How do I interpret these linear mixed model coefficients from r?

Question

I've fitted a mixed model with participants and vowels as random factors and language (Tamil and French) as the fixed factor. The dependent variable is durations of prolongations (of a phoneme). The question I'm trying to answer: are the mean durations of prolongations in the two language significantly different from one another? I considered a T-test, but realized that it may not capture the effects of variations in the population or even vowels (some vowels in english or german are more easily prolonged than consonants, for example).

model1<-lmer(log(Duration) ~ Language + (1|Participant) + (1|Vowel), data = lmmdf, REML = TRUE)
summary(model1)

Output:

Linear mixed model fit by REML. t-tests use Satterthwaite's  method
 [lmerModLmerTest]
Formula: log(Duration) ~ Language + (1 | Participant) + (1 | Vowel)
   Data: lmmdf
REML criterion at convergence: 363.8
Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-3.03238 -0.65717 -0.05817  0.67908  2.42541
Random effects:
 Groups      Name        Variance Std.Dev.
 Vowel       (Intercept) 0.030179 0.17372 
 Participant (Intercept) 0.009809 0.09904 
 Residual                0.178592 0.42260 
Number of obs: 297, groups:  Vowel, 26; Participant, 18
Fixed effects:
              Estimate Std. Error       df t value
(Intercept)   -1.43762    0.07990 21.75516 -17.992
LanguageTamil  0.03926    0.09777 20.19149   0.402
                        Pr(>|t|)

(Intercept)   0.0000000000000151 ***
LanguageTamil              0.692

Signif. codes:  0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
            (Intr)
LanguageTml -0.667

I can see that Language is not a statistically significant predicting factor for Prolongations. But how do I interpret the estimate of 'LanguageTamil'(0.3926) when it is a categorical variable (French or Tamil) and there is no "1 unit increase" between the two. Does this actually mean that when the participant spoke in Tamil, they prolonged 0.03926ms longer than when they did in French (but not stat significant)? Also, what does the negative sign in the 'Correlation of Fixed Effects' mean? (-0.667).

Answer 1

how do I interpret the estimate of 'LanguageTamil'(0.3926) when it is a categorical variable (French or Tamil) and there is no "1 unit increase" between the two.

Categorical variables get converted into numerical variables, one for each category, with values zero or one depending on the category of the specific row.

For example

$$\begin{array}{c|c|c|c} \text{Observation id} & \text{Language} & \text{LanguageTamil} & \text{LanguageFrench} \\ \hline 1 & \text{French} & 0 & 1\\ 2 & \text{Tamil} & 1 & 0\\ 3 & \text{French} & 0 & 1\\ 4 & \text{French} & 0 & 1\\ 5 & \text{Tamil} & 1 & 0\\ \vdots & \vdots & \vdots & \vdots\\ n & \text{French} & 0 & 1\\ \end{array}$$

These variables are also called dummy variables.

I your regression only one of the dummy variables is used. The value of the category French coincides with the intercept, and the value of Tamil coincides with the intercept plus the coefficient for the dummy variable LanguageTamil.

Related: Why does glm in R create new variables upon training?

Also, what does the negative sign in the 'Correlation of Fixed Effects' mean? (-0.667).

The estimates will differ for different experiments, and also these differences correlate. That is, the errors in the coefficients are not independent. In your case it is a negative correlation and the estimates of the intercept and the effect have a negative correlation. If the estimate of the intercept is too high then the estimate of the coefficient for Tamil will relatively often be too low.

In this question Why and how does adding an interaction term affects the confidence interval of a main effect? you see an example of regression where the estimates of the coefficients correlate.

The ellipse shows how the parameter estimates are estimated to be distributed. In the left image they have some correlation.

This is also a reason why the application of a t-distribution in order to estimate the significance is not always so great. On the other hand, in the image you see that this relates mostly to the intercept and the error of the slope parameter is the same in both left/right images.