How to interpret lme() outcome with a mix of cat/non cat variables

Question

I apologize in advance for my question that can seem redundant, but I am still struggling to interpret the outcome of my lme model, as the other posts mainly deal with several categorical variables and I get really confused ...

I have three continuous variables, Delta (my measurement), Date, and Light and two categorical variables, Identity (names) and Treatment applied to the subjects, which has 3 non ordered levels.

I understand how to interpret betas when the parameters are all categorical/all continuous - but the mix makes me perplex.

I use the lme model:

model.lme = lme(Delta ~ Treatment + Light * Date, 
random =~1|Identity, na.action = na.omit,
control = lmeControl(optimum ="opt"))

and I get :

Random effects:
 Formula: ~1 | ID
        (Intercept)  Residual
StdDev:  0.08049669 0.3757045

Fixed effects: Delta ~ Treatment + Light * Date 
                  Value  Std.Error     DF    t-value p-value
(Intercept)  0.27590485 0.03358550 350400   8.214999  0.0000
Treatment2  -0.06606907 0.04418992 350400  -1.495116  0.1349
Treatment3  -0.01027265 0.04861505     16  -0.211306  0.8353
Light        0.00088466 0.00009181 350400   9.635306  0.0000
Date        -0.00083441 0.00004336 350400 -19.241967  0.0000
Light:Date   0.00000134 0.00000050 350400   2.673734  0.0075
 Correlation: 
           (Intr) Trtmn2 Trtmn3 Light  DOY   
Treatment2 -0.709                            
Treatment3 -0.654  0.490                     
Light      -0.109  0.000  0.000              
DOY        -0.235  0.004  0.007  0.459       
Light:DOY   0.109  0.000  0.000 -0.995 -0.462

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-5.28753179 -0.23911026 -0.09499650  0.05150716 43.17908373 

Number of Observations: 350422
Number of Groups: 18

If I am right, Intercept is the base effect, the value of Delta when the others parameters = 0. But somehow in what I have read I understand that I should take intercept as Treatment1 ?

Does it mean that Treatment1 is taken as a reference to which other parameters are compared ? It happens that Treatment1 is a control condition, but I am not sure whether or not this is scientifically valid as I would like all treatments to be considered 'equals'. If this is not right, what exactly is Treatment1 here ?

Also, should I interpret the value as a linear relation, as the model assumes so ? eg. Date has a significant effect on Delta, and its value is negative. So the further we are in time the lower is Delta ? Is it correct to say that Delta decreases following a slope of 0,0008 ? Or is it more accurate to say that the slope is (0,2759 - 0,0008)?

I guess that if I have no proof that there is a linear relation (which is not very likely here I think) the only thing I can say with this model is 'Delta is significantly affected by Light and Date' ?

Answer 1

Is there any particular reason why you use lme and not lmer ? The latter is more recent and better in many situations.

You are correct in your interpretation of the intercept. To avoid confusion I will refer to this as the global intercept, since your model also has random intercepts. To be slightly more formal about it, the global intercept is the mean value of Delta when the numerical covariates are zero and the categorical covariates are at their reference level(s), in your case when Treatment is 1, for the "average" group (ID). However, each group has it's own intercept, which will be an offset from the global intercept.

The estimate for Treatment2 is the difference in the mean of Delta when treatment is 2, compared to when it is 1, for observations within the same group and holding the other covariates constant. Likewise for Treatment3. If you want to compare Treatment2 with Treatment3 then you could recode your data with either of those as the reference level instead, using the relevel function.

This is a linear model, so, since the estimate for Date is negative this means there is a linear association between Delta and Date, where the estimate for Date is the slope.

Note that in the presence of an interaction, the main effects of the variables do not have the usual interpretation. For numerical variables, the main effect is the association of that variable with the outcome when the other variable is zero. With numerical variables this often makes no sense (when zero is not a plausible and/or when zero falls far outside the observed range), so centering the data makes interpretation more sensible. In your case, the estimate for the interaction is several orders of magnitude smaller than the main effects and not significant at the 0.05 level, so you could consider removing it.

Your statement "Delta is significantly affected by Light and Date" I would alter to "There is a statistically significant linear association between Delta and both Light and Date" and quote the confidence intervals for each estimate. Try to avoid causal language such as "Y is affected by X". If you have a research hypothesis that such a causal effect exists then you can say that your model supports this hypothesis.

If you think there might be nonlinear associations, then you could investigate this by including higher order terms, such as Date^2 and Date^3 in addition to the linear term. Plotting the data before modelling it is good practice and often informs possibly non-linearities.

You should also check the model assumptions, particularly that the residuals and random intercepts are plausibly normally distributed, homoscedastic, and without autocorrelation.