# Intercept in mixed model with multiple multilevel factors

I am using a mixed model to predict the effect of certain environmental exposures on brain region measures. So temp[, j] gives the brain region our regressor and temp[, var] gives the environmental variable of the predictor. Height, weight, intracranial volume, age are controlled for and random effects for family relation and scanner site are controlled for. The terms that matter are temp[,var] and its interactions with sex and race_ethnicity.

fml<-lmer(temp[,j]~ interview_age+rel_relationship+ehi_y_ss_scoreb+
scale(smri_vol_scs_intracranialv)+
(1|mri_info_deviceserialnumber)+
(1|mri_info_deviceserialnumber:rel_family_id)+
anthroheightcalc + anthroweightcalc + sex + race_ethnicity +
temp[,var] + temp[,var]:sex + temp[,var]:race_ethnicity,
data = temp)


We are interested in how race as a social construct and gender may influence them so they have been modelled. sex has two levels race_ethnicity in our case has 3 levels. The coefficients for the fixed effects are:

> coef(summary(fml))
Estimate Std. Error        df    t value  Pr(>|t|)
(Intercept)                       4207.6374706 80.6146186 8102.8607 52.1944722 0.0000000
interview_age                       -0.6941963  0.5300313 8904.7643 -1.3097271 0.1903220
rel_relationship                    -2.8646617  4.6742776  219.7843 -0.6128566 0.5406048
ehi_y_ss_scoreb                     -1.7937162  5.0437752 9048.6169 -0.3556297 0.7221262
scale(smri_vol_scs_intracranialv)  280.7948583  3.5021661 9074.3988 80.1774825 0.0000000
anthroheightcalc                    -0.5141972  1.6238005 8889.1745 -0.3166628 0.7515069
anthroweightcalc                     0.1612428  0.2037307 8738.9575  0.7914507 0.4287025
sexM                                 4.4027100 12.5976755 9078.9926  0.3494859 0.7267327
race_ethnicity2                      5.0497778 22.2093470 5893.2108  0.2273717 0.8201426
race_ethnicity3                     14.9754508 16.2265245 2172.1264  0.9228995 0.3561621
temp[, var]                          0.1745047  0.2581049 1963.3899  0.6760999 0.4990568
sexM:temp[, var]                    -0.3622916  0.2689064 9076.3253 -1.3472776 0.1779245
race_ethnicity2:temp[, var]         -0.1747920  0.3647274 7669.2146 -0.4792401 0.6317815
race_ethnicity3:temp[, var]         -0.2288939  0.3619681 4730.3581 -0.6323591 0.5271828


What does the intercept represent in this case? Is it the the mean of temp[, var] at sex F and race_ethnicity 1? Then the coefficients are deviations from this grand mean?

I think the answers in the posts here and here are relevant but given the more complex design of my model I think this warrants a separate answer.