1 Overview

There really is no good reason I can think of to switch reference (base) levels for a contrast in the context of a hierarchical sequence of LMMs or the change the order of factor levels. So, in this sense, The exercise in this RPub is a bit of a silly exercise. Moreover, for the maximum LMM the choice of contrasts and the order of factor levels is irrelevant; they are mathematically equivalent reparameterizations of the same LMM. And, of course, estimates of variance parameters (VPs) and correlation parameters (CPs) in the random-effect structure do depend on type of contrast and the order of levels of the group factor.

However, when we reduce model complexity, mathematical equivalence at different levels of model complexity will be lost, because for reduced models goodness-of-fit depends on the specific VP(s) or CP(s) that are no longer in the LMM. In other words, as soon as we remove one or several VPs or CPs from the model, the mathematical equivalence may no longer hold. Thus, the above model sequences make only sense if the same contrast and the same ordering of factor levels holds for all models in a sequence.

Thus, we cannot change the order of factor levels or the base or reference levels within a specific type of contrast when we move within model sequences such as the ones determined in the previous RPubs and reproduced here.

(1a) RE1: max_LMM -> prsm1_LMM -> zcp_LMM -> prsm2_LMM -> min1_LMM  
(1b) RE0: max_LMM_RE0 -> prsm1_LMM_RE0 -> zcp_LMM_RE0 -> prsm2_LMM_RE0 -> min1_LMM

(2a) max_LMM -> int_LMM -> min1_LMM 
(2b)                    -> min2_LMM 

(3a)         -> fm5     -> min1_LM
(3b)                    -> min2_LM

This script demonstrates the effects of re-ordering factor levels and different reference (base) levels for treatment, sum, and sliding contrasts for model sequence (1a).

2 Setup

The demonstration uses the Machines data which are part of the MEMSS package [efficiency scores of 6 workers (random factor) tested 3 times on each of 3 machines (fixed factor) ].

knitr::opts_chunk$set(echo = TRUE)
suppressMessages(library("lme4"))
library("RePsychLing")
library(sjPlot)

## Install package "strengejacke" from GitHub (`devtools::install_github("strengejacke/strengejacke")`) to load all sj-packages at once!

suppressMessages(library("tidyverse"))
library(modelr)

3 Numeric covariates

data("Machines", package = "MEMSS")
Machines <- Machines[do.call(order, Machines), ] # reorder by Worker, then Machine

# Extract numeric covariates for treatment contrasts
## ...  with reference A
contrasts(Machines$Machine) <- contr.treatment(3, base=1)
mm1a <- model.matrix(~ 1 + Machine, Machines)
dBA <- mm1a[, 2]
dCA <- mm1a[, 3]

## ...  with reference B
contrasts(Machines$Machine) <- contr.treatment(3, base=2)
mm1b <- model.matrix(~ 1 + Machine, Machines)
dAB <- mm1b[, 2]
dCB <- mm1b[, 3]

## ...  with reference C
contrasts(Machines$Machine) <- contr.treatment(3, base=3)
mm1c <- model.matrix(~ 1 + Machine, Machines)
dAC <- mm1c[, 2]
dBC <- mm1c[, 3]

# Extract numeric covariates for sum contrasts
## ... leave out C - default order A-B-C
contrasts(Machines$Machine) <- contr.sum(3)
mm1d <- model.matrix(~ 1 + Machine, Machines)
dA_M <- mm1d[, 2]
dB_M <- mm1d[, 3]

## ... leave out B
Machines$Machine <- factor(Machines$Machine, levels=c("A", "C", "B"))
contrasts(Machines$Machine) <- contr.sum(3)
mm1e <- model.matrix(~ 1 + Machine, Machines)
dC_M <- mm1e[, 3]

# Extract numeric covariates for sliding contrasts
## ... order A-B-C - default
Machines$Machine <- factor(Machines$Machine, levels=c("A", "B", "C"))
contrasts(Machines$Machine) <- MASS::contr.sdif(3)
mm1f <- model.matrix(~ 1 + Machine, Machines)
dB_A <- mm1f[, 2]
dC_B <- mm1f[, 3]

## ... order A-C-B
Machines$Machine <- factor(Machines$Machine, levels=c("C", "A", "B"))
contrasts(Machines$Machine) <- MASS::contr.sdif(3)
mm1g <- model.matrix(~ 1 + Machine, Machines)
dA_C <- mm1g[, 2]

