I have been using mixed effect models to analyzing neuroimaging datasets with multiple scanning sessions per participant. All my previous models only included a random intercept, and it was not until recently that I heard about random slopes.
I am still uncertain about when I should be using a random intercept and slope in my mixed effect models? Any practical or theoretical insight would be greatly appreciated.
This is a general question, but for the sake of example lets say I am testing a hypothesis that there will be a negative relationship between total brain volume and depression severity symptoms (both variables are numeric). In this case, I am trying to control for age at scan since there are inconsistent intervals between scan sessions across all participants. Therefore, should I include a random intercept and random slope of age?
gamm4(GreyMatterVol ~ s(AgeAtScan, k=4) + DepressionSeverity, random=as.formula(~(1|sub), data=alltimepoints, REML=T)$gam
I am assuming that you specifically want to compare a standard random intercept model (participants allowed their own intercepts only) to a standard random coefficient model (participants are allowed their own intercept AND slope). In other words we are not talking about other types of models (e.g. shared intercept, but random slope)
There are two basic approaches to choosing between these two models.
The first approach (purely empirical) is to compare the model fit of a random intercept model to that of a random coefficients model (i.e. random intercepts AND random slopes) using something like AIC or BIC. The advantage of these measures (and other IC statistics) is that they penalize for model complexity. The basic idea is that if the AIC (or BIC) is lower for a random coefficient model compared to a random intercept model then the gain in fit is worth the extra model complexity.
Personally, I think this approach alone is a little too 'black box'. I like to think about the problem in terms of the data (and research problem itself). For a random intercept model we are assuming that participants are allowed to have their own baseline values (and these are normally distributed around the average intercept). BTW remember this is AFTER we account for differences in participants due to treatment group and/or risk factors (i.e. we can assume the fixed effect predictors don't interfere with these decisions). The subtext for the random intercept model is that we assume (despite there different starting points) that participants respond to time in exactly the same way (in other words, all of the individual participants' regression lines are parallel). So the important question is: Do we think such an assumption is valid? If we assume that participants may respond to time differently (e.g. different patients may respond to therapy in a different way...remember though, beyond that explain by the fixed effect predictors in the model).To summarize, if you had to fit a regression line individually for each participant, would you constrain them to a share slope (random intercept) or not (random coefficient). In other words, have look at your spaghetti plot.
Hope this helps
There are some limitations when the number of measurements per person is low ( https://www.frontiersin.org/articles
/10.3389/feduc.2017.00058/full ). I see in your example that you are treating AgeAtScan with a spline. I assume that is the variable that you want to have random slopes for. Would this be for all effects of it? If so, it would be important to check that all have realistic variances (i.e., not near zero or huge).
Kelvyn Jones has an excellent paper that might be helpful. I've attached a copy. Good luck, David Booth
One useful practical tip (OK cheat!) is that if you estimate a standard error of a fixed parameter in the usual and robust manner, and they are quite different then this is a good indication that you may want to consider allowing the fixed parameter to have variable slopes
see page 80 in this Book Developing multilevel models for analysing contextuality, he...
Generally with repeated measures data I would expect to see random slopes for the Age or Time variable otherwise you would be assuming that differences (say) at the start for an individual do not change with time, which may be implausible. The covariance is a key term as it will indicate whether heterogeneity between individuals is increasing or decreasing; here is a simple tutorial
http://www.bristol.ac.uk/media-library/sites/cmm/migrated/documents/repeated-measures.pdf
and here is a cornucopia of mixed mode resources
http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html
I am assuming that you specifically want to compare a standard random intercept model (participants allowed their own intercepts only) to a standard random coefficient model (participants are allowed their own intercept AND slope). In other words we are not talking about other types of models (e.g. shared intercept, but random slope)
There are two basic approaches to choosing between these two models.
The first approach (purely empirical) is to compare the model fit of a random intercept model to that of a random coefficients model (i.e. random intercepts AND random slopes) using something like AIC or BIC. The advantage of these measures (and other IC statistics) is that they penalize for model complexity. The basic idea is that if the AIC (or BIC) is lower for a random coefficient model compared to a random intercept model then the gain in fit is worth the extra model complexity.
Personally, I think this approach alone is a little too 'black box'. I like to think about the problem in terms of the data (and research problem itself). For a random intercept model we are assuming that participants are allowed to have their own baseline values (and these are normally distributed around the average intercept). BTW remember this is AFTER we account for differences in participants due to treatment group and/or risk factors (i.e. we can assume the fixed effect predictors don't interfere with these decisions). The subtext for the random intercept model is that we assume (despite there different starting points) that participants respond to time in exactly the same way (in other words, all of the individual participants' regression lines are parallel). So the important question is: Do we think such an assumption is valid? If we assume that participants may respond to time differently (e.g. different patients may respond to therapy in a different way...remember though, beyond that explain by the fixed effect predictors in the model).To summarize, if you had to fit a regression line individually for each participant, would you constrain them to a share slope (random intercept) or not (random coefficient). In other words, have look at your spaghetti plot.
Hope this helps
As answered before, you can always use a biological approach and give meaning to your parameters, and therefore attribute random or fixed effect for each of them.
But it might not hold. So the ideal is test your model: introduce random effects as you go;
Mod= fixed;
Mod2= random intercept;
Mod3= random slope;
Mod3= random intercept + slope;
Them you check the distribution of your residuals (predicted vs observed), best model gives more homogeneous and close to 0 residuals;
To confirm the best model you can use the mean squared prediction error (RMSE) and bias; The best model is the one with the smallest prediction error.
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