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MAST90084: Statistical Modelling Assignment
创建日期：2024-04-12 16:42:17
所属分类：数学mathematics
标签： MAST90084
Statistical Modelling Assignment

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MAST90084: Statistical Modelling Assignment 

 1. Suppose the data consist of repeated observations (yit,x T it), t = 1, · · · , T , for each individual i = 1, · · · , n. Here yit is the response and xit is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of xit is of the fol- lowing form yi = Ziβ +Wibi + εi, where yi = (yi1, · · · , yiT )T ; Zi is a T × p design matrix built from {xit} for the fixed effects β; Wi is a T × q design matrix for the random effects bi; {bi} are i.i.d. MVN(0, Q) random vectors with Q > 0; {εi} are i.i.d. MVN(0, σ2I) random vectors with σ2 > 0 and I being an identity matrix; and {bi} and {εi} are mutually uncorrelated. We will assume all Zi,Wi are of full rank, and for now the variance-covariance components θ = (σ2, Q) are also assumed to be fixed. When β is treated as fixed, Harville (1976) has derived the formulae βˆ = ( n∑ i=1 ZTi V −1 i Zi )−1 n∑ i=1 ZTi V −1 i yi (1) and bˆi = QW T i V −1 i (yi − Ziβˆ), i = 1, . . . , n, (2) as the best linear unbiased estimator (BLUE) for δ := (βT ,bT1 , . . . ,b T n ) T , where Vi = Iσ 2 + WiQW T i . Alternatively, consider a Bayesian formulation where β is considered random and indepen- dent of all bi’s, with an improper “flat” prior imposed on it (i.e., the coordinates of β are i.i.d. and each uniformly distributed on the whole real line ). Hence, the posterior density of δ satisfies the proportionality relationship f(δ|Y ; θ) ∝ n∏ i=1 f(yi|bi,β;σ2) n∏ i=1 p(bi;Q), (3) where Y = {y1, . . . ,yn} is the full dataset, f(yi|bi,β;σ2) is the density of yi given (bi,β) and p(bi;Q) is the MVN(0, Q) prior on bi. This problem aims to derive that the posterior mode δ˜ := (β˜, b˜1, . . . , b˜n) ≡ arg max δ f(δ|Y ; θ) is given by (1) and (2) as well. In the following problems, this general fact may be useful: If X is a random vector with an invertible covariance matrix Σ and has a density f(X) satisfying the proportionaliity relationship f(X) ∼ exp ( −1 2 XTΣ−1X +XT c ) , for some vector c with the same dimension as X, then X must be multivariate normal with mean Σc and covariance Σ. MAST90084 Statistical Modelling Assignment 4 Semester 1, 2021 (a) Using the proportionality in (3), show that the posterior density f(δ|Y ;Q, σ) is a mul- tivariate normal; in particular, demonstrate that the posterior precision matrix (i.e. the inverse of the covariance matrix) of δ has the form Cov[δ|Y ; θ]−1 =  σ−2 ∑n i=1 Z T i Zi σ −2ZT1 W1 · · · σ−2ZTnWn σ−2W T1 Z1 σ −2W T1 W1 +Q −1 · · · 0 ... ... . . . ... σ−2W Tn Zn 0 · · · σ−2W TnWn +Q−1  , where the lower right submatrix is block-diagonal. From the posterior normality of δ, one can deduce that δ˜ must be the same as the posterior mean. [5] (b) Show that β˜ has the form in (1). [5] Hint: Consider the marginalized version of the linear mixed model, i.e. for a given β, yi’s are independent and each distributed as yi ∼MVN(Ziβ, Vi) (c) Show that b˜i has the form in (2) for i = 1, . . . , n. Hint: You showed in (a) that p(b1, . . . ,bn,β|Y ; θ) is multivariate normal. This implies that p(b1, . . . ,bn|Y,β; θ), the conditional density of b1, . . . ,bn given (Y,β), is also multivariate normal. You may find the following Woodbury matrix identity useful: (σ−2W Ti Wi +Q −1)−1 = Q−QW Ti (σ2I +WiQW Ti )−1WiQ. [6] 2 MAST90084 Statistical Modelling Assignment 4 Semester 1, 2021 2. This question is a continuation of the previous one. Now suppose θ = (σ2, Q) is unknown. Let y = (yT1 , . . . ,y