Abstract
We have covered a variety of statistical models in this book, but all have shared a common feature: The criterion and error term were treated as random variables, but all of the predictors were assumed to be fixed. In this chapter, we will consider models that include a broader mixture of fixed and random variables. For obvious reasons, these models are called mixed-effects models or mixed models.
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Notes
- 1.
Although many mixed models have a hierarchical structure, not all do, so the two terms should not be used interchangeably.
- 2.
An incidence matrix is different than a matrix with dummy coded vectors. The elements of both matrices are 0 and 1, but an incidence matrix has as many vectors as there are levels of the variable it represents (q), whereas a matrix using dummy coding has one vector less than the levels of the variable it represents (q − 1).
- 3.
The Residual Maximum Likelihood Function is also called the Restricted Maximum Likelihood Function or the Reduced Maximum Likelihood Function.
- 4.
A fourth possibility is to include a random slope without including a random intercept. This configuration is rarely used and will not be considered.
- 5.
This issue is the subject of considerable controversy, and readers wishing more information can visit: https://stat.ethz.ch/pipermail/r-help/2006-May/094765.html.
- 6.
\( {\mathcal{R}}^{\prime }s \) nlme package is an earlier version of its lme4 package. The two packages use slightly different optimization routines so sometimes produce slightly different results.
- 7.
These values also appear in the 2nd column of Table 14.2.
- 8.
The default specification in \( \mathcal{R} \) assumes correlated terms by using a single bar in model 1d. To specify uncorrelated terms, we use double bars, as in model 1c. Alternatively, we can type: lmer(y1 ~ x + (1 | schl) + (0 + x | schl)).
- 9.
To aid in the interpretation of the intercept, it is advantageous to recode days to begin at 0 rather than at 1.
References
Bates, B., Mäechler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67, 1–48. https://doi.org/10.18637/jss.v067.i01.
Demidenko, E. (2013). Mixed models: Theory and applications with R (2nd ed.). New York: Wiley.
Henderson, C. R. (1953). Estimation of variance and covariance components. Biometrics, 9, 226–252.
Pinheiro, J., & Bates, D. (2000). Mixed-effects models in S and S-PLUS. Berlin, Germany: Springer.
Searle, S. R. (1994). On mixed models, REML and BLUP. https://ecommons.library.cornell.edu/bitstream/1813/31843/1/BU-1256-M.pdf .
Searle, S. R., Casessla, G., & McCulloch, C. E. (1992). Variance components. New York: Wiley.
Wood, S. N. (2003). Generalized additive models: An introduction with R. London: Taylor & Francis.
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Brown, J.D. (2018). Mixed-Effects Models. In: Advanced Statistics for the Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93549-2_14
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