Anova and unbalanced designs

Dear John, 

thank you again! You replicated the type III result I got in SPSS! When I
calculate Anova() type II:

Univariate Type II Repeated-Measures ANOVA Assuming Sphericity

                    SS num Df Error SS den Df      F  Pr(>F)  
between         4.8000      1   9.0000      8 4.2667 0.07273 .
within          0.2000      1  10.6667      8 0.1500 0.70864  
between:within  2.1333      1  10.6667      8 1.6000 0.24150  
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 

I see the exact same values as you had written. 
However, and now I am really lost, type III (I did not change anything else)
leads to the following: 

Univariate Type III Repeated-Measures ANOVA Assuming Sphericity

                              SS num Df Error SS den Df       F    Pr(>F)    
(Intercept)               72.000      1    9.000      8 64.0000 4.367e-05 ***
between                    4.800      1    9.000      8  4.2667   0.07273 .  
as.factor(within)          2.000      1   10.667      8  1.5000   0.25551    
between:as.factor(within)  2.133      1   10.667      8  1.6000   0.24150    
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 

How is this possible? 

This looks like a contrast parametrization issue: If we look at the 
per-group mean within-differences and their SE, we get

 > summary(lm(within1-within2~between - 1))
..
Coefficients:
          Estimate Std. Error t value Pr(>|t|)
between1  -1.0000     0.8165  -1.225    0.256
between2   0.3333     0.6667   0.500    0.631
..
 > table(between)
between
1 2
4 6

Now, the type II F test is based on weighting the two means as you would 
after testing for no interaction

 > (4*-1+6*.3333)^2/(4^2*0.8165^2+6^2*0.6667^2)
[1] 0.1500205

and type III is to weight them as if there had been equal counts

 > (5*-1+5*.3333)^2/(5^2*0.8165^2+5^2*0.6667^2)
[1] 0.400022

However, the result above corresponds to looking at group1 only

 > (-1)^2/(0.8165^2)
[1] 1.499987

It helps if you choose orhtogonal contrast parametrizations:

 > options(contrasts=c("contr.sum","contr.helmert"))
 > betweenanova <- lm(values ~ between)> Anova(betweenanova, idata=with, 
idesign= ~as.factor(within), type = "III" )

Type III Repeated Measures MANOVA Tests: Pillai test statistic
                           Df test stat approx F num Df den Df    Pr(>F)
(Intercept)                1     0.963  209.067      1      8 5.121e-07 ***
between                    1     0.348    4.267      1      8   0.07273 .
as.factor(within)          1     0.048    0.400      1      8   0.54474
between:as.factor(within)  1     0.167    1.600      1      8   0.24150
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
   c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
  (*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907