T n ) T , Z = (ZT1 , . . . , Z T n ) T and V be a block diagonal matrix such that V1, . . . Vn are on its diagonal. For a fixed value of β, the marginalized version of the linear mixed effect model takes the form y ∼MVN(Zβ, V ) after integrating out the bi’s. Since Z is of rank p, there exists a matrix A of dimensions (nT − p) × (nT ) such that AZ = 0 , AAT = I and ATA = I − Z(ZTZ)−1ZT . Precisely, the rows of A comprise a set of orthonormal vectors spanning col(Z)⊥, i.e., the orthogonal complement of the column space of Z. Hence, the distribution of αˆ := Ay, conditional on β or not, is normal and doesn’t depend on β, and we denote this density as f(αˆ|θ) to show its dependence on θ alone. It is also not hard to see that βˆ can be equivalently expressed as βˆ = (ZTV −1Z)−1ZTV −1︸ ︷︷ ︸ B y = By (a) Show that given a fixed value of β, βˆ is independent of αˆ := Ay. [3] (b) Show that up to additive constants not involving θ, log f(αˆ|θ) has the REML form 1 2 { − log |V | − log |ZTV −1Z| − (y − Zβˆ)TV −1(y − Zβˆ) } , where | · | denotes the determinant of a matrix. [7] Hint: Let C := [AT |BT ]T , a square matrix stacking A on top of B. The map y 7→ (αˆ, βˆ) is a one-to-one transformation involving the Jacobian determinant |C|. Use the result from (a), as well as the facts (y − Zβ)TV −1(y − Zβ) = (y − Zβˆ)TV −1(y − Zβˆ) + (β − βˆ)T (ZTV −1Z)(β − βˆ). and |C| = |ZTZ|−1/2. (The equality for |C| can be obtained by a trick that uses the Schur complement formula for determinant, but its derivation is irrelevant to solving this problem.) 3 MAST90084 Statistical Modelling Assignment 4 Semester 1, 2021 3. This question concerns parametric Weibull regression for survival analysis. Let T be Weibull- distributed and suppose the proportional hazard (PH) model describes its dependence on covariates as follows: For a given covariate vector x, the hazard function of T has the form h(t|x) = h0(t)φ(x) for the baseline hazard h0(t) = λα(λt) α−1 and some positive function φ(·). (a) Show that there exists a function ψ(·) so that h(t|x) = ψ(x)h0(ψ(x)t), i.e. T ’s dependence on x also follows a general accelerated failure time (AFT) model. [4] (b) Consider the gehan dataset that assesses the ability of 6-mercaptopurine (6MP) in maintaining remission time (in weeks) among leukemia patients, available in the MASS package. library(MASS) help(gehan) head(gehan) ## pair time cens treat ## 1 1 1 1 control ## 2 1 10 1 6-MP ## 3 2 22 1 control ## 4 2 7 1 6-MP ## 5 3 3 1 control ## 6 3 32 0 6-MP i. Read the documentation pages for survreg() from the package survival, as well as Modern Applied Statistics with S by Venables and Ripley, 4th Edition, p.359- 361 (ebook available from unimelb library). Use survreg() to fit the Weibull PH model with φ(x) = exp(γx), where x = 1 for the treatment group and x = 0 for the control group. Report the estimates for γ, α, and λ. [7] Hint: You may find the following useful: a standard exponential distribution has the density f(x) = e−x. and a standard Gumbel distribution has the density e−(x+e −x). Moreover, If X is a standard exponential random variable, − log(X) has a standard Gumbel random variable. 4 MAST90084 Statistical Modelling Assignment 4 Semester 1, 2021 ii. For each treatment group, obtain an estimate of the probability of having remission time more than 18 weeks. [4] iii. Form a 95% confidence interval for the PH regression coefficient γ. [6] iv. Form a 95% confidence interval for the hazard ratio h(t|x=1) h(t|x=0) . [3] Total marks = 50 5

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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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