I'm just responding in case this might be useful for anyone else.
Turns out, I still have not found a clear resource that distinguishes between these two options in a theoretical manner - i.e. distinguishes between them not as an optimization but as being different model structures that are best suited for capturing different kinds of experimental designs.
Ideally, you should be able to rely upon your knowledge of the properties / structure of the study / data sample / data elicitation technique - and, make decisions about the random effects model before you have looked at the data.
After all, random effects should be determined a priori based on the theoretical assumptions.
That part of the question remains unanswered in a satisfactory manner
What are the theory-driven / experimental design based features that should allows us to determine whether Option 1 or Option 2 is the appropriate choice.
OPTION 1: random slope-intercept with no constraints
(G)LMER---|[ m.1 ]|---[Y ~ 1 + B + (1 + B | A)]|
OPTION 2: random slope-intercept with uncorrelated random effects
(G)LMER---|[ m.2 ]|---[Y ~ 1 + B + (1 | A) + (0 + B | A)]|
PARTIAL SOLUTION IN PRACTICE
However, I did find a a tutorial by Douglas Bates that may help. Around slide 73 onwards, he covers this topic. Essentially, this response is inspired by and often reproduces those slides. If you would like more detail, head there.
1. Inspect Your Random Effects Plots
Bates suggest that if visual inspection of the data plots gives you "little indication of a systematic relationship between a subject’s random effect for slope and his/her random effect for the intercept," we may want to consider using a model with uncorrelated random effects.
2. MODEL COMPARISON
2(a) Build Option 2 from above
First, we construct the model with the uncorrelated random effects. To express this we use two random-effects terms with the same grouping factor and different left-hand sides.
TWO GROUPING FACTORS:
- (1 | A)-----------[Random Intercept]
- (0 + B | A)-------[Random Slope, no intercept]
- Since the distinct random effects terms are modeled as being independent, by design, this imposes the constraint that the random
intercept (1) from above is independent of the slope (2) conditional
2(b) Compare the models using ANOVA
Using ANOVA for model comparison
Model m.1 contains m.2 in the sense that:
If the parameter values for model m.1 were constrained so as to force the correlation (and, hence the covariance) to be zero and we could get the model to re-fit, we would get m.2
Use a likelihood ratio test to determine if m.1 adds something substantial and statistically significant;
If not, use the preference for parsimonious models (i.e. "smaller is better") and prefer the simpler, more constrained model;
Since the value 0 to which the correlation is constrained is not on
the boundary of the allowable parameter values, a likelihood ratio test and a reference distribution of a χ2 on 1 degree of freedom is suitable.
3. Likelihood ratio tests on variance component
As for the case of a covariance, we can fit the model with and
without the variance component and compare the quality of
The likelihood ratio is a reasonable test statistic for the comparison but the “asymptotic” reference distribution of a χ2 does not apply because the parameter value being tested is on the boundary.
The p-value computed using the χ2 reference distribution should be conservative (i.e. greater than the p-value that would be obtained through simulation).
4. References and Resources
For additional resources, please see the following two useful links:
Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random
effects structure for confirmatory hypothesis testing: Keep it
maximal. Journal of Memory and Language, 68(3)
Also, search the subsequent article by Barr ("Random effects
structure for testing interactions in linear mixed-effects models")
if you are testing any interactions in your models).