'
>
> local ICC=(`sigA')/(`sigA'+`sigB'+`sigAB'+`MSE')
> display `ICC'
>
> *---------------------
>
> Notice, we have to work a bit harder to get the mean squares for each 
> of
> the factors this time, since they are not in the saved results after
> -anova-. But, that is where -test- comes in handy, because they are in
> the results after -test-.
Secondly, I came across a more general formula for the intraclass 
correlation in a text by Kirk:

phat_I = (F - 1) / ( (n - 1) + F)

I could not quite understand what to put in "n" for above. In a one 
random factor ANOVA it's the number of observations in a given level of 
the random factor. So if you have 10 levels of a random factor with 5 
obs per level, "n" is 5. Does this mean that when random factors A and 
B are present, that "n" is the number of observations per level of A, 
per level of B, and per A*B combination when using this formula? So 
that say we have the following table, where animal is random and side 
is fixed, then "n" for animal is 4 while "n" for side*animal is 2 when 
using the formula above?

-------------------------------
          |        side
   animal |  left  right  Total
----------+--------------------
        1 |     2      2      4
        3 |     2      2      4
        4 |     2      2      4
        5 |     2      2      4
        6 |     2      2      4
        8 |     2      2      4
        9 |     2      2      4
       10 |     2      2      4
          |
    Total |    16     16     32
-------------------------------

-Dave


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