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== Session 1:  Multilevel (a.k.a. mixed or hierarchical) models ==

This session will cover the basics of multilevel or mixed models, both linear and logit models, how to fit them in R, and how to interpret the output. 

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=== Materials ===
 * [attachment:ranef_sim.R simulated data]
 * [attachment:lmer.R linear mixed models]
 * [attachment:multilevellogitmodel.R mixed logit models]

=== Reading ===
|| Baa08 || Chapter 7 (pp. 263-309) || Grouped data, functions, lmer ||
|| Jaeger08 || [http://dx.doi.org/10.1016/j.jml.2007.11.007 Categorical data analysis ] (pp. 442ff.) || Multilevel logit models vs. ANOVA ||

Optional reading:
|| G&H07 || Sections 1.1-1.3 (pp. 1-3) || Intro, examples, motivation ||
|| || Chapter 11 (pp. 237-248) || Multilevel structures ||
|| || Chapter 12 (pp. 251-277) || Multilevel linear models:  the basics ||
|| G&H07 || Chapter 14 (pp. 301-321) || Multilevel logistic regression ||

=== Q&A ===
 * Q: What is REML - restricted maximum likelihood? 
 * A: '''Restricted/residual maximum likelihood (REML)''': Mixed ''linear'' models in R are fitted using REML rather than ML (which, we learned, is standardly used to fit ordinary linear regression). Mixed model fitting implies fitting the variance-covariance matrix for the random effects (and the residuals), based on the available sample. REML, unlike ML, can be used to derive ''unbiased'' estimates of variances and covariances. [[BR]]
 A biased estimate is pretty much what one would think it is (see this  [http://en.wikipedia.org/wiki/Bias_of_an_estimator wiki article on the notion of a statistical bias in the estimation of a parameter]). Recall that in fitting linear models, the goal is to derive the (best) estimates for the parameters in our model. In a ordinary linear models, these are the coefficients; in a mixed linear model, the parameters also include the estimates of the random effect variances. We want these estimates of the variances (which are, of course, based on our ''sample'') to be unbiased estimates of the true underlying population variance. When you read the above wiki article, consider that the example it gives for variance estimation, is an example of maximum likelihood estimation (the given estimate given the sample is the ML estimate of the sample's variance and it's a [downward] biased estimate of the population's variance).

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=== Assignments ===
|| Baa08 || Section 4.7 (p. 126) || Exercises 3 and 7* ||
|| || Section 6.7 (p. 260) || Exercise 1, 8 ||


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== Possible topics ==