[R-sig-ME] lmer specification for random effects with several within-subject factors
Gabriel Baud-Bovy
baud-bovy.gabriel at hsr.it
Mon Oct 7 23:03:40 CEST 2013
Hi,
In my previous post lmer specification for maximal random effects
structure and one-between and two within-subject factors), I had
actually two quite different questions, which is not a very good idea.
Thanks to Tobias, I got an answer to my first question. To restate the
second question, I'll use a repeated-measure design with two crossed
within subject factors (A: 3 levels and B:2 levels) and replications as
an imaginery example. I assume that A and B are coded with sum contrasts.
I found many instances where one is told to use (1|S) + (0+A|S) or
(1|S) + (A-1|S) to simplify the correlation structure (I have done it
myself) but I am not actually sure to fully understand what is
happening. More precisely, while the random effects specified by the
formula (A*B|S) yields the results that I intuitively expected, I find
the results obtained with (1|S) + (A-1|S) + (B-1|S) + (A:B-1|S) or
(1|S) + (A-1|S) + (B-1|S) + (A:B-1|S) confusing. In these cases, I am
actually not sure how to interpet the random effects yielded by lmer.
Also, the design matrix is redundant and this redundancy makes these
models more difficult to fit, which is somewhat paradoxical since the
intention is to simplify the correlation structure.
I think that I understand how the design matrix Z is produced (i.e., by
binding together the terms that are specifed between parenthesis) even
though
there are some cases where I am not sure because the results for fixed
and random effects are different (see the example with the interaction
below).
My question is whether the design matrix Z specified by these formulae
make sense in this context and how to interpret the results if it does?
I could not find any discussion of these issues behond the simple case
of removing correlation betweenintercept and slope of a continuous
covariate.
As I said above, fitting a model maximal random effect structure with
Y~A*B+(A*B|S) yields the expected variance-covariance matrix
Groups Name Std.Dev. Corr
S (Intercept) 1.90664
A1 0.71938 0.710
A2 0.63624 -1.000 -0.727
B1 0.38072 -0.613 -0.992 0.632
A1:B1 0.51257 -0.394 -0.927 0.416 0.968
A2:B1 0.62905 -0.972 -0.855 0.978 0.781 0.598
Residual 2.07002
The corresponding design matrix Z for one subject is
(Intercept) A1 A2 B1 A1:B1 A2:B1
[1,] 1 1 . 1 1 .
[2,] 1 1 . 1 1 .
[3,] 1 1 . -1 -1 .
[4,] 1 1 . -1 -1 .
[5,] 1 . 1 1 . 1
[6,] 1 . 1 1 . 1
[7,] 1 . 1 -1 . -1
[8,] 1 . 1 -1 . -1
[9,] 1 -1 -1 1 -1 -1
[10,] 1 -1 -1 1 -1 -1
[11,] 1 -1 -1 -1 1 1
[12,] 1 -1 -1 -1 1 1
where columns A1, A2 and B1 corresponds to the expected sum contrasts.
This matrix is the same as the design matrix for the fixed effects.
To remove correlations between the intercept A, B and A:B interaction
terms, I tried Y~A*B+(1|S) + (A|S) + (B|S) + (A:B|S).
Groups Name Std.Dev. Corr
S (Intercept) 2.5421e-08
S.1 (Intercept) 2.1751e-07
A1 9.1658e-08 0.992
A2 3.1944e-07 -0.991 -0.970
S.2 (Intercept) 6.5355e-08
B1 1.2692e-07 1.000
S.3 (Intercept) 1.8430e+00
A1:B1 1.8613e-01 0.828
A2:B1 1.4158e+00 -0.959 -0.802
A3:B1 7.9928e-01 0.957 0.901 -0.981
A1:B2 1.8497e+00 0.433 0.457 -0.668 0.640
A2:B2 7.5347e-01 0.252 0.347 -0.509 0.491 0.980
A3:B2 6.7337e-01 -0.439 -0.353 0.674 -0.610 -0.981 -0.947
Residual 2.0700e+00 1.8845e+00
I did not get the expected results. This removed correlation between the
terms but also additional (redundant) intercepts terms. Moreover, the
interaction terms includes not only an intercept but also every
combination between the factor A and B. The design matrix is
(Intercept) (Intercept) A1 A2 (Intercept) B1 (Intercept) A1:B1
A2:B1 A3:B1 A1:B2 A2:B2 A3:B2
[1,] 1 1 1 . 1 1 1 1
. . . . .
[2,] 1 1 1 . 1 1 1 1
. . . . .
[3,] 1 1 1 . 1 -1 1
. . . 1 . .
[4,] 1 1 1 . 1 -1 1
. . . 1 . .
