[R] two-factor linear models with missing cells

[Peter Dalgaard]

Murray Jorgensen wrote:
> I am wondering how to interpret the parameter estimates that lm()
> reports in this sort of situation:
> 
> y = round(rnorm(n=24,mean=5,sd=2),2)
> A = gl(3,2,24,labels=c("one","two","three"))
> B = gl(4,6,24,labels=c("i","ii","iii","iv"))
> # Make both observations for A=1, B=4 missing
> y[19] = NA
> y[20] = NA
> data.frame(y,A,B)
> nonadd = lm(y ~ A * B)
> 
> 
>> summary(nonadd)
> 
> Call:
> lm(formula = y ~ A * B)
> 
> Residuals:
> Min 1Q Median 3Q Max
> -3.555e+00 -7.675e-01 -6.939e-17 7.675e-01 3.555e+00
> 
> Coefficients: (1 not defined because of singularities)
> Estimate Std. Error t value Pr(>|t|)
> (Intercept) 3.755 1.667 2.252 0.0457 *
> Atwo 1.655 2.358 0.702 0.4974
> Athree 3.330 2.358 1.412 0.1856
> Bii 1.435 2.358 0.609 0.5552
> Biii 2.055 2.358 0.871 0.4021
> Biv -1.635 2.358 -0.693 0.5025
> Atwo:Bii -1.145 3.335 -0.343 0.7378
> Athree:Bii -4.535 3.335 -1.360 0.2011
> Atwo:Biii -3.230 3.335 -0.969 0.3536
> Athree:Biii -2.105 3.335 -0.631 0.5408
> Atwo:Biv 1.655 3.335 0.496 0.6295
> Athree:Biv NA NA NA NA
> ---
> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
> Residual standard error: 2.358 on 11 degrees of freedom
> (2 observations deleted due to missingness)
> Multiple R-squared: 0.2797, Adjusted R-squared: -0.3752
> F-statistic: 0.4271 on 10 and 11 DF, p-value: 0.9044
> 
>> fitted(nonadd)
> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 21
> 3.755 3.755 5.410 5.410 7.085 7.085 5.190 5.190 5.700 5.700 3.985 3.985
> 5.810 5.810 4.235 4.235 7.035 7.035 5.430
> 22 23 24
> 5.430 5.450 5.450
>> t(model.matrix(nonadd)%*%coef(nonadd))
> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 21 22 23 24
> [1,] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
> 
> I guess that the parameter estimates reported are linear combinations of
> the cell means, but which linear combinations and how does lm() decide
> what parameters to report?
> 
> Cheers, Murray
> 

What's the problem? The parameters are defined as usual for the two-way 
layout:

The intercept is the fitted value in the top left corner
The A coefficients are the fitted values in the first column minus the 
intercept.
The B coefficients vice versa.
The interaction coefficients are the fitted values minus the sum of the 
the intercept and the corresponding A and B coefficients.

One interaction coefficient is set missing because you have no data, but 
except for that, the fitted values equal the cell means.

-- 
    O__  ---- Peter Dalgaard             Øster 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