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ISU STAT 511 - Methods Statistics

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Methods 2 Statistics Ph.D. Prelim 2008 page 1 of 7 (Majors and Co-Majors) The following scenario is based on a real study, though most details have been changed and the data provided are not real. Attached to this question is a set of R output that you will want to consult at appropriate points in answering the following questions about this scenario and the analysis of data from it. An experiment is done to determine whether the progressive effects of a particular disease can be detected in an electrical response of rat eyes to flashes of light. Four animals (rats) are infected with the disease and a particular electrical response to a "standard" light flash ( V)yμ= Two measurements are made on each infected rat at one month, two months, and then three months after infection, for a total of 6 observations per infected rat. To account for the fact that (unintended) changes in lab conditions and/or measurement equipment might occur over time, 2 measurements were also made on 2 uninfected animals at each measurement period. But, these were different animals each month. Thus there were a total of 422210+++= rats involved in the study. Four of these were infected and observed at multiple periods and 6 were control animals that were observed at only a single point in time. The questions of greatest scientific interest in this study are questions such as "Is there a detectable difference in electrical response between rats that have been infected and those that have not?" and "Can changes in electrical response over time be detected for infected rats?" 1) In light of the basic goals of the study, discuss in qualitative terms any additional utility that would have been provided had the study a) included measurement of the infected animals at month 0 (just before infection). b) included measurement of each control animal at all of months 0,1,2, and 3 . Label the infected rats 1, 2, 3, 4i = and the control rats 5,6, ,9,10i=… . Suppose control rats 5 and 6 are tested at month 1, rats 7 and 8 are tested at month 2, and rats 9 and 10 are tested at month 3. Let the th response measured on rat at period ijkykij= For 0123a mean response for an uninfected rata month 1 mean repsonse for an infected rata month 2 mean repsonse for an infected rata month 3 mean repsonse for an infected ratμμμμ==== and a rat effect for rat , 1,2, ,10a month effect for month , 1,2,3ijriimjj====… we will initially consider models for this situation of the basic form [][]044ijk j i j ijkyIi Iirmμμε=>+≤+++ (1) for all 36n = relevant combinations of , , and ij k.Methods 2 Statistics Ph.D. Prelim 2008 page 2 of 7 (Majors and Co-Majors) First, for the sake of simplicity, consider only the first of the two replicate measurements made on only rats 1,2,5,7, and 9 at each time period, and an ordinary (fixed effects) Gauss-Markov linear model version of (1). Let ()111 121 131 211 221 231 511 721 931,,,,,,,,yyyyyyyyy′=Y and ()01231257912 3, , , ,,,,,, , ,rrrrrmm mμμμμ′=β 2) Find a matrix X so that model (1) for these observations can be written in the usual linear model form =+YXβε Is the resulting model of full rank? (Argue this carefully one way or the other.) 3) Is the quantity 10μμ− estimable in this model? (Argue this carefully one way or another.) It is probably both more reasonable and also more effective in terms of statistical inference to interpret the rat and month effects in equation (1) as random rather than fixed. So henceforth suppose that the 'sir in (1) are iid ()2N0,rσ independent of 'sjm that are iid ()2N0,mσ, all of which are independent of 'sijkε that are iid ()2N0,σ. Continue for the present to consider only the first of the two replicate measurements made on only rats 1,2,5,7, and 9 at each time period and Y exactly as listed at the top of this page. 4) Find matrices and XZ and vectors and β u so that model (1) for these observations can be written in the usual mixed linear model form =++YXβ Zu ε Is the resulting model of full rank? (Argue this carefully one way or the other.) Are the scientifically interesting quantities 0for 1,2,3iiμμ−= and 2132 31, , and μμμ μ μ μ−−− all estimable from these data? Why or why not?Methods 2 Statistics Ph.D. Prelim 2008 page 3 of 7 (Majors and Co-Majors) Now begin to consider the whole data set from this study. 5) Based on the mixed effects model described by (1) and the distributional assumptions indicated just after 3) above, evaluate the following in terms of model parameters: a) ()Varijky for any set of indices , ,ijk in the data set b) ()111 112Corr ,yy c) () ()111 211 111 121Corr , and Corr ,yy yy d) ()111 221Corr ,yy 6) Each pair ()12,ij ijyy in the data set can be used to compute a sample variance, say 2ijs . How does this sample variance compare to ()22.1121ijk ijkεε=−−∑ (the "sample variance" of the pair ()12,ij ijεε)? In light of this, what is the mean of the average sample variance, namely 21E18ijijs⎛⎞⎜⎟⎝⎠∑ ? Using (1) write 1.. 2.. 3.. 4.., , , and yyy y (the simple averages of the rat 1,2,3, and 4 responses) in terms of averages of appropriate fixed and random effects and random errors. Based on this, what is the expected value of the sample variance of these four sample means? What is the expected value of the sample variance of rat 5 and 6 averages? Of rat 7 and 8 averages? Of rat 9 and 10 averages? Call the sample variances for groups of rat means referred to above by the respective names ( )()()()222 21,2,3,4 , 5,6 , 7,8 , and 9,10sss s. What are sufficient conditions on constants 0 1234 56 78 9 10, ,,,cc c c c under which ( ) () () ( )22 2 2 2 20 1234 56 78 9 101E 1,2,3,4 5,6 7,8 9,1018ij rijcscs cscscsσ⎛⎞⎛⎞+ +++ =⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠∑Methods 2 Statistics Ph.D. Prelim 2008 page 4 of 7 (Majors and Co-Majors) 7) The current version of the lmer() function in the lme4 package of Bates produces point estimates of variance components (and their square roots) and Bayes credible intervals (presumably based on Jeffreys priors) for the "error" standard deviation in a mixed linear model and ratios of the other model standard deviations to the error standard deviation. Here the credible intervals are $sigma lower upper [1,]


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ISU STAT 511 - Methods Statistics

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