[5,] 1 1 . 1 1 1 1 .
1 . . . .
[6,] 1 1 . 1 1 1 1 .
1 . . . .
[7,] 1 1 . 1 1 -1 1
. . . . 1 .
[8,] 1 1 . 1 1 -1 1
. . . . 1 .
[9,] 1 1 -1 -1 1 1 1 .
. 1 . . .
[10,] 1 1 -1 -1 1 1 1 .
. 1 . . .
[11,] 1 1 -1 -1 1 -1 1 .
. . . . .
[12,] 1 1 -1 -1 1 -1 1 .
. . . . .
Note that column A1, B2, B1 correspond to the contrasts. For the
intercation term, a different coding (dummy variables)
is used and an intercept term is added (this is quite surprising since
A:B for fixed effect yields a different matrix).
If this matrix makes sense, how should I interpret the variance estimate
associated with the intercept? Should simply
consider the sum as the estimate of the between subject variability? Is
the split between the different terms
well defined ?
To remove intercepts, I tried Y~A*B+(1|S) + (A-1|S) + (B-1|S) +
(A:B-1|S). This yields
Groups Name Std.Dev. Corr
S (Intercept) 3.2644e-07
S.1 A1 0.0000e+00
A2 1.0269e-07 0.992
A3 2.6214e-07 -0.9920.366
S.2 B1 0.0000e+00
B2 5.2044e-08 0.992
S.3 A1:B1 1.9998e+00
A2:B1 6.2886e-01 0.768
A3:B1 2.6180e+00 0.998 0.724
A1:B2 3.1261e+00 0.850 0.316 0.883
A2:B2 2.1594e+00 0.944 0.516 0.964 0.976
A3:B2 1.6618e+00 0.930 0.949 0.904 0.598 0.758
Residual 2.0700e+00
and
(Intercept) A1 A2 A3 B1 B2 A1:B1 A2:B1 A3:B1 A1:B2 A2:B2 A3:B2
[1,] 1 1 . . 1 . 1 . . . . .
[2,] 1 1 . . 1 . 1 . . . . .
[3,] 1 1 . . . 1 . . . 1 . .
[4,] 1 1 . . . 1 . . . 1 . .
[5,] 1 . 1 . 1 . . 1 . . . .
[6,] 1 . 1 . 1 . . 1 . . . .
[7,] 1 . 1 . . 1 . . . . 1 .
[8,] 1 . 1 . . 1 . . . . 1 .
[9,] 1 . . 1 1 . . . 1 . . .
[10,] 1 . . 1 1 . . . 1 . . .
[11,] 1 . . 1 . 1 . . . . . 1
[12,] 1 . . 1 . 1 . . . . . 1
The intercept are indeed removed but each factor is dummy coded.
Contrasts are not used anymore. This matrix is still rank deficient
and this model is still more difficult to fit than the original (A*B|S)
model.
To summarize, I find the random effects of lmer difficult to interpret
when using the formula (1|S) + (A-1|S) + (B-1|S) + (A:B-1|S) or (1|S)
+ (A|S) + (B|S) + (A:B|S) . Moreover, their redundancy make them more
difficult to fit than the (A*B|S).
I expected that (1|S) + (A-1|S) + (B-1|S) + (A:B-1|S) would all yield
random effect structure that correspond to the fixed effects like when
using (A*B|S) (see above) with a correlation structure like
Groups Name Std.Dev. Corr
S (Intercept) 3.2644e-07
S.1 A1 0.0000e+00
A2 1.0269e-07 0.992
S.2 B1 0.0000e+00
S.3 A1:B1 1.9998e+00
A2:B1 6.2886e-01 0.768
Residual 2.0700e+00
I was not sure what I was expecting with (A|S) + (B|S) + (A:B|S) but
it would be nice if there was a way to specificy a covariance
Groups Name Std.Dev. Corr
S (Intercept) 3.2644e-07
A1 1.0269e-07 0.992
A2 2.6214e-07 -0.9920.366
S.1 B1 5.2044e-08 0.992
S.2 A1:B1 6.2886e-01 0.768
A2:B1 2.6180e+00 0.998 0.724
Residual 2.0700e+00
This would allow one to specify correlated random effects between the
subject intercept and the various conditions while excluding
correlated random effects between the conditions (this appear quite fare
from what is possible to do with lmer syntax(.
Best,
Gabriel
--
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Gabriel Baud-Bovy tel.: (+39) 02 2643 4839 (office)
UHSR University (+39) 02 2643 3429 (laboratory)
via Olgettina, 58 (+39) 02 2643 4891 (secretary)
20132 Milan, Italy fax: (+39) 02 2643 4892
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