[R-sig-ME] How to select (and nest) covariates in a crossed design model?
Dennis Murphy
djmuser at gmail.com
Fri Jul 15 13:23:01 CEST 2011
Hi:
Your terminology is slightly amiss. A (completely) crossed design for
two treatment factors is one in which all levels of one factor occur
in combination with all levels of the second factor. In a crossed
design, experimental units are randomly assigned to factor
combinations. Your description conforms to that of a 2 x 2 crossover
design, in which the treatment assignments within subjects at time 1
are reversed at time 2; i.e., treatments comprise one within-subject
factor and time another. In a mixed model context, the treatments and
periods would normally be treated as fixed effect factors.
Some things are not clear to me and may have an effect on the form of
the model you eventually fit:
(1) Do you have replicated sets of matched pairs in areas A and B or
is this a completely 'within-subject' design with the two areas as
experimental units? If replicated,
* what are the individual experimental units and how does the
matching take place between areas A and B?
* what randomization, if any, occurred within pairs?
* are individuals 'naturally' associated with areas or are they
[randomly] assigned to areas?
(2) The 2 x 2 crossover occurs at the Area (top) level. Do you have a
time variable defined that corresponds to when the regimes were
switched for each pair?
(3) What is the role of 'before' and why is it measured in each of
three sessions? I presume that 'before' is intended to represent a
baseline measure of some sort, but if that's the case, what is the
point of the crossover and what then is the role of the control? [This
is not meant to be a rhetorical or provocative question---my
presumptions may well be wrong.]
(4) Is there a sufficient time lag between the switching of regimes to
wash out the treatment effect of those assigned the treatment
condition at time 1? This is a major concern in crossover studies as
it has an impact on the validity of inferences about treatments
assigned within subject.
>From what you've described, it appears that session is nested* within
area (or subject) and rounds are nested within session. Sessions could
be treated as a repeated measures factor. The rounds appear to occur
within the same session, so those measurements are basically
triplicates ('pseudo-replication'). They would be useful to quantify
measurement error within session, but not much else.
[* Factor B is said to be nested within factor A if the levels of
factor B occur in combination with only one level of factor A. By
definition, nested factors cannot interact with one another in the
context of a statistical model.]
The point of these questions is that it seems inappropriate to
postulate a reasonable model on the basis of what you've presented
thus far. My concern is that if areas A and B are the only two
experimental units in your study (i.e., the only two units to which
treatments were assigned), you have a big problem because you would
have zero degrees of freedom for error to test the treatment or time
effects. That's what prompted the questions posed above.
HTH,
Dennis
On Fri, Jul 15, 2011 at 12:26 AM, Mari Laine <mslain at utu.fi> wrote:
> Hi all,
>
> I have difficulties in building a proper model for my experimental data. The study was done in two areas, A and B. A received a test treatment and B acted as a control, and then the experiment was repeated with inversed treatments, making it a crossed design. Response was measured in three sessions per treatment per area: before, during and after the treatment. Each session included 3 consecutive observations (called rounds).
>
> I'm interested in the impact of treatment (test vs. control), and whether it is related to sessions or not. I've built a generalized linear mixed model, and placed "treatment", "session" and their interaction as fixed factors. My problem with the model is that, in addition to "experiment", I'm not quite sure what to include in random (or repeated?) factors and whether to nest these factors or not. I've browsed the web for examples that would include using a 2x3 (or 2x2) crossed design with multiple observations within a cell, but I've found none.
>
>
> Here is a data table resembling the one I'm using in my analysis:
>
> Area A
> experim treatm before during after
> 1 c 1.80 0.44 0.71 2.08 1.10 1.48 2.45 1.64 3.88
> 2 e 6.11 4.66 5.40 6.98 4.57 5.39 7.83 6.31 7.25
>
> Area B
> experim treatm before during after
> 1 e 1.96 2.08 2.20 0.92 1.34 1.42 0.59 1.02 1.76
> 2 c 1.45 2.41 4.62 2.55 5.31 3.75 5.03 8.24 1.28
>
>
> And the same data in an importable form:
> area experim treatment session sessnro round response
> A 1 c before 1 1 1.80
> A 1 c before 1 2 0.44
> A 1 c before 1 3 0.71
> A 1 c during 2 4 2.08
> A 1 c during 2 5 1.10
> A 1 c during 2 6 1.48
> A 1 c after 3 7 2.45
> A 1 c after 3 8 1.64
> A 1 c after 3 9 3.88
> A 2 e before 4 10 6.11
> A 2 e before 4 11 4.66
> A 2 e before 4 12 5.40
> A 2 e during 5 13 6.98
> A 2 e during 5 14 4.57
> A 2 e during 5 15 5.39
> A 2 e after 6 16 7.83
> A 2 e after 6 17 6.31
> A 2 e after 6 18 7.25
> B 1 e before 1 1 1.96
> B 1 e before 1 2 2.08
> B 1 e before 1 3 2.20
> B 1 e during 2 4 0.92
> B 1 e during 2 5 1.34
> B 1 e during 2 6 1.42
> B 1 e after 3 7 0.59
> B 1 e after 3 8 1.02
> B 1 e after 3 9 1.76
> B 2 c before 4 10 1.45
> B 2 c before 4 11 2.41
> B 2 c before 4 12 4.62
> B 2 c during 5 13 2.55
> B 2 c during 5 14 5.31
> B 2 c during 5 15 3.75
> B 2 c after 6 16 5.03
> B 2 c after 6 17 8.24
> B 2 c after 6 18 1.28
>
>
> Thanks,
> Mari Laine
>
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