# set back to default order of levels and contrast
Machines$Machine <- factor(Machines$Machine, levels=c("A", "B", "C"))
contrasts(Machines$Machine) <- contr.treatment(3)

4 Treatment contrasts

# max_LMM 
# ... base = A
m1a <- lmer(score ~ 1 + Machine + (1 + dBA + dCA | Worker), Machines, REML=FALSE)
#  ... base = B
m1b <- lmer(score ~ 1 + Machine + (1 + dAB + dCB | Worker), Machines, REML=FALSE)
#  ... base = C
m1c <- lmer(score ~ 1 + Machine + (1 + dAC + dBC | Worker), Machines, REML=FALSE)


# prsm1_LMM
# ... base = A
m2a <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dBA + dCA | Worker), Machines, REML=FALSE)
# ... base = B
m2b <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dAB + dCB | Worker), Machines, REML=FALSE)
# ... base = C
m2c <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dAC + dBC | Worker), Machines, REML=FALSE)

# zcp_LMM
# ... base = A
m3a <- lmer(score ~ 1 + Machine + (1 + dBA + dBA || Worker), Machines, REML=FALSE)
# ... base = B
m3b <- lmer(score ~ 1 + Machine + (1 + dAB + dCB || Worker), Machines, REML=FALSE)
# ... base = C
m3c <- lmer(score ~ 1 + Machine + (1 + dAC + dBC || Worker), Machines, REML=FALSE)

# prsm2_LMM
# ... base = A
m4a <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dBA || Worker), Machines, REML=FALSE)
# ... base = B
m4b <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dCB || Worker), Machines, REML=FALSE)
# ... base = C
m4c <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dAC || Worker), Machines, REML=FALSE)
# min1_LMM
m5 <- lmer(score ~ 1 + Machine + (1 | Worker), Machines, REML=FALSE)

# Deviances for contrasts within models 4 and 3
suppressMessages(tab_model(m5, m4a, m4b, m4c, m3a, m3b, m3c,
         dv.labels = c("m5", "m4a", "m4b", "m4c", "m3a", "m3b", "m3c"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4a	m4b	m4c	m3a	m3b	m3c
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	3.32	7.16	7.31	3.32	0.93	0.93
τ₀₀	21.93 _Worker	12.19 _Worker	27.22 _Worker	27.70 _Worker	12.19 _Worker	60.32 _Worker	15.82 _Worker
		26.05 _Worker.1	8.91 _Worker.1	8.26 _Worker.1	26.05 _Worker.1	27.80 _Worker.1	10.86 _Worker.1
						28.42 _Worker.2	29.31 _Worker.2
Observations	54	54	54	54	54	54	54
Deviance	293.703	256.127	288.471	289.158	256.127	235.387	222.047

# Deviances for contrasts within model 2 and 1
suppressMessages(tab_model(m2a, m2b, m2c, m1a, m1b, m1c,
         dv.labels = c("m2a", "m2b", "m2c", "m1a", "m1b", "m1c"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m2a	m2b	m2c	m1a	m1b	m1c
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	0.93	0.93	0.93	0.92	0.92	0.92
τ₀₀	13.72 _Worker	61.05 _Worker	15.77 _Worker	13.82 _Worker	61.95 _Worker	16.00 _Worker
	29.11 _Worker.1	28.30 _Worker.1	10.98 _Worker.1
τ₁₁	11.04 _Worker.1.dCA	28.93 _Worker.1.dCB	29.63 _Worker.1.dBC	28.69 _Worker.dBA	28.69 _Worker.dAB	11.24 _Worker.dAC
				11.24 _Worker.dCA	29.31 _Worker.dCB	29.31 _Worker.dBC
ρ₀₁	0.30 _Worker.1	0.80 _Worker.1	0.33 _Worker.1	0.49 _Worker.dBA	-0.91 _Worker.dAB	-0.50 _Worker.dAC
				-0.36 _Worker.dCA	-0.88 _Worker.dCB	0.38 _Worker.dBC
Observations	54	54	54	54	54	54
Deviance	220.576	229.434	221.398	216.418	216.418	216.418

