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Stat 511 HW 6 S09

Stat 511 HW 6 S09 - Stat 511 HW#6 Spring 2009 This...

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1 Stat 511 HW#6 Spring 2009 This assignment consists of problems on mixed linear models. Most are repeats from HW #10 of 2003 and HW#8 of 2004. All of these problems requiring computing should be done using both the lme() function in the nlme package in R (used in 2003 and 2004 versions of Stat 511) and the lmer() function in the lme4 package used first in 2008. (See Chapter 8 of Faraway's Extending the Linear Model with R for examples of the use of the lmer() function.) The syntax of the new function is easier to understand and use than that of the earlier one, but the earlier package has more useable methods associated with it. 1. Below is a very small set of fake unbalanced 2-level nested data. Consider the analysis of these under the mixed effects model ijk i ij ijk y μ αβε = ++ + where the only fixed effect is , the random effects are all independent with the () 2 iid N 0, i α σ , the ( ) 2 iid N 0, ij β , and the ( ) 2 iid N 0, ijk ε . Level of A Level of B within A Response 1 1 6.0, 6.1 2 8.6, 7.1, 6.5, 7.4 2 1 9.4, 9.9 2 9.5, 7.5 3 6.4, 9.1, 8.7 After loading the MASS , nlme , and lme4 packages and specifying the use of the sum restrictions as per > options(contrasts=c("contr.sum","contr.sum")) use the lme() and lmer() functions to do an analysis of these data. (See Section 8.6 of Faraway for something parallel to this problem.) a) Run summary() on the results of your lme() and lmer() calls. Then do what is necessary to get "95% intervals" here for both fixed effects and for the random effects or their ratios. In the first case, you may simply use the intervals() function to get approximate confidence intervals. In the second case, the current version of the lmer() function produces point estimates of variance components (and their square roots). One must do more to get a (Bayes credible) interval (presumably based on "Jeffreys priors") for the "error" standard deviation in a mixed linear model (presented directly) and multipliers (listed as elements of $ST ) to be applied to those end points in order to produce intervals for the other model standard deviations . (The elements of $ST then portray "relative standard deviations" or the ratios of the other model standard deviations to .) (Credible intervals are not quite confidence intervals but can be thought of as roughly comparable.) You may employ code like > sims <- mcmcsamp(lmer.fit, 50000) > HPDinterval(sims)
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2 Then compute an exact confidence interval for σ based on the mean square error (or pooled variance from the 5 samples of sizes 2,4,2,2, and 3). How do these limits compare to what R provides for the mixed model in this analysis? b) A fixed effects analysis can be made here (treating the and ii j α β as unknown fixed parameters). Do this using the lm() function. Run summary() and confint() on the result of your call. Note that the estimate of produced by this analysis is exactly the one based on the mean square error in a).
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Stat 511 HW 6 S09 - Stat 511 HW#6 Spring 2009 This...

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