'random intercept only', 'random slope only' and 'random intercept and slope' models

Question

I have a terminological question about the use of the terms random intercept models, random slope models and random intercept and slope models.

Through my readings, I find that most authors use the terms random slopes, and random intercepts and slopes interchangeably. At least, this is my understanding so far.

For example, Zuur et. al (2009) use random intercept and slope models in contrast to random intercepts only models whereas Barr et. al (2013) and Winter (2014) use random slopes models vs. random intercepts only models.

So, my question is are the terms random 'intercepts and slopes models' the same as 'random slopes models' or do we have three separate types of models: 'intercepts only', 'slopes only', and 'intercepts and slopes models' (maximal models)?

Thanking you,

PS. work cited is below.

Zuur, A. F., Ieno, E. N., Walker, N., Saveliev, A. A., & Smith, G. M. (2009). Mixed effects models and extensions in ecology with R. New York, NY: Springer Science+Business Media, LLC.

Winter, B. (2014). Linear models and linear mixed effects models in R with linguistic applications. arXiv:1308.5499. [arxiv.org/pdf/1308.5499.pdf]

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), 255-278. ncbi.nlm.nih.gov/pmc/articles/PMC3881361/pdf/nihms533954.pdf

Answer 1

EDITED

I am not an expert (so someone correct me if I am wrong) but when you have a model with random effects, you get to choose which parameters will be treated as random. Usually your intercept will always be random (but you can make it fixed) and you can also specify if you want your slope to be random or not. The terms authors use to explain what is going on on their models/methods may vary but that is the bottom line --- So in summary to answer your question, yes, you can have models with only a random intercept, only a random slope, or both random intercept and slope (and just for the record, you can run all three types for the same dataset and find pretty similar results, depending on how complex your data/model is).