Random slopes
Section of Random slopes, and a binary mixed model, part of Correlated data
Random slopes
| Type | Section |
| Taught in | Random slopes, and a binary mixed model |
| Part | Correlated data |
| Questions using it | 4 |
Write {\boldsymbol{Z}}_{ji} for the covariates whose effects are allowed to vary by block. They may be the same as {\boldsymbol{X}}_{ji}, or a subset, or something else. Y_{ji} ={\boldsymbol{X}}_{ji}^\top\boldsymbol{\beta} +{\boldsymbol{Z}}_{ji}^\top\boldsymbol{\alpha}_j +\varepsilon_{ji}, \boldsymbol{\alpha}_j\stackrel{\mathrm{ind}}{\sim}\mathrm{N}({\boldsymbol{0}},{\color{blue}\boldsymbol{\Sigma}}_A), \qquad \varepsilon_{ji}\stackrel{\mathrm{ind}}{\sim}\mathrm{N}(0,\sigma^2). {\color{blue}\boldsymbol{\Sigma}}_A holds the variances of the random intercept and the random slopes, and the covariances between them. Dropping every random slope returns the model of the previous chapter. We start from the larger model when a covariate might interact with the block: some hospitals are better at some treatments, some drivers deteriorate faster when they are not allowed to sleep.
A fixed effect is a constant. Repeat the experiment and it is the same constant. A random effect is a draw. Repeat the sleep study and eighteen different drivers are drawn, and the eighteen intercepts and slopes are new. We estimate the mean and the covariance of that population. Predicting the eighteen draws we happened to get is a different task, and it is a prediction of a random variable, not an estimate of a fixed parameter. With the variances known, that prediction is the best linear unbiased predictor. With the variances estimated, it is the empirical version of the same thing. Software computes it. We do not derive it by hand.
Used in
- Taught in Random slopes, and a binary mixed model
- Also used in Repeated measures in R
- Questions that use it (4) — Random slopes and changed conclusions (answer) · Toenail infection GLMM (answer) · Orthodontic growth curves (answer) · Random intercepts and slopes
- All 6 questions on Random slopes, and a binary mixed model