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ISU STAT 511 - Homework # 6

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1Stat 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 ijkyμαβε=++ + where the only fixed effect isμ, the random effects are all independent with the()2 iid N 0,iαασ, the()2 iid N 0,ijββσ, and the()2 iid N 0,ijkεσ. 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 σ 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 σ.) (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)2Then 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 iijαβ 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 σ produced by this analysis is exactly the one based on the mean square error in a). c) Run predict() or fitted() (and random.effects() or ranef() for the mixed model analyses) on the results of the calls from parts b) and c) above. Identify the predictions from the fixed effects analysis as simple functions of the data values. (What are these predictions?) Notice that the predictions from the mixed effects analysis are substantially different from those based on the fixed effects model. 2. The article "Variability of Sliver Weights at Different Carding Stages and a Suggested Sampling Plan for Jute Processing" by A. Lahiri (Journal of the Textile Institute, 1990) concerns the partitioning of variability in "sliver weight." (A sliver is a continuous strand of loose, untwisted wool, cotton, etc., produced along the way to making yarn.) For a particular mill, 3 (of many) machines were studied, using 5 (10 mm) pieces of sliver cut from each of 5 rolls produced on the machines. The weights of the (75) pieces of sliver were determined and a standard hierarchical (balanced data) ANOVA table was produced as below. (The units of weight were not given in the original article.) Source SS df Machines 1966 2 Rolls 644 12 Pieces 280 60 Total 2890 74 Use the same mixed effects model as in Problem 1 and do the following. a) Make 95% confidence intervals for each of the 3 standard deviations , and αβσσσ. Based on these, where do you judge the largest part of variation in measured weight to come from? (From differences between pieces for a given roll? From differences between rolls from a given machine? From differences between machines?) (Use Cochran- Satterthwaite for and αβσσ and give an exact interval for σ.) b) Suppose for sake of illustration that the grand average of all 75 weight measurements was ...35.0y = . Use this and make a 95% confidence interval for the model parameterμ. 3. Consider the fake unbalanced 33× factorial data of Problem 3 on HW 4. Here do a random effects analyses of those th response at level of A and level of Bijkyk i j= based on a two-way random effects model without interaction ijk i j ijyμαβε=++ +3where μ is an unknown constant, the iα are iid ()2N0,ασ, the jβ are iid ()2N0,βσ, the ijε are iid ()2N0,σ and all the sets of random effects are all independent. After loading the MASS, nlme, and lme4 packages (and specifying the use of the sum restrictions), do an analysis of these data. (See Section 8.7 of Faraway for something parallel to this problem.) Run the summary()function on the results of your lme() and lmer() calls. Make appropriate function calls to get predictions/fitted values and 95% intervals. What are 95% limits for the model parameters and μσ? What do the intervals you are able to produce tell you about the relative sizes of the standard deviations , , and αβσσσ? 4. The data set below is taken from page 54 of Statistical Quality Assurance Methods for Engineers by Vardeman and Jobe. It gives burst strength measurements made by 5 different technicians on small pieces of paper cut from 2 different large sheets (each technician tested 2 small pieces from both sheets). Technician (B) 1 2 3 4 5 1 13.5, 14.8 10.5, 11.7 12.9, 12.0 8.8, 13.5 12.4, 16.0Sheet (A) 2 11.3, 12.0 14.0, 12.5 13.0, 13.1 12.6, 12.7 11.0, 10.6 Vardeman and Jobe present an ANOVA table for these data based on a two-way random effects model with interaction ijk i j ij ijkyμαβαβε=+ + + + (where μ is the only fixed effect). Source SS dfEMS Sheet (A)


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