# r-linear mixed model with the random effect variable be continuous

Suppose there are $$N$$ subjects under study, with subject $$i$$ contribution $$n_i$$ observations, for $$i =1,...,N$$. And let $$y_{ij}$$ denote a response variable for subject $$i$$ at observation $$j$$. Let $$x_{ij}$$ denote a $$p\times 1$$ vector of predictors, and let $$z_{ij}$$ denote a $$q\times 1$$ vector of predictors. In general, the linear mixed effects model is $$y_i=X_i\alpha+Z_i\beta_i+\epsilon_i$$ where $$y_i=(y_{i1},...,y_{in_i})^T$$, $$X_i=(x_{i1}^T,...,x_{in_i}^T)^T$$, $$Z_i=(z_{i1}^T,...,z_{in_i}^T)^T$$, $$\alpha$$ is a $$p \times 1$$ vector of unknown population parameters, $$\beta_i$$ is a $$q \times 1$$ vector of unknown subject-specific random effects with $$\beta_i \sim N(0, D)$$ and the ellements of the residual vector, $$\epsilon_i$$, are $$N(0, \sigma^2I)$$.

I was trying to use the package lme4 in R which allows we to deal with the hierarchical data and consider the variability between subjects. But in this case, the data usually looks like

And we can use like

lme4(frequency ~ attitude + (1|subject) + (1|scenario), data=politeness)

to specify which variable has fixed effect, attitude, and which has a random effect, the subject and scenario are two variable with random effect.

But my questions are:

1. In our example, the variable which has a random effect is $$Z_i$$. But according to the error I got when I use lme4, the level of random effect variable should be more than 1 and smaller than the number of observations. How should we deal with it when our $$Z_i$$ here is continuous?

2. We actually know the distribution of the random effect coefficients $$\beta$$, how should we using this information here? i.e. where to put the information in $$D$$?

Many thanks to any comments!

• If you want to fix the covariance matrix of the random effects $$\beta_i$$ you should look at packages that fit kinship mixed models, e.g., the lmekin() function in the coxme package. There are also other packages available with this capability.