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| * '''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). REML is used because mixed models estimate variance and covariances in the fitting process and REML, unlike ML, can be used to derive ''unbiased'' estimates of variances and covariances. 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] 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 biased estimate of the population's variance). | * '''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). REML is used because mixed models estimate variance and covariances in the fitting process and REML, unlike ML, can be used to derive ''unbiased'' estimates of variances and covariances. 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] 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). |
Session 3: Multilevel (a.k.a. Hierarchical, a.k.a. Mixed ) Linear Models
June 10 2008
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 |
Baa08 |
Chapter 7 (pp. 263-282) |
Grouped data, functions, lmer |
Notes on the reading
In G&H07, the first two examples (Section 11.2-3) that leads to the motivation of multilevel models is a logit model, which we haven't yet talked about. Just ignore that detail and focus on the conceptual argument made in that section. Think of the logit model as predicting the likely outcome (here: treatment success vs. failure) given the predictors we put into the model, just like for linear regression. In reading the Chapter 11, ask yourself, in a classical ANOVA Latin-square design, e.g. in a priming study where each subject sees say 24 items in one of its 4 conditions - what are the individuals and what are the groups ?
Additional terminology
Feel free to add terms you want clarified in class:
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). REML is used because mixed models estimate variance and covariances in the fitting process and REML, unlike ML, can be used to derive unbiased estimates of variances and covariances. 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] 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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