[R] two-factor linear models with missing cells

[Murray Jorgensen]

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

-- 
Dr Murray Jorgensen      http://www.stats.waikato.ac.nz/Staff/maj.html
Department of Statistics, University of Waikato, Hamilton, New Zealand
Email: maj at waikato.ac.nz                                Fax 7 838 4155
Phone  +64 7 838 4773 wk    Home +64 7 825 0441   Mobile 021 0200 8350