# Understanding the output of lme

I am having some difficulties interpreting the results of an analysis perfomed using lme. I conducted an experiment where the subjects had to estimate the time elapsed in a task involving a spatial measure (e.g. subjects watched a video game where a car travels a certain distance). My goal is to determine if there is a linear relation between perceived time and the space traveled by the car. Subject underwent 2 different conditions (e.g. the car was traveling two distances), and each condition was repeated twice. Therefore, given the nature of the experimental design involving repeated measures, I cannot use a simple linear model but I have to use a mixed effects linear model with a random intercept for subject.

I use the R language, and I adopted this formula to solve my problem

library(nlme)
summary(lme(Time ~ Distance, data = my_Table, random = ~1 | Subject))


The output that I get is:

Linear mixed-effects model fit by REML
Data: my_Table
AIC      BIC    logLik
608.315 618.4454 -300.1575

Random effects:
Formula: ~1 | Subject
(Intercept) Residual
StdDev:    2.964139 4.919044

Fixed effects: Time ~ Distance
Value Std.Error DF  t-value p-value
(Intercept)                            5.518714 0.8212930 64 6.719543  0.0000
Distance                              0.013092 0.0053225 64 2.459718  0.0166
Correlation:
(Intr)
Distance                              -0.415

Standardized Within-Group Residuals:
Min         Q1        Med         Q3        Max
-1.2904862 -0.4302117 -0.2593047  0.2081889  5.0914269

Number of Observations: 95
Number of Groups: 30


Now, what is this output telling me? As far as I understand from studying the literature that I found online (I am not a statistician...), if I am rght these results are telling me that there is a linear relation between time and space, since b = 0.013092 and p-value is significant.

Now, if the p-value was not significant, this would have meant that there is no linear relation between time and space?

-