1 Stat 511 Exam 2 April 7, 2008 Prof. Vardeman I have neither given nor received unauthorized assistance on this exam. ________________________________________________________ Name _______________________________________________________________ Name Printed21. An experimental data set in a set of slides due to S.A. Jenekhe found on the University of Washington Chemistry Department web site of Prof. Lawrence Ricker concerns 2CO solubility in a glassy polymer. Given are the pressures, p, and corresponding concentrations, c , of 2CO below. Pressure, p (atm) 2.74 6.10 9.76 14.45 18.92 26.74 33.28 42.23 Concentration, c (33cm (STP)/cm polymer ) 36.6 51.4 64.3 78.7 91.5 110.3 122.9 143.7 A standard deterministic model for gas solubility in a polymer is aca1LpcHpLLp⎛⎞=+⎜⎟+⎝⎠ for constants H (the Henry's law constant), cL (the Langmuir capacity constant), and aL (the Langmuir affinity constant). Below is a plot of these data and a fitted concentration versus pressure curve. Note that for large pressure this (fitted) curve is nearly linear with slope H and intercept cL , while the derivative of concentration with respect to pressure at 0 pressure is caHLL+⋅. There is an R printout at the end of this exam from the session in which this plot was generated. Use it to answer the following questions about a nonlinear regression analysis of this situation based on a model aca1iii iiLpcHpLLpε⎛⎞=++⎜⎟⎜⎟+⎝⎠ (*) for iid ()2N0,σ errors , 1,2, ,8iiε= … .3a) What are approximate 95% confidence limits for the standard deviation of concentration at any fixed pressure according to the model (*)? (If you need some percentage point(s) of a distribution that you don't have, say very carefully/completely exactly what you need.) Plug into any formula you provide. b) Under what conditions on the parameters of model (*) is the mean concentration a simple multiple of pressure? Is there definitive evidence in these data that such a ("single mode") model is too simple and so the full complexity of model (*) is justified? Explain in terms of some measures of statistical significance.4c) What are approximate 95% confidence limits for the derivative of mean concentration with respect to pressure at 0 pressure? (Plug into an appropriate formula. You don't need to do arithmetic, but you must plug in, and if you don't have necessary percentage points of a distribution, say very carefully/completely exactly what you need.) 2. In a typical industrial "gauge R&R study," each of I different parts from some process is measured m times by each of J different operators, as a way of studying the consistency of measurement using a single gauge. We will here consider a case where the operators are "fixed" (being the only ones a company will ever use to do such measuring) while parts are "random" (representing ongoing production of such parts) and for the th measurement obtained on part by operator ijkyk i j= model as ijk i j ij ijkyμαβαβε=+++ + (**) where and the jμβ are unknown constants and the , , and iij ijkααβ ε are independent random variables, with ()2iid N 0,iαασ∼, ()2iid N 0,ijαβαβ σ∼, and ()2iid N 0,ijkεσ∼. (Here the jβ might be thought of as consistent operator biases and the ijαβ might be thought of as so-called operator "nonlinearities of measurement.") To begin, first consider a small/toy case where 2IJm=== (there are 2 parts, 2 operators, and each part is measured 2 times by each operator).5a) For 8 observations written down in dictionary order, show how to write out model (**) in mixed linear model matrix form (by providing the elements of =++YXβ Zu ε indicated below). 111112121122211212221222yyyyyyyy⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠Y =X =β =Z =u b) Write out the following in terms of model (**) parameters. 111Var y = ________________________ ()111 112Cov ,yy=________________________ ()111 121Cov ,yy= ___________________ ()111 211Cov ,yy=________________________ At the end of this exam, there is an R printout for an analysis based on model (**) of a modification of a real data set from an R&R study based on 4I= parts, 3J= operators, and 2m = measurements per part. (These are measured heights of some steel punches in 310− inch.) Use it to answer the next two questions.6c) Based on the results on the printout • do you find clear evidence of differences in operator measurement "biases," and • do operator "nonlinearities" appear to play a large role in measurement of these punch heights? (Return to the parenthetical remark following model statement (**) for use of these terms.) Explain using appropriate values from the printout. d) What are approximate BLUPs for • 11 11μαβαβ+++ (a long-run average of measurements of part 1 by operator 1) (give a numerical value) • 5 1 51 511μαβαβ ε+++ + (a measurement on a new punch by operator #1) Here, give both the BLUP AND an appropriate standard error (give numerical values).73. Suppose that for 1,2 and 1,2ij==, ij i ijyμαε=++ for independent variables ()2 iid N 0,iαασ∼ and ()2 iid N 0,ijεσ∼. Take =WBY for ()11 12 21 22,,,yyyy′=Y and 11 1 1110000 1 1−−⎛⎞⎜⎟=−⎜⎟⎜⎟−⎝⎠B . a) Argue that REML estimation of 22and ασσ can be based on W and write out explicitly the function of 22123, , , , and wwwασσ that can be maximized as a function of 22 and ασσ to produce REML estimates. b) The restricted loglikelihood from a) can be written as a function of log-variances 21logγσ= and 22logαγσ= . For a particular W , this is maximized at 1ˆ.6931γ= and 2ˆ1.3863γ=. The Hessian (matrix of second partial derivatives) of this function at its maximizer is1.5 112−−⎛⎞⎜⎟−−⎝⎠. Find approximate 95% confidence limits for ασ based on this information. (Plug in and evaluate.)84. Suppose that for 1, 2, 3i = independent observations ()2 iid N ,ij i iyμσ∼ for 1, ,ijn= … (that is, we have independent samples of sizes in from three different normal distributions). For constants 12 3,, and cc c, the random variable 11 2 2 33cy cy cy++ has variance 222222312123123cccVnnnσσσ=++ Based on variances for the 3 samples (22 212 3,, and ss s) use the
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