# Deviances for base A
suppressMessages (tab_model(m5, m4a, m3a, m2a, m1a, 
         dv.labels = c("m5", "m4a", "m3a", "m2a", "m1a"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4a	m3a	m2a	m1a
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	3.32	3.32	0.93	0.92
τ₀₀	21.93 _Worker	12.19 _Worker	12.19 _Worker	13.72 _Worker	13.82 _Worker
		26.05 _Worker.1	26.05 _Worker.1	29.11 _Worker.1
τ₁₁				11.04 _Worker.1.dCA	28.69 _Worker.dBA
					11.24 _Worker.dCA
ρ₀₁				0.30 _Worker.1	0.49 _Worker.dBA
					-0.36 _Worker.dCA
Observations	54	54	54	54	54
Deviance	293.703	256.127	256.127	220.576	216.418

# Deviances for base B
suppressMessages (tab_model(m5, m4b, m3b, m2b, m1b, 
         dv.labels = c("m5", "m4b", "m3b", "m2b", "m1b"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4b	m3b	m2b	m1b
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	7.16	0.93	0.93	0.92
τ₀₀	21.93 _Worker	27.22 _Worker	60.32 _Worker	61.05 _Worker	61.95 _Worker
		8.91 _Worker.1	27.80 _Worker.1	28.30 _Worker.1
			28.42 _Worker.2
τ₁₁				28.93 _Worker.1.dCB	28.69 _Worker.dAB
					29.31 _Worker.dCB
ρ₀₁				0.80 _Worker.1	-0.91 _Worker.dAB
					-0.88 _Worker.dCB
Observations	54	54	54	54	54
Deviance	293.703	288.471	235.387	229.434	216.418

# Deviances for base C
suppressMessages (tab_model(m5, m4c, m3c, m2c, m1c,
         dv.labels = c("m5", "m4c ", "m3c", "m2c", "m1c"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4c	m3c	m2c	m1c
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	7.31	0.93	0.93	0.92
τ₀₀	21.93 _Worker	27.70 _Worker	15.82 _Worker	15.77 _Worker	16.00 _Worker
		8.26 _Worker.1	10.86 _Worker.1	10.98 _Worker.1
			29.31 _Worker.2
τ₁₁				29.63 _Worker.1.dBC	11.24 _Worker.dAC
					29.31 _Worker.dBC
ρ₀₁				0.33 _Worker.1	-0.50 _Worker.dAC
					0.38 _Worker.dBC
Observations	54	54	54	54	54
Deviance	293.703	289.158	222.047	221.398	216.418

5 Sum contrasts

# max_LMM
# ... no A
m1am <- lmer(score ~ 1 + Machine + (1 + dB_M + dC_M | Worker), Machines, REML=FALSE)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : unable to evaluate scaled gradient

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge: degenerate Hessian with 2
## negative eigenvalues

# ... no B
m1bm <- lmer(score ~ 1 + Machine + (1 + dA_M + dC_M | Worker), Machines, REML=FALSE)
# ... no C
m1cm <- lmer(score ~ 1 + Machine + (1 + dA_M + dB_M | Worker), Machines, REML=FALSE)


# prsm1_LMM
# ... no A
m2am <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dB_M + dC_M | Worker), Machines, REML=FALSE)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: large eigenvalue ratio
##  - Rescale variables?

# ... no B
m2bm <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_M + dC_M | Worker), Machines, REML=FALSE)
# ... no C
m2cm <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_M + dB_M | Worker), Machines, REML=FALSE)

# zcp_LMM
# ... no A
m3am <- lmer(score ~ 1 + Machine + (1 + dB_M + dC_M || Worker), Machines, REML=FALSE)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: large eigenvalue ratio
##  - Rescale variables?

