This is a beginner level question, and I apologise in advance if the answer is obvious.
If a by-subject random slope for a predictor "X" improves the model fit*, but the predictor "X" is not significant (with and without the by-subject random slope for predictor "X"), is there any point in adding the random slope for predictor "X"? The reason why I am asking this is because I always thought that the point of adding random slopes was to minimise the risk of Type I error, but there is no risk of Type I error with regards to predictor "X" if it is not significant in the first place.
*as judged by the pairwise comparisons using the LRT
Here is an illustration of what I mean:
ordinal_Eiii <- clmm(Understanding ~ zAge + EnglishProficiency + zSTOFHLA+Education + zFRES + zRDL2 + (1|Text) + (1|Subject), data = Study2.pairs, link = "probit", threshold = "flexible")
In the above model:
Estimate Std. Error z value Pr(>|z|) zRDL2 -0.02217 0.13923 -0.159 0.87350
After the addition of a random slope:
ordinal_Eii <- clmm(Understanding ~ zAge + EnglishProficiency + zSTOFHLA+Education + zFRES + zRDL2 + (1|Text) + (zRDL2 + 1|Subject), data = Study2.pairs, link = "probit", threshold = "flexible")
Estimate Std. Error z value Pr(>|z|) zRDL2 -0.18864 0.18454 -1.022 0.30669
Pairwise comparisons using the LRT:
no.par AIC logLik LR.stat df Pr(>Chisq)
ordinal_Eiii 19 1440.1 -701.06
ordinal_Eii 21 1430.6 -694.31 13.503 2 0.001169 **
In addition to the above, the inclusion of the random slope for zRDL2 changes the estimates of the other predictors. Thus, is the inclusion of random slope for zRDL2 warranted since it decreases the risk of committing a Type I error with regards to the remaining predictors? Am I understanding this right?