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ISU STAT 511 - MS Exam -2003

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1Stat 511 MS Exam, Spring 2003 Page 1 of 3 This question concerns several analyses of a small set of data on the operation of a Butane Hydrogenolysis Reactor. The response variable percent conversion of butaney = is to be understood as a function of the chemical reactor process variables a total feed flow (cc/sec at STP)feed ratio (Hydrogen/Butane)the reactor wall temperature ( F)flowratiotemp=== The data are shown in the following table. Run, i y flow ratio temp day Setup, s(i) 1 82 115 6 495 1 12 91 50 4 470 1 23 75 180 8 520 1 34 98 50 4 520 2 45 39 180 8 470 2 56 77 115 6 495 2 17 95 50 8 520 3 68 61 180 4 470 3 79 81 115 6 495 3 110 76 50 8 470 4 811 92 180 4 520 4 912 82 115 6 495 4 1 Twelve runs were made over a period of four days, three runs being made each day. Nine process setups (corresponding to combinations of levels of the flow, ratio, and temp factors) were used. Note that one setup was repeated on each of the four days. I. Begin by considering analysis based on a cell means model ()for 1,2, ,12isiiyiµε=+ =… (*) where12 9,,,µµµ… are unknown constants (the 9 mean responses for the different setups of the process), theiεare iid()2N0,σ, and we use the notation ( ) the setup number employed in the th run of the processsi i= (For example, when 8i = for the 8th run, (8) 7s= to indicate that setup 7 was used.) (a) Write this linear model out in matrix form. (What are and X β here? Use the ordering of the elements of Y employed in the table.) (b) Give 90% confidence limits forσin this model. (Display a formula, insert numerical values, but don't complete the calculations.)2Stat 511 MS Exam, Spring 2003 Page 2 of 3 (c) Setup #1 is a "center point" for the set of(),,flow ratio temp combinations in the data set. The other 8 setups form a 222×× factorial structure. In this 3 way factorial structure, the quantity ()()3579 24681144µµµµ µµµµ+++ − +++ is potentially of interest. In the jargon of factorial analysis, what is this quantity? Give 95% confidence limits for it based on the cell means model (*). (d) Give 95% prediction limits for an additional observation under process setup #2 under this model. There is an R printout attached to this question that you should consult for the rest of this question. When you use something off this printout in making an answer, be very explicit about where you got it (give page numbers and use the names employed on the printout). II. A second analysis of these data can be made on the basis of a regression model 01 2 3iiiiiy flow ratio tempβββ βε=+ + + + (**) (a) The model (**) is a "reduced model" version of model (*). Give the value and degrees of freedom for anF statistic that can be used to test the hypothesis that the regression model (**) is adequate. (b) Under the regression model (**) the linear combination of 8 setup means considered in part I.(c) can be expressed in terms of the parametersβof the regression model. Do so, and make 95% confidence limits for this quantity (under model (**)). III. To this point we have ignored the fact that these data were collected on 4 different days. It is probably sensible to think of "day" effects as random. So consider an analysis of the data based on a model equation 0()1 2 3iki i i iiy flow ratio tempβδβ β β ε=+ + + + + (***) where ( ) the day number on which the th run of the process was madeki i= under the interpretation that1234,,,δδδδare iid()2N0,δσ, independent of theiε. This is a mixed linear model. (a) What are and Zuhere in the usual matrix formulation of the mixed model =++YXβ Zu ε ? (Use the ordering of the elements of Y employed on the printout and in the table on page 1.)3Stat 511 MS Exam, Spring 2003 Page 3 of 3 (b) Give an approximate 95% confidence interval forδσin the mixed model (***). (c) What is an approximate BLUE of012 350 4 470βββ β+++ ? (Give a numerical value.) Then write down a standard error for your estimate. (Display a formula, insert numerical values, but don't complete the calculations.) (d) Consider the 4 observations from setup #1. In the mixed model (***), these all have the same mean, namely 012 3115 6 495βββ β+++, and differ only in that each one has a different (),δεpair added to the mean. Consider the sample variance of these 4 observations. What function of the mixed model parameters does this estimate? Find an estimate of this function of the mixed model parameters based on the REML estimates. How does this REML-based estimate compare to the sample variance? Comments?1Printout for Stat 511 MS Exam Spring 2003 Page 1 of 3 > day<-c(1,1,1,2,2,2,3,3,3,4,4,4) > flow<-c(115,50,180,50,180,115,50,180,115,50,180,115) > temp<-c(495,470,520,520,470,495,520,470,495,470,520,495) > ratio<-c(6,4,8,4,8,6,8,4,6,8,4,6) > y<-c(82,91,75,98,39,77,95,61,81,76,92,82) > Day<-as.factor(day) > Flow<-as.factor(flow) > Temp<-as.factor(temp) > Ratio<-as.factor(ratio) > options(contrasts=c("contr.sum","contr.sum")) > setup<-c(1,2,3,4,5,1,6,7,1,8,9,1) > D<-data.frame(y,flow,ratio,temp,day,setup) > D<-data.frame(y,flow,ratio,temp,day,setup) > D y flow ratio temp day setup 1 82 115 6 495 1 1 2 91 50 4 470 1 2 3 75 180 8 520 1 3 4 98 50 4 520 2 4 5 39 180 8 470 2 5 6 77 115 6 495 2 1 7 95 50 8 520 3 6 8 61 180 4 470 3 7 9 81 115 6 495 3 1 10 76 50 8 470 4 8 11 92 180 4 520 4 9 12 82 115 6 495 4 1 > lmout1<-lm(y~flow+ratio+temp) > summary(lmout1) Call: lm(formula = y ~ flow + ratio + temp) Residuals: Min 1Q Median 3Q Max -11.458 -1.990 2.417 3.292 5.792 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -109.14936 46.13914 -2.366 0.045554 * flow -0.17885 0.03528 -5.069 0.000966 *** ratio -3.56250 1.14657 -3.107 0.014509 * temp 0.46500 0.09173 5.069 0.000966 *** --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 6.486 on 8 degrees of freedom Multiple R-Squared: 0.8841, Adjusted R-squared: 0.8407 F-statistic: 20.35 on 3 and 8 DF, p-value: 0.00042232> anova(lmout1) Page 2 of 3 Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F)


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ISU STAT 511 - MS Exam -2003

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