Interpretation of lme4 output with different contrasts for variables

I have data assessing reaction time (RT) with 2 variables: Variable 1: 'HighLow', with 2 levels ('High' and 'Low') Variable 2: 'Condition', with 3 levels ('Predicted', 'Implausible', 'Plaus/Unpred')

I am learning about contrasts and have therefore run 2 mixed models that differ based on the contrasts used.

Contrasts for 1st Mixed Model:

HighLow is left as default coding (dummy coded with 'High' representing the intercept). Condition is simple coded with the following contrast matrix:

Predicted     1   1/3  1/3
Implausible   1   -2/3 1/3
Plaus/Unpred  1    1/3 -2/3


The first contrast (cH01) therefore represents Predicted-Implausible, and the second contrast represents Predicted-Plaus/unpred.

Contrasts for 2nd mixed model:

Contrasts for Condition is the same as the first mixed model (simple coded). The contrast for the HighLow variable is sum coded (High=-0.5, Low=0.5).

The mixed model in both cases is

lmer(RT ~ Condition*HighLow+(Condition|Pt_ID) + (Condition|SentNumb)


Fixed Effects Output For Mixed Model 1

      Estimate Std. Error        df
(Intercept)                0.77357    0.03029  84.70182
ConditioncH01             -0.08738    0.01886 115.78438
ConditioncH02             -0.35944    0.03105  99.02833
HighLowLow                -0.01422    0.01439  75.30162
ConditioncH01:HighLowLow   0.05955    0.02043  76.36148
ConditioncH02:HighLowLow   0.17244    0.04049  74.76725
t value Pr(>|t|)
(Intercept)               25.539  < 2e-16 ***
ConditioncH01             -4.634 9.47e-06 ***
ConditioncH02            -11.576  < 2e-16 ***
HighLowLow                -0.988  0.32632
ConditioncH01:HighLowLow   2.915  0.00467 **
ConditioncH02:HighLowLow   4.259 5.90e-05 ***


Fixed Effects Output For Mixed Model 2

Fixed effects:
Estimate Std. Error        df
(Intercept)              0.76647    0.02943  76.32919
ConditioncH01           -0.05760    0.01592  95.00997
ConditioncH02           -0.27322    0.02365 109.77111
HighLow1                -0.01422    0.01439  75.30164
ConditioncH01:HighLow1   0.05955    0.02043  76.36134
ConditioncH02:HighLow1   0.17244    0.04049  74.77345


My interpretation is as follows:

Model 1

(Intercept)   : mean of High
ConditioncH01: Predicted,High-Implausible,High
ConditioncH02: Predicted,High- Plaus/unpred,high
HighLowLow: Low-High
ConditioncH01:HighLowLow: Predicted,low-Implausible,low
ConditioncH02:HighLowLow: Predicted,low- Plaus/unpred,low


Model 2

(Intercept)   : grand mean across all levels of both variables
ConditioncH01: Predicted-Implausible (high and low combined)
ConditioncH02: Predicted- Plaus/unpred (high and low combined)
HighLowLow: Low-High/2
ConditioncH01:HighLowLow: (Predicted-Implausible) for high- (Predicted-Implausible) for low
ConditioncH02:HighLowLow: (Predicted-plaus/unpred) for high- (Predicted-plaus/unpred)low


However, looking at the means (see below) my assumptions do not all seem to be correct? In particular, I am confused about the interpretation of HighLow in both outputs and the interactions (which are the same in both outputs)?

Means

Predicted,High      0.6272715
Implausible,Hig     0.7055791
Plaus/Unpred,High   0.9423626
Predicted,Low       0.6861870
Implausible,Low     0.7123400
Plaus/Unpred,Low    0.8428695

Predicted               0.6564936
Implausible             0.7089290
Plaus/Unpred            0.8896377

High    0.7421962
Low     0.7426768


The unique rows of your fixed effects design matrix under your first contrast specification can be obtained via

library(MASS)

condition <- gl(3, 1, length = 6, labels = c("predicted", "implausible", "plausible"))
contrasts(condition) <- cbind(c(1/3, -2/3, 1/3), c(1/3, 1/3, -2/3))

highlow <- gl(2, 3, labels = c("high", "low"))

