# [R] Adjusted means and generalized chain block designs

Giles gilescrane at verizon.net
Mon Feb 15 20:47:29 CET 2010

```Dear Colleagues,

John Mandel ( Chain block designs with two-way elimination of heterogeneity.
Biometrics 10, 251-272 ,1954).
extended the class of chain block designs (Youden & Conner (1953)
to elimination of both row and column (blocks) effects.

These experimental designs can be useful in engineering
and other fields.

I am having difficulty obtaining his adjusted treatment means
in his example, shown below, by the use of lm(), or glm(), or ols().
However, I can obtain Mandel's analysis of variance.

# Mandel example, op. cit. page 263
# y are logs of treadwear, blocks are wheel positions,
# and rows are runs.
blocks = factor( rep(1:4, 4) )    # blocks
rows = factor(c( 1,1,1,1, 2,2,2,2, 3,3,3,3, 4,4,4,4)  )  #pos
trt = factor( c(1:8, 7,8,2,1,3,4,6,5) )  # tires
y = c(1.802, 1.862, 1.173, 1.762,
1.935, 2.072, 1.703, 1.935,
1.610, 1.568, 1.267, 1.522,
1.816, 1.935, 1.418, 1.594)

xtabs(y ~ rows + blocks)    # tabulate by row and column

# One of my attempts using lm() from the stats package:
m1 = lm( y ~  trt + blocks + rows)

Mandel obtrain the following adjusted means
mandel = c(1.728, 1.817, 1,588, 1.785,
1.512, 1. 678, 1.720, 1.658)

Can anyone shed light on how to obtain  in R
the correct treatment means,
adjusted for both row and column effects (blocks)
in these designs?

Cordially,
Giles Crane
gilescrane at verizon.net

```