In relation to my previous question on different output results and their interpretation based on a model with or without an interaction term, this is a follow up question on how to report such results.
Based on the following model formula and its output (see below), am I correct at interpreting this no-interaction model as follows;
For treatment1 at time6, we have amp.sqrt = 115.184
For treatment2 at time6, we have amp.sqrt = 115.184 + 2.644
For treatment3 at time6, we have amp.sqrt = 115.184 + 23.365
For treatment1 at time7, we have amp.sqrt = 115.184 + 13.958
For treatment2 at time7, we have amp.sqrt = 115.184 + 13.958 + 2.644
For treatment3 at time7, we have amp.sqrt = 115.184 + 13.958 + 23.365
etc..
I had thought of reporting the results as: "there was a positive effect of treatment, with amp.sqrt increasing from treatment 1, to 2 (2.644) and 3 (23.365). Also a positive effect of time was observed, increasing from time 1 to 2 (13.958) and 3 (21.799). The main effect of axis increased from 1 to 2 and 3 as well."
However, now that I have my model with the interaction term, and the main effects on their own seem strange to report (as I wrote above), should I instead report the overall trends as seen by your plot (i.e. adding the fixed effects and the interaction term and then presenting that value) or should I still present the results as above, even if the main effect outputs on their own are counterintuitive unless specified in relation with the interaction term results?
(NO-INTERACTION TERM MODEL)
mTEST1<- lmer(amp.sqrt~ time + treatment + axis + (1+treatment|ID))
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 115.184 7.546 36.300 15.265 < 2e-16 ***
time7 13.958 4.707 474.800 2.965 0.00318 **
time8 21.799 4.787 478.500 4.554 6.7e-06 ***
treatment2 2.644 8.571 18.400 0.308 0.76117
treatment3 23.365 6.139 19.200 3.806 0.00117 **
axis2 60.458 4.746 474.800 12.737 < 2e-16 ***
axis3 128.456 4.746 474.800 27.063 < 2e-16 ***
---
(INTERACTION-TERM MODEL)
mTEST2<- lmer(amp.sqrt~ time * treatment + axis + (1+treatment|ID))
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 130.587 8.417 55.500 15.515 < 2e-16 ***
time7 -7.697 8.120 471.000 -0.948 0.3436
time8 -2.628 8.120 471.000 -0.324 0.7464
treatment2 -3.766 10.713 44.500 -0.352 0.7269
treatment3 -14.929 8.851 83.600 -1.687 0.0954 .
axis2 60.458 4.569 471.000 13.232 < 2e-16 ***
axis3 128.456 4.569 471.000 28.113 < 2e-16 ***
time7:treatment2 9.697 11.206 471.000 0.865 0.3873
time8:treatment2 8.554 11.396 473.700 0.751 0.4532
time7:treatment3 53.206 11.206 471.000 4.748 2.73e-06 ***
time8:treatment3 62.411 11.289 473.300 5.528 5.35e-08 ***
Best Answer
First of all, yes, you are correct in the way you are interpreting the fixed effects.
However, note that we are only dealing here with fixed effects. Your model also has random effects, and in particular you have random coefficients for
treatment
which means that each subject has their own individualtreatment
effect. The calculations with the fixed effects therefore represent averages across all subjects.Although the findings are largely the same, I would present your findings based on the model with the interactions. We can view the model with no interactions with this plot:
While the model with interactions looks like this:
Formally, you can do a likelihood ratio test using the
anova()
function to test which model is better (you will have to re-run your models using theREML=FALSE
option because likelihood-based methods cannot be used to compare models with different fixed effects).I would focus on treatment 3 being associated with higher values of
amp.sqrt
at time 7 and further at time 8, and these differences are also statistically significant at the 5% level. Treatment 3 is associated with lower values at time 6 (although this difference is not statistically significant at the 5% level). Also, there is very little differences between treatments 1 and 2 at all time points (and these are also not statistically significant). Moreover there appears to be no time trend for treatments 1 and 2. You might be interested in a test of whether treatment 2 is different between years 6 and 8 since there is a small upward trend. Personally this looks negligible to me, compared with treatment 3 but if you wanted to test this you could use a post-hoc test such as Dunnett's.