# ... no B
m3bm <- lmer(score ~ 1 + Machine + (1 + dA_M + dC_M || Worker), Machines, REML=FALSE)
# ... no C
m3cm <- lmer(score ~ 1 + Machine + (1 + dA_M + dB_M || Worker), Machines, REML=FALSE)

# prsm2_LMM
m4am <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_M || Worker), Machines, REML=FALSE)
m4bm <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dB_M || Worker), Machines, REML=FALSE)
m4cm <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dC_M || Worker), Machines, REML=FALSE)

# Deviances for contrasts within models 4 and 3
suppressMessages (tab_model(m5, m4am, m4bm, m4cm, m3am, m3bm, m3cm,
         dv.labels = c("m5", "m4am", " m4bm", "m4cm", "m3am", " m3bm", "m3cm"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4am	m4bm	m4cm	m3am	m3bm	m3cm
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	8.41	4.54	4.54	4.54	0.93	0.93
τ₀₀	21.93 _Worker	22.06 _Worker	22.49 _Worker	22.49 _Worker	22.49 _Worker	22.90 _Worker	22.90 _Worker
		1.56 _Worker.1	6.73 _Worker.1	6.73 _Worker.1	5.82 _Worker.1	5.51 _Worker.1	5.51 _Worker.1
					0.90 _Worker.2	11.43 _Worker.2	11.43 _Worker.2
Observations	54	54	54	54	54	54	54
Deviance	293.703	291.932	271.565	271.565	271.565	227.491	227.491

# Deviances for contrasts within models 2 and 1
suppressMessages (tab_model(m2am, m2bm, m2cm, m1am, m1bm, m1cm, 
         dv.labels = c("m2am", " m2bm", "m2cm", "m1am", " m1bm", "m1cm"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m2am	m2bm	m2cm	m1am	m1bm	m1cm
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	4.54	0.92	0.92	4.54	0.92	0.92
τ₀₀	22.49 _Worker	22.90 _Worker	22.90 _Worker	22.49 _Worker	22.90 _Worker	22.90 _Worker
	8.90 _Worker.1	5.62 _Worker.1	5.62 _Worker.1
τ₁₁	11.85 _{Worker.1.dC_M}	11.64 _{Worker.1.dC_M}	11.64 _{Worker.1.dB_M}	17.47 _{Worker.dB_M}	5.62 _{Worker.dA_M}	5.62 _{Worker.dA_M}
				6.76 _{Worker.dC_M}	11.64 _{Worker.dC_M}	11.64 _{Worker.dB_M}
ρ₀₁	0.68 _Worker.1	0.71 _Worker.1	-0.71 _Worker.1	0.34 _{Worker.dB_M}	-0.65 _{Worker.dA_M}	-0.65 _{Worker.dA_M}
				-0.27 _{Worker.dC_M}	-0.84 _{Worker.dC_M}	0.84 _{Worker.dB_M}
Observations	54	54	54	54	54	54
Deviance	271.565	223.545	223.545	266.291	216.418	216.418

# Deviances for models am
suppressMessages(tab_model(m5, m4am, m3am, m2am, m1am, 
         dv.labels = c("m5", "m4am", "m3am", "m2am", "m1am"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4am	m3am	m2am	m1am
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	8.41	4.54	4.54	4.54
τ₀₀	21.93 _Worker	22.06 _Worker	22.49 _Worker	22.49 _Worker	22.49 _Worker
		1.56 _Worker.1	5.82 _Worker.1	8.90 _Worker.1
			0.90 _Worker.2
τ₁₁				11.85 _{Worker.1.dC_M}	17.47 _{Worker.dB_M}
					6.76 _{Worker.dC_M}
ρ₀₁				0.68 _Worker.1	0.34 _{Worker.dB_M}
					-0.27 _{Worker.dC_M}
Observations	54	54	54	54	54
Deviance	293.703	291.932	271.565	271.565	266.291

# Deviances for models bm
suppressMessages(tab_model(m5, m4bm, m3bm, m2bm, m1bm, 
         dv.labels = c("m5", "m4bm", "m3bm", "m2bm", "m1bm"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4bm	m3bm	m2bm	m1bm
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	4.54	0.93	0.92	0.92
τ₀₀	21.93 _Worker	22.49 _Worker	22.90 _Worker	22.90 _Worker	22.90 _Worker
		6.73 _Worker.1	5.51 _Worker.1	5.62 _Worker.1
			11.43 _Worker.2
τ₁₁				11.64 _{Worker.1.dC_M}	5.62 _{Worker.dA_M}
					11.64 _{Worker.dC_M}
ρ₀₁				0.71 _Worker.1	-0.65 _{Worker.dA_M}
					-0.84 _{Worker.dC_M}
Observations	54	54	54	54	54
Deviance	293.703	271.565	227.491	223.545	216.418

