From: John Wilkinson [[log in to unmask]]
Analysis of Variance for Random Effects
-----------------------------------------
As Winbugs' random effects models produce a "Variance components model",
there is a difference between this and a classical " Analysis of Variance
for Random Effects" models.
For Winbugs results to be reconciled with the AOV in "R" (or S+) results,
the Winbugs results; 'within' and 'between' "Variance Components" need to be
linearly combined with the factors' respective replication cell counts
according to the " Sampling Expectation Of Mean Square"; (E.M.S.)
rules,(Box and Tiao Bayesian Inference.p 245,from whence the Winbugs "Dyes"
example is taken,and their table 6.2.1,). Here the Dyes example (6
categories of Factor each with cells containing 5 replicate random
samples),is comparing its "Variance components" with those derived by Box
and Tiao from Classical randomised AOV using the E.M.S rules.
Thus Winbugs'example "Dyes" Results are----
node mean
sigma2.btw 2231.0
sigma2.with 3023.0
--------------------------------------------------------------------
The "R" Analysis of variance result summary for" Random Effects Model"---
----------------------------------------------------------------------
> batch.aov<-aov(yield~batch)
> summary(batch.aov)
df Sum Sq Mean Sq F value Pr(>F)
batch 5 56358 11272 4.5983 0.004398 **
Residuals 24 58830 2451
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
---------------------------------------------------------------
Winbugs' "Variance Components" reconciliation with Mean Squares (
according to Box and Tiao p 247)------
------------------------------------------------------------------
Between Batches Sampling Expectation of mean square (E.M.S.) =
(sigma2.with+5*sigma2.btw)
Between Batches E.M.S =(3023+5*2231) = 14178
Within Batches E.M.S.=sigma2.with = 3023
F value = E.M.S.(between)/E.M.S.(within) , d.f., (5,24) =4.69
F Prob (> F(5,24) =4.69) = 0.003964 ** (from tables)
This compares favourably with the "R" analysis result 0.004398**
above, and shows the E.M.S rules In action.
The EMS rules become increasingly important when considering " Multi-Way
A.O.V with Interactions" for random effects models.
For references in implimenting the E.M.S rules ,when doing A.O.V in Winbugs
with interactions,
I refer to Box and Tiao's Table 6.2.1.
For an explicit method in constructing an E.M.S table for interactions
I refer to "Ott and Langnecker -Data Analysis ", page1000.
The main result of "Variance Components" produced naturally by Winbugs is to
see the
proportion of the model's Total Variance allocted between the components.
Whilst this gives the individual weightings of the components, unlike the F
test it does
not give their significane level.
Since the "F"test is calculated differently for random and fixed effects
A.O.V models,
( particularly with interactions),and since the "F" test does not apply to
"Variance Components",
I would be interested to learn how to test these components directly for
levels of significance
codes, instead of working backwards from Winbugs to obtain F values from the
E.M.S.rules.
With thanks for any help with this,
John Wilkinson
-------------------------------------------------------------------
This list is for discussion of modelling issues and the BUGS software.
For help with crashes and error messages, first mail [log in to unmask]
To mail the BUGS list, mail to [log in to unmask]
Before mailing, please check the archive at www.jiscmail.ac.uk/lists/bugs.html
Please do not mail attachments to the list.
To leave the BUGS list, send LEAVE BUGS to [log in to unmask]
If this fails, mail [log in to unmask], NOT the whole list
|