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
Best Answer
There is no standard terminology here and each author might use the terms differently or interchangebly. You'll need to examine the context and model specified in context in order to determine what each author means. That being said, nearly always, the terms are generally used as follows:
A random slopes model typically refers to models in which the coefficients associated with variables in the model (i.e. $b_{ik}$, $\forall k\ne0$) are set to be random. For example: \begin{eqnarray*} Y_{ij}=\beta_0+\beta_1X_{ij1}+b_{i1}Z_{ij1}+\epsilon_{ij}, \end{eqnarray*} where $b_{i1}$ is a random effect corresponding to variable $Z_{ij1}$.
A random intercept model typically refers to models in which each subject (or observation) has a random intercept term specified (i.e. $b_{i0}$), typically in addition to the fixed intercept term (i.e. $\beta_{0}$). For example: \begin{eqnarray*} Y_{ij}=\beta_0+\beta_1X_{ij1}+b_{i0}+\epsilon_{ij}, \end{eqnarray*} where $b_{i0}$ adds a random effect to the subject's intercept term, $\beta_0$. There is really no other meaning for this model. If an author writes "random intercept" model, this is what he means.
A random intercepts and slopes model refers to models in which each subject (or observation) has a random intercept term specified (i.e. $b_{i0}$), typically in addition to the fixed intercept term (i.e. $\beta_{0}$) as well as one or more random effects associated with other variables in the model (i.e. $\beta_1, \beta_2$, etc.). For example: \begin{eqnarray*} Y_{ij}=\beta_0+\beta_1X_{ij1}+b_{i0}+b_{i1}Z_{ij1}+b_{i2}Z_{ij2}+\epsilon_{ij}, \end{eqnarray*} where $b_{i0}$ adds a random effect to the subject's fixed effect intercept term, $\beta_0$; $b_{i1}$ is a random effect (slope) associated with $Z_{ij1}$; and $b_{i2}$ is a random effect associated with $Z_{ij2}$.
On a very rare occasion, I have seen some authors refer generally to random slopes to any model that may include random intercepts AND/OR random slopes as defined in the bulleted list above. Again, you'll need to review the models in context as presented by the author to be sure to which model the author is referring. But, lacking any context, you are probably safe using the definitions provided in my answer above.