# Deviances for models cm
suppressMessages(tab_model(m5, m4cm, m3cm, m2cm, m1cm, 
         dv.labels = c("m5", "m4cm", "m3cm", "m2cm", "m1cm"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4cm	m3cm	m2cm	m1cm
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	4.54	0.93	0.92	0.92
τ₀₀	21.93 _Worker	22.49 _Worker	22.90 _Worker	22.90 _Worker	22.90 _Worker
		6.73 _Worker.1	5.51 _Worker.1	5.62 _Worker.1
			11.43 _Worker.2
τ₁₁				11.64 _{Worker.1.dB_M}	5.62 _{Worker.dA_M}
					11.64 _{Worker.dB_M}
ρ₀₁				-0.71 _Worker.1	-0.65 _{Worker.dA_M}
					0.84 _{Worker.dB_M}
Observations	54	54	54	54	54
Deviance	293.703	271.565	227.491	223.545	216.418

Models m1am, m2am, and m3am did not converge properly. Therefore, the mathematical equalivance of m1am with m1bm and m1cm fails. (Other comparisons involving these LMMs are also invalid.)

6 Sliding contrasts

# max_LMM
# ... no A_C
m1ac <- lmer(score ~ 1 + Machine + (1 + dB_A + dC_B | Worker), Machines, REML=FALSE)
# ... no B_A
m1ba <- lmer(score ~ 1 + Machine + (1 + dA_C + dC_B | Worker), Machines, REML=FALSE)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Hessian is numerically singular: parameters are not
## uniquely determined

# ... no C_B
m1cb <- lmer(score ~ 1 + Machine + (1 + dA_C + dB_A | Worker), Machines, REML=FALSE)

# prsm1_LMM
# ... no A_C
m2ac <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dB_A + dC_B | Worker), Machines, REML=FALSE)
# ... no B_A
m2ba <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_C + dC_B | Worker), Machines, REML=FALSE)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: large eigenvalue ratio
##  - Rescale variables?

# ... no C_B
m2cb <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_C + dB_A | Worker), Machines, REML=FALSE)

# zcp_LMM
# ... no A_C
m3ac <- lmer(score ~ 1 + Machine + (1 + dB_A + dC_B || Worker), Machines, REML=FALSE)
# ... no B_A
m3ba <- lmer(score ~ 1 + Machine + (1 + dA_C + dC_B || Worker), Machines, REML=FALSE)
# ... no C_B
m3cb <- lmer(score ~ 1 + Machine + (1 + dA_C + dB_A || Worker), Machines, REML=FALSE)

# prsm2_LMM
m4ac <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dA_C || Worker), Machines, REML=FALSE)
m4ba <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dB_A || Worker), Machines, REML=FALSE)
m4cb <- lmer(score ~ 1 + Machine + (1 | Worker) + (0 + dC_B || Worker), Machines, REML=FALSE)

# Deviances for contrasts within models 4 and 3
suppressMessages(tab_model(m5, m4ba, m4cb, m4ac, m3ba, m3cb, m3ac,      
                           dv.labels = c("m5", "m4ba", " m4cb", "m4ac", "m3ba", " m3cb", "m3ac"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4ba	m4cb	m4ac	m3ba	m3cb	m3ac
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	7.21	7.07	7.07	7.07	0.93	0.93
τ₀₀	21.93 _Worker	22.20 _Worker	22.21 _Worker	22.21 _Worker	22.21 _Worker	22.90 _Worker	22.90 _Worker
		9.50 _Worker.1	10.03 _Worker.1	10.03 _Worker.1	4.78 _Worker.1	28.81 _Worker.1	28.19 _Worker.1
					5.25 _Worker.2	28.19 _Worker.2	28.81 _Worker.2
Observations	54	54	54	54	54	54	54
Deviance	293.703	287.778	287.202	287.202	287.202	229.569	229.569