X_1 <- model.matrix(~ condition * highlow)
rownames(X_1) <- paste(condition, highlow, sep = "_")
fractions(X_1)
#                  (Intercept) condition1 condition2 highlowlow condition1:highlowlow condition2:highlowlow
# predicted_high      1         1/3        1/3          0          0                     0
# implausible_high    1        -2/3        1/3          0          0                     0
# plausible_high      1         1/3       -2/3          0          0                     0
# predicted_low       1         1/3        1/3          1        1/3                   1/3
# implausible_low     1        -2/3        1/3          1       -2/3                   1/3
# plausible_low       1         1/3       -2/3          1        1/3                  -2/3


Each row of X_1 shows how the corresponding (population) cell mean, indicated by the row name, is a linear combination of the (fixed) regression coefficients.

To interpret the regression coefficients in terms of the cell means, we need the inverse of X_1

fractions(solve(X_1))
#                       predicted_high implausible_high plausible_high predicted_low implausible_low plausible_low
# (Intercept)            1/3            1/3              1/3              0             0               0
# condition1               1             -1                0              0             0               0
# condition2               1              0               -1              0             0               0
# highlowlow            -1/3           -1/3             -1/3            1/3           1/3             1/3
# condition1:highlowlow   -1              1                0              1            -1               0
# condition2:highlowlow   -1              0                1              1             0              -1


Now, each row shows the representation of the corresponding regression coefficient as a linear combination of the cell means.

This confirms your output interpretation except for the estimated interaction coefficients, which (reading off from the corresponding rows) are estimates of \begin{align} &-1 \cdot \mu_\text{predicted, high} + 1 \cdot \mu_\text{implausible, high} + 1 \cdot \mu_\text{predicted, low} -1 \cdot \mu_\text{implausible, low} \\ &= \left(\mu_\text{implausible, high} - \mu_\text{predicted, high} \right) - \left(\mu_\text{implausible, low} - \mu_\text{predicted, low} \right) \end{align} and \begin{align} &-1 \cdot \mu_\text{predicted, high} + 1 \cdot \mu_\text{plausible, high} + 1 \cdot \mu_\text{predicted, low} -1 \cdot \mu_\text{plausible, low} \\ &= \left(\mu_\text{plausible, high} - \mu_\text{predicted, high} \right) - \left(\mu_\text{plausible, low} - \mu_\text{predicted, low} \right), \end{align} respectively.

The output for your second contrast specification

contrasts(highlow) <- c(-0.5, 0.5)

X_2 <- model.matrix(~ condition * highlow)
rownames(X_2) <- paste(condition, highlow, sep = "_")
fractions(X_2)
#                  (Intercept) condition1 condition2 highlow1 condition1:highlow1 condition2:highlow1
# predicted_high      1         1/3        1/3       -1/2     -1/6                -1/6
# implausible_high    1        -2/3        1/3       -1/2      1/3                -1/6
# plausible_high      1         1/3       -2/3       -1/2     -1/6                 1/3
# predicted_low       1         1/3        1/3        1/2      1/6                 1/6
# implausible_low     1        -2/3        1/3        1/2     -1/3                 1/6
# plausible_low       1         1/3       -2/3        1/2      1/6                -1/3

fractions(solve(X_2))
#                     predicted_high implausible_high plausible_high predicted_low implausible_low plausible_low
# (Intercept)          1/6            1/6              1/6            1/6           1/6             1/6
# condition1           1/2           -1/2                0            1/2          -1/2               0
# condition2           1/2              0             -1/2            1/2             0            -1/2
# highlow1            -1/3           -1/3             -1/3            1/3           1/3             1/3
# condition1:highlow1   -1              1                0              1            -1               0
# condition2:highlow1   -1              0                1              1             0              -1


shows that your interpretation of the coefficient estimate HighLowLow (which I interpret as HighLow1) is off by a factor of $$2$$ (it should be the same as with the first contrast specification) and your interpretation of the estimated interaction coefficients is off by a factor of $$-1$$ (it should, again, be the same as with the first contrast specification).