Pts consider the heteroscedastic regression through

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Problem 3 [20 pts]Consider theheteroscedasticregression through the origin modelYi=βxi+i,ı = 1, . . . , n,iindN(0,σ2i),wherewi=1σ2iandσ2iis known fori= 1, . . . , n.Part I (10 pts)Derive the weighted least squares estimatorˆβwfor the slope parameterβ. You can use amatrix approach or a scalar approach for this problem. You cannot begin with the weightedleast squares solution. You must solve the optimization problem from scratch.weanuseLSorMLE.5
Part II (5 pts)Show that the weighted least squares estimatorˆβwis an unbiased estimator ofβ.6
Part III (5 pts)Derive an expression for the variance ofˆβw. Simplify the expression completely.
Methods ComponentProblem 4 [15 pts]Patients who suer from moderate to severe migraine headache took part in a double-blindclinical trial to assess an experimental surgery.A group of 79 patients were randomlyassigned to receive either the real surgery in migraine trigger sites (m= 53) or a shamsurgery (n= 26) in which an incision was made but no further procedure was performed.The surgeons hoped that patients would experience “a substantial reduction in migraineheadaches,” which we will label as “success.” A substantial reduction means at least 50%reduction in migraine headache frequency, intensity, or duration when compared with base-line (presurgery) values.SurgeryNo successSuccessReal1241Sham1115Consider the following logistic regression output related to the above dataset.> Success <- c(rep(1,41+15),rep(0,12+11))> Real <- c(rep(1,41),rep(0,15),rep(1,12),rep(0,11))> summary(glm(Success~Real,family=binomial(link="logit")))Call:glm(formula = Success ~ Real, family = binomial(link = "logit"))Coefficients:Estimate Std.Errorz-valuePr(>|z|)(Intercept)0.31020.39700.7810.4346RealMissing0.5151Missing0.0745 .Part I (10 pts)Compute the two values missing from the aboveRoutput.8
Part II (5 pts)Run the appropriate test to see if successful reduction of migraine headache was morecommon among patients who received the real surgery than among those who received thesham surgery.Problem 5 [10 pts]A university medical center urology group was interested in the association between prostate-specific antigen (PSA) and a number of prognostic clinical measurements in men with ad-vanced prostate cancer. Data were collected on 97 men who were about to undergo radicalprostatectomies. The 8 variables are:VariableVariable NameDescriptionX1PSA levelSerum prostate-specific antigen level (mg/ml)X2Cancer volumeEstimate of prostate cancer volume (cc)X3WeightProstate weight (gm)X4AgeAge of patient (years)X5Benign prostatic hyperplasiaAmount of benign prostatic hyperplasia (cm2)X6Seminal vesicle invasionPresence or absence of seminal vesicle invasionX7Capsular penetrationDegree of capsular penetration (cm)YGleason scorePathologically determined grade of disease

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