# Deviances for contrasts within models 2 and 1
suppressMessages(tab_model(m2ba, m2cb, m2ac, m1ba, m1cb, m1ac,        
                           dv.labels = c(" m2ba", "m2cb", "m2ac", "m1ba", "m1cb", "m1ac"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m2ba	m2cb	m2ac	m1ba	m1cb	m1ac
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	7.07	0.92	0.92	7.07	0.92	0.92
τ₀₀	22.21 _Worker	22.90 _Worker	22.90 _Worker	22.21 _Worker	22.90 _Worker	22.90 _Worker
	21.44 _Worker.1	29.31 _Worker.1	28.69 _Worker.1
τ₁₁	8.80 _{Worker.1.dC_B}	28.69 _{Worker.1.dB_A}	29.31 _{Worker.1.dC_B}	2.42 _{Worker.dA_C}	29.31 _{Worker.dA_C}	28.69 _{Worker.dB_A}
				7.58 _{Worker.dC_B}	28.69 _{Worker.dB_A}	29.31 _{Worker.dC_B}
ρ₀₁	0.74 _Worker.1	0.81 _Worker.1	-0.81 _Worker.1	-0.12 _{Worker.dA_C}	0.77 _{Worker.dA_C}	0.82 _{Worker.dB_A}
				-0.80 _{Worker.dC_B}	0.82 _{Worker.dB_A}	-0.77 _{Worker.dC_B}
Observations	54	54	54	54	54	54
Deviance	287.202	223.545	223.545	285.135	216.418	216.418

#  Deviances for models ba
suppressMessages (tab_model(m5, m4ba, m3ba, m2ba, m1ba, 
         dv.labels = c("m5", "m4ba", " m3ba", "m2ba", "m1ba"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4ba	m3ba	m2ba	m1ba
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	7.21	7.07	7.07	7.07
τ₀₀	21.93 _Worker	22.20 _Worker	22.21 _Worker	22.21 _Worker	22.21 _Worker
		9.50 _Worker.1	4.78 _Worker.1	21.44 _Worker.1
			5.25 _Worker.2
τ₁₁				8.80 _{Worker.1.dC_B}	2.42 _{Worker.dA_C}
					7.58 _{Worker.dC_B}
ρ₀₁				0.74 _Worker.1	-0.12 _{Worker.dA_C}
					-0.80 _{Worker.dC_B}
Observations	54	54	54	54	54
Deviance	293.703	287.778	287.202	287.202	285.135

# Deviances for models cb
suppressMessages (tab_model(m5, m4cb, m3cb, m2cb, m1cb, 
         dv.labels = c("m5", "m4cb", " m3cb", "m2cb", "m1cb"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4cb	m3cb	m2cb	m1cb
Predictors	Estimates	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92	13.92
Random Effects
σ²	9.58	7.07	0.93	0.92	0.92
τ₀₀	21.93 _Worker	22.21 _Worker	22.90 _Worker	22.90 _Worker	22.90 _Worker
		10.03 _Worker.1	28.81 _Worker.1	29.31 _Worker.1
			28.19 _Worker.2
τ₁₁				28.69 _{Worker.1.dB_A}	29.31 _{Worker.dA_C}
					28.69 _{Worker.dB_A}
ρ₀₁				0.81 _Worker.1	0.77 _{Worker.dA_C}
					0.82 _{Worker.dB_A}
Observations	54	54	54	54	54
Deviance	293.703	287.202	229.569	223.545	216.418

# Deviances fir nodels ac
suppressMessages (tab_model(m4ac, m3ac, m2ac, m1ac, 
         dv.labels = c("m5", "m4ac", " m3ac", "m2ac", "m1ac"), 
         show.ci = FALSE, show.icc = FALSE,  show.p = FALSE, show.r2 = FALSE,
         show.dev = TRUE ))

	m5	m4ac	m3ac	m2ac
Predictors	Estimates	Estimates	Estimates	Estimates
(Intercept)	52.36	52.36	52.36	52.36
B	7.97	7.97	7.97	7.97
C	13.92	13.92	13.92	13.92
Random Effects
σ²	7.07	0.93	0.92	0.92
τ₀₀	22.21 _Worker	22.90 _Worker	22.90 _Worker	22.90 _Worker
	10.03 _Worker.1	28.19 _Worker.1	28.69 _Worker.1
		28.81 _Worker.2
τ₁₁			29.31 _{Worker.1.dC_B}	28.69 _{Worker.dB_A}
				29.31 _{Worker.dC_B}
ρ₀₁			-0.81 _Worker.1	0.82 _{Worker.dB_A}
				-0.77 _{Worker.dC_B}
Observations	54	54	54	54
Deviance	287.202	229.569	223.545	216.418

Models m1ba and m2ba did not converge properly. Therefore, the mathematical equalivance of m1ba with m1cb and m1ac fails. (Other comparisons involving m1ba and m2ba are also invalid.)

