[R-sig-ME] lmer for IV by design interaction?

[Michelle Ashburner]

Greetings,

Is it possible/appropriate to use lme4::lmer() to compare the effect of an
independent variable across two designs: within-subjects and
between-subjects?

Data below from Erlebacher (1977), used to illustrate his methodology:

###
dataset <- NULL

dataset$A <- c(rep.int(1, 20), rep.int(2, 20), rep.int(1, 20), rep.int(2,
20))
#A is the independent variable.
#1, 2 represent the two levels of the IV

dataset$D <- c(rep.int(1, 40), rep.int(2, 40))
#D is the design factor.
#1 represents a within-ss measurement; 2 a between-ss measurement.

dataset$S <- c(60, 73, 93, 10, 90, 80, 83, 37, 83, 70, 77, 7, 100, 70, 100,
43, 43, 83, 40, 73, 36, 53, 66, 0, 73, 43, 20, 10, 26, 40, 60, 3, 53, 26,
63, 6, 3, 30, 7, 10, 53, 77, 2, 38, 68, 92, 3, 15, 67, 53, 58, 20, 17, 40,
85, 60, 25, 3, 82, 67, 62, 0, 57, 42, 3, 55, 22, 28, 45, 47, 52, 75, 38,
45, 65, 50, 2, 0, 10, 60)

dataset <- as.data.frame(dataset)
###

Erlebacher's analysis on these data can be computed using code developed by
Merritt, Cook, and Wang (2014):
 https://www.researchgate.net/publication/264158186_Erlebacher's_Method_for_Contrasting_the_Within_and_Between-Subjects_Manipulation_of_the_Independent_Variable_using_R_and_SPSS
<https://www.researchgate.net/publication/264158186_Erlebacher's_Method_for_Contrasting_the_Within_and_Between-Subjects_Manipulation_of_the_Independent_Variable_using_R_and_SPSS>

The output of an Erlebacher's ANOVA for these data is:
Effect of A: F(1, 51) = 21.25,
Effect of D: F(1, 42) = 0.89
Effect of A x D = F(1, 51) = 7.88
(df obtained via Satterthwaite's (1946) Method)

Some have suggested a multilevel model with the IV and the design as fixed
effects; subject as a random effect, instead of the Erlebacher's ANOVA. For
example, this Stack Exchange discussion:
https://stats.stackexchange.com/questions/414995/statistically-testing-the-impact-of-a-within-subject-vs-between-subject-design

While the following gives similar results, I am unable to determine if this
is the correct approach:

###
dataset$A <- as.factor(dataset$A)
dataset$D <- as.factor(dataset$D)
dataset$subject <- c(rep(1:20, times = 2), 21:60)

library(lme4)
library(lmerTest)
anova(lmer(S ~ A + D + A*D + (1|subject),
           dataset,
           contrasts = list(A = "contr.sum", D = "contr.sum")))
###
Which outputs:
-------
Type III Analysis of Variance Table with Satterthwaite's method
     Sum Sq    Mean Sq    NumDF  DenDF    F value       Pr(>F)
A   3097.70    3097.70         1         75.004   21.7904   0.00001304 ***
D    123.74     123.74           1         53.030   0.8704     0.355067
A:D 1148.50    1148.50        1         75.004   8.0790     0.005765 **
-------

As of yet, I am unable to manage a theoretical manipulation of Erlebacher's
model to fit a multilevel model like the one above, which adds to my
confusion regarding whether one can use a MLM approach for this type of
data.

Thank you in advance for any advice.

-- 
 Michelle Ashburner, MMATH, MA (Psych), BEd

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