random intercept, random slope, what's next?

Question

Consider a mixed effects model with a random intercept. This means

$$y_{ij} = b_{0i} + \dots \text{ fixed effects } \dots +e_{ij}.$$ Now, suppose that we group the observations with respect to time $$y_{ij} = b_{0i} + b_{1i}t_{ij} + \dots \text{ fixed effects } \dots +e_{ij}.$$ This is a random slope model.

If I use higher orders in the expansion of $t$, what are those models called? This is

$$y_{ij} = b_{0i} + b_{1i}t_{ij} + b_{2i}t_{ij}^2 + \dots \text{ fixed effects } \dots +e_{ij}.$$ ... $$y_{ij} = b_{0i} + b_{1i}t_{ij} + b_{2i}t_{ij}^2 + ... + b_{pi}t_{ij}^p + \dots \text{ fixed effects } \dots +e_{ij}.$$

Answer 1

The models that include random effects for the higher order terms can still be referred to as a random slopes (and intercepts model). They can also be referred to as random coefficients (and intercepts) models. The latter is slightly more intuitive:

$$y_{ij} = b_{0i} + \dots \text{ fixed effects } \dots +e_{ij} \tag{1}$$

$$y_{ij} = b_{0i} + b_{1i}t_{ij} + \dots \text{ fixed effects } \dots +e_{ij} \tag{2}$$

$$y_{ij} = b_{0i} + b_{1i}t_{ij} + b_{2i}t_{ij}^2 + \dots \text{ fixed effects } \dots +e_{ij}\tag{3}$$

$$y_{ij} = b_{0i} + b_{1i}t_{ij} + b_{2i}t_{ij}^2 + ... + b_{pi}t_{ij}^p + \dots \text{ fixed effects } \dots +e_{ij}\tag{4}$$

$(1)$ is a random intercepts-only model.

$(2)$ is a random intercepts and random slopes model (note that it is possible to have random slopes without random intercepts, so it can be a little confusing to call it just a random slopes model). This makes intuitive sense because each subject has it's own slope for $t$.

$(3)$ and $(4)$ are also a random intercepts and random slopes models, however coefficients for the higher order terms control the shape of the curve, so they might, more intuitively, be called random random intercepts, random slopes and random shapes models.

More generally, $(2)$, $(3)$ and $(4)$ can be called random intercepts and random coefficients models