Related RPubs:

Kliegl, R. (2018a). Reduction of complexity of linear mixed models with double-bar syntax (Version 2)
Kliegl, R. (2018b). Factor contrasts and factor levels in linear mixed models
Kliegl, R. (2018c). Hierarchical random-effect structures in linear mixed models
Kliegl, R. (2018d). Images of Contrast-Dependent Random-Effect Structures in Linear Mixed Models
Kliegl, R. (2018e). Linear Mixed Models with Contrast-Based Measures
Kliegl, R. (2018f). Decompositon of a Linear Mixed Model with Treatment Contrasts
Kliegl, R. (2018g). Decompositon of Linear Mixed Models with Sliding and Sum Contrasts Contrasts

7 Appendix

sessionInfo()

## R version 3.5.1 (2018-07-02)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS  10.14
## 
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] bindrcpp_0.2.2    modelr_0.1.2      forcats_0.3.0    
##  [4] stringr_1.3.1     dplyr_0.7.6       purrr_0.2.5      
##  [7] readr_1.1.1       tidyr_0.8.1       tibble_1.4.2     
## [10] ggplot2_3.0.0     tidyverse_1.2.1   sjPlot_2.6.0     
## [13] RePsychLing_0.0.4 lme4_1.1-18.9000  Matrix_1.2-14    
## 
## loaded via a namespace (and not attached):
##  [1] nlme_3.1-137       lubridate_1.7.4    httr_1.3.1        
##  [4] tools_3.5.1        TMB_1.7.14         backports_1.1.2   
##  [7] R6_2.2.2           sjlabelled_1.0.14  lazyeval_0.2.1    
## [10] colorspace_1.3-2   nnet_7.3-12        withr_2.1.2       
## [13] tidyselect_0.2.4   mnormt_1.5-5       emmeans_1.2.4     
## [16] compiler_3.5.1     cli_1.0.1          rvest_0.3.2       
## [19] xml2_1.2.0         sandwich_2.5-0     effects_4.0-3     
## [22] scales_1.0.0       mvtnorm_1.0-8      psych_1.8.4       
## [25] ggridges_0.5.1     digest_0.6.17      foreign_0.8-71    
## [28] minqa_1.2.4        rmarkdown_1.10.11  stringdist_0.9.5.1
## [31] pkgconfig_2.0.2    htmltools_0.3.6    pwr_1.2-2         
## [34] rlang_0.2.2        readxl_1.1.0       htmldeps_0.1.1    
## [37] rstudioapi_0.7     bindr_0.1.1        zoo_1.8-4         
## [40] jsonlite_1.5       magrittr_1.5       modeltools_0.2-22 
## [43] bayesplot_1.6.0    Rcpp_0.12.18       munsell_0.5.0     
## [46] prediction_0.3.10  stringi_1.2.4      multcomp_1.4-8    
## [49] yaml_2.2.0         snakecase_0.9.2    carData_3.0-1     
## [52] MASS_7.3-50        plyr_1.8.4         grid_3.5.1        
## [55] parallel_3.5.1     sjmisc_2.7.5       crayon_1.3.4      
## [58] lattice_0.20-35    ggeffects_0.5.0    haven_1.1.2       
## [61] splines_3.5.1      sjstats_0.17.0     hms_0.4.2         
## [64] knitr_1.20         pillar_1.3.0.9000  estimability_1.3  
## [67] codetools_0.2-15   stats4_3.5.1       glue_1.3.0        
## [70] evaluate_0.11      data.table_1.11.6  nloptr_1.0.4      
## [73] cellranger_1.1.0   gtable_0.2.0       assertthat_0.2.0  
## [76] coin_1.2-2         xtable_1.8-3       broom_0.5.0       
## [79] survey_3.33-2      coda_0.19-1        survival_2.42-6   
## [82] glmmTMB_0.2.2.0    TH.data_1.0-9

Linear Mixed Models for Variations within Types of Contrast

Reinhold Kliegl

2018-10-02 (revised: 2018-10-02 23:24:02)