How can we predict differences in a slope, given that the random component was significant?

Using R lme function, I found that both fixed and random effects of
variable
A on variable B are significant.  

It would be good if you could tell us how you found out that "the random
effects" were significant. I must have missed something here.

Now, I'd like to analyze what variables are predicting differences in the
slope.  

In the slightly modified standard example coming with lme, the line
age:SexFemale tells us that "girls grow slower".

Dieter

library(nlme)
fm2 <- lme(distance ~ age * Sex, data = Orthodont, random = ~ 1)

Fixed effects: distance ~ age * Sex 
              Value Std.Error DF t-value p-value
(Intercept)    16.3      0.98 79    16.7   0.000
age             0.8      0.08 79    10.1   0.000
SexFemale       1.0      1.54 25     0.7   0.508
age:SexFemale  -0.3      0.12 79    -2.5   0.014

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