R - lme4 - Help on repeat measures over time, continuous predictor interactions

Question

I've created an exmaple data set below to hopefully help answer a question in regards to a lme4 in R and measurements over time.

The data is 4 plots, measured over 4 years, with the dependent variable being "Yield". The first year, the level of disturbance ("Dist") was measured, and never changed over all three years.

I want to know if "Dist" was a significant predictor of Yield over the three years measured.

dat<-data.frame(Year=c(1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4), 
            Plot=c("P1","P2","P3","P4","P1","P2","P3","P4","P1","P2","P3","P4","P1","P2","P3","P4"), 
            Yield=c(8,12,20,2,6,6,7,1,1,2,2,1,0.5,1,1.5,0.5), 
            Dist=c(10,20,40,5,10,20,40,5,10,20,40,5,10,20,40,5))

As I understand it, using lme4 in R, I should be able to do as follows:

d1<-lmer(Yield ~ Dist*Year+(1+Year|Plot), data=dat)

And using the lmerTest package, determine the significance of each predictor:

library(lmerTest)
summary(d1)

Random effects:
Groups   Name        Variance Std.Dev. Corr 
Plot     (Intercept) 3.8282   1.9566        
      Year        0.2654   0.5152   -1.00
Residual             4.9637   2.2279        
Number of obs: 16, groups:  Plot, 4

Fixed effects:
        Estimate Std. Error       df t value Pr(>|t|)  
(Intercept)  2.73913    2.88665  2.32566   0.949   0.4305  
Dist         0.52391    0.12524  2.32566   4.183   0.0403 *
Year        -0.55217    0.96433  3.75418  -0.573   0.5995  
Dist:Year   -0.14322    0.04184  3.75418  -3.423   0.0295 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
      (Intr) Dist   Year  
Dist      -0.813              
Year      -0.927  0.754       
Dist:Year  0.754 -0.927 -0.813

Would I be correct in interpreting that this means at year 1, "Dist" was significant, and over time the effect of "Dist:Year" is significantly decreasing?

Thanks!

Answer 1

I always like view the equations implied by the R syntax. So let's start with the level 1 equation. In your model, this represents the average effect of time. You allow random effects though for your intercept and slope - which is basically saying that sure there is an overall effect of time (year) but the properties of change varying across each plot. (Side note I recommend giving 0 a meaningful value in your time variable. It gives your grand intercept meaning. Currently your (Intercept) is an estimate for Yield when Year is equal to 0, which does not occur in your data.

So in notation Raudenbush & Bryk (2002) notation you have:

$$ Yield_{ti}= \pi_{0i}+\pi_{1i}(Year_t - 1) + e_{ti} $$

I added the $- 1$ so as a way of noting that you should center the time variable at 0 It is fairly common practice in these linear growth models to center time at the initial time point, but there may be reasons to center at other time points.

Now we add in the level 2 equations.

$$ \pi_{0i} = \beta_{00}+\beta_{01}(Distance_i-\bar{Distance})+r_{0i} $$

In your model above, $\beta_{00} =$ 2.793 and $\beta_{01} =$ 0.524. Here is another place in the model where the value of centering becomes apparent again. It gives $\beta_{00}$ a more obvious interpretation. It is the predicted intercept in your model when Dist = mean(Dist). The coefficient $\beta_{01}$ then represents the expected change in the grand intercept as a function of a one-unit change in distance scores. If you had centered your time variable at 0, you can specifically tie this back to the expected difference in Yield score at the first year of observation as a function of Dist.

Now for the last equation implied by your model.

$$ \pi_{1i} = \beta_{10} + \beta_{11}(Distance_i-\bar{Distance}) + r_{1i} $$

As before, centering improves interpretation here as your estimate for $\beta_{10}$ would then represent the expected change in the Year slope for a plot with an average distance score. From your model, $\beta_{10} =$ -0.552 and $\beta_{11} =$ -0.143.

So all of this is to show how your variables in your model interact and the interpretations you can make from them. Those interpretations would be greatly helped if 0 in each of your variables has a meaning. There are two components to your top level model - an intercept and a slope with random effects. You then evaluate the degree to which your plot-level variable of Dist predicts changes in these top-level parameters.