# lmer model interaction with 1 intercept and 1 random effect

I am very new to statistical analysis, but I am trying to understand the mixed-effect model.

The data that I have is the mass volume of different rats across different days. Each rat has different time points where they took the measurement of that volume. There are 20 rats with volume measurements and two groups; ten rats came from Chile and ten from England.

I would like to assess if there is an effect on inter variability of the rats and check how the growth is behaving (if it is slower on the rats in Chile and faster in England or vice-versa) :

m1 < - lmer(lVolume ~ Country * Day + (1 | Rat))

Linear mixed model fit by REML ['lmerMod']
Formula: lVolume ~ Country * Day + (1 | Rat)

REML criterion at convergence: 117.7

Scaled residuals:
Min      1Q  Median      3Q     Max
-3.6272 -0.3786  0.0727  0.5431  1.8786

Random effects:
Groups    Name        Variance Std.Dev.
Rat       (Intercept) 0.23205  0.4817
Residual              0.07364  0.2714
Number of obs: 145, groups:  Rat, 20

Fixed effects:
Estimate Std. Error t value
(Intercept)            1.905378   0.256483   7.429
CountryEngland         1.374471   0.321670   4.273
Day                    0.040673   0.001894  21.474
CountryEngland:Day    -0.005562   0.002354  -2.363

Correlation of Fixed Effects:
(Intr) CtyEngland Day
CtyEngland   -0.797
Day          -0.794  0.633
CEngland:     0.638 -0.723       -0.805


But I am really confused about how to interpret these values.

Does the 0.23205 that came from the random effect on rats mean that there is significance inter population variability that affects the growth?

What does it mean the values on the fixed effect values?

In addition, I plotted the model:

plot_model(m1, show.values = TRUE, value.offset = .3)



Do the values mean significance in terms of the fixed or random effect or perhaps both?

Could I have some feedback for this, please?

Thanks.

Firstly, you may be able to make more sense of your model, at least the fixed effects, by plotting predicted values rather than coefficients. The plot_model function you use can do this.

Interpreting the fixed effects:

1. The intercept is the average volume of Chilean rats at baseline (day 0).
2. The coefficient CountryEngland says that, at baseline (day 0), rats from England have on average 1.37 units greater mass than rats from Chile.
3. The coefficient for Day tells you that Chilean rats, on average, increase their volume by 0.04 units per day.
4. Lastly, the CountryEngland:Day interaction coefficient tells you that the effect of Day (the rate of change of Volume) is on average -0.005 units lower amongst English rats compared to Chilean rats.

Interpreting the random effects is a little more complicated. The standard deviation of the random effect Rat, 0.48, tells you how much between-rat variation there is in Volume, after accounting for your predictors. In other words, conditional on your predictors, you would expect most (68%) rats to be within around $$\pm$$ 0.48 units of volume.

The statistical significance of this variation I don't think is really relevant - but consider that the standard deviation of the 'between-rat' variation (0.48) is considerably larger than the 'within-rat' variation (0.27), suggesting most variation in Volume is at the between-rat level.

Note that your model can also be extended by allowing the rate of change in Volume to vary between rats. This would introduce a random effect of Day:

m1 < - lmer(lVolume ~ Country * Day + (1 + Day | Rat))


This way, you will also be able to estimate the amount of variation in the rate of change of volume between rats. Whether or not that's valuable for your experiment I don't know.

• Hi @Lachlan, first of all, thanks for the clarification, it is very helpful. I am trying to digest the interpretation. Regarding the plot_model(). I updated the question. I plotted (plot_model() as you suggested before)before but I interpreted this plot as the fixed effect intercept of both countries. Telling me that the growth rate of the mice in the country Chile is slower than the mice in England. Is this correct? In addition, why in the last plot CountryEngland is in red with a negative -0.01 and not confidence interval as CountryEngland (1.37***)Do these values represent something else? Feb 28, 2022 at 11:23
• Not a problem. I would interpret the plot as saying that English mice start with a larger volume but tend to grow at a slightly slower rate than Chilean mice (you can see that by day 180, there is only a very small difference between the two). For your second question, I think the -0.01 you're referring to is the coefficient for the interaction. This tells you that the rate of growth is -0.01 units slower in English mice than Chilean ones. Mar 2, 2022 at 0:49
• Thank you very much @Lachlan ! Last question if I may, the 68% you mentioned is because of the ICC estimated by the model? The thing is that I don't get the difference in "between-rat" and "within-rat". The between I would expect that is the variability that the tumour growth rat of each rat is bringing to the model, but within I do not understand it. Sorry... The ICC for my original model is Adjusted: 0.759 Conditional 0.229 and for the model you suggested Adj 0.924 Cond: 0.309. So in the second it is quite high, does this suggests growth curves are flat and there is little change over time? Mar 2, 2022 at 10:46
• I would recommend reading the first four chapters of the book 'Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence' by Willett and Singer. This will help you make sense of the random effects Mar 3, 2022 at 1:34