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MAST90084: Statistical Modelling Assignment
创建日期：2024-04-06 15:39:46
所属分类：统计学statistics
标签： MAST90084
Statistical Modelling Assignment

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MAST90084: Statistical Modelling Assignment 1
1. Let X and Y be two categorical random variables, X with I different categories identified with the set
{1, . . . , I} and Y with J different categories identified with the set {1, . . . , J}. Suppose observations of
the variable pair (X,Y ) are tabulated in a I × J contingency table. Using standard notations, for a given
(i, j) ∈ {1, . . . , I} × {1, . . . , J}, nij is the entry in the (i, j)-th cell that denotes the count of observations
with X equal to its i-th category and Y equal to its j-th category. A Poisson sampling model for the
contingency table assumes that the nij ’s are independently distributed with
nij ∼ Poi(µij),
where µij denotes the Poisson mean for the cell count nij .
(a) Derive the conditional joint distribution of {nij}(i,j)∈{1,...,I}×{1,...J} given n. Identify the name of this
distribution, and explicitly state what its parameter values are in terms of {µij}(i,j)∈{1,...,I}×{1,...J}
and n. [5]
(b) Let I = J = 2. The quantity µ11/µ12µ21/µ22 , also known as the odds ratio, measures the association between
X and Y . What should be the value of the odds ratio if X and Y are independent and why? [3]
2. Data in the following 2× 2× 3 contingency table were used to study the effect of passive smoking on lung
cancer. The table summarizes the results of case-control studies from 3 countries for nonsmoking women
married to smokers. (Source: Blot and Fraumeni, J. Nat. Cancer Inst., 77:993-1000 (1986) and Agresti
(1996).)
Country Spouse Smoked Cases Controls
Japan No 21 82
Yes 73 188
UK No 5 16
Yes 19 38
USA No 71 249
Yes 137 363
(a) A log-linear model mod1 can be fitted to the data, with the results being given in the following R
output. Give the mathematical formula of form ln(µ) = · · · for the mean model of mod1, where µ is
the mean of the response. Any dummy variables in your formula should be explicitly defined. [5]
> pasSmoking.dat=data.frame(freq=c(21,73,5,19,71,137,82,188,16,38,249,363))
> pasSmoking.dat$Cnt=factor(rep(c("Japan","UK", "USA"), times=2, each=2))
> pasSmoking.dat$Smo=factor(rep(c("No","Yes"), times=6))
> pasSmoking.dat$Can=factor(rep(c("Case","Control"), each=6))
> pasSmoking.dat
freq Cnt Smo Can
1 21 Japan No Case
2 73 Japan Yes Case
3 5 UK No Case
4 19 UK Yes Case
5 71 USA No Case
6 137 USA Yes Case
7 82 Japan No Control
8 188 Japan Yes Control
9 16 UK No Control
10 38 UK Yes Control
11 249 USA No Control
12 363 USA Yes Control
MAST90084 Statistical Modelling Assignment 1 Semester 1, 2021
> mod1=glm(freq~Cnt+Smo+Can+Cnt:Smo+Cnt:Can+Smo:Can, family=poisson, data=pasSmoking.dat)
> anova(mod1, test="Chisq")
Analysis of Deviance Table; Model: poisson; Link: log; Response: freq
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev P(>|Chi|)
NULL 11 1168.85
Cnt 2 726.43 9 442.42 < 2.2e-16
Smo 1 112.52 8 329.90 < 2.2e-16
Can 1 307.56 7 22.34 < 2.2e-16
Cnt:Smo 2 15.50 5 6.84 0.0004316
Cnt:Can 2 1.05 3 5.80 0.5919109
Smo:Can 1 5.56 2 0.24 0.0184215
> 1-pchisq(0.24,2)
[1] 0.8869204
> 1-pchisq(5.80,3)
[1] 0.1217566
(b) Expanding the notations from Question 1, for the current contingency table we can also use nijk to
denote the count in each cell, where i ∈ {1, 2}, j ∈ {1, 2}, k ∈ {1, 2, 3} are indices corresponding
to Can (variable X), Smo (variable Y ) and Cnt (variable Z) respectively. Moreover, if nijk are
independently distributed with
nijk ∼ Poi(µijk),
one can, for any k ∈ {1, 2, 3}, define the odd ratios θXY (k) = µ11kµ22kµ12kµ21k for the partial table with Z = k.
The table is said to have homogeneous XY association when θXY (1) = θXY (2) = θXY (3). Explain why
the model in part (a) has XY homogenous association. [5]
(c) Based on the displayed R output in (a), test the significance of the interaction effect Smo:Can at
significance level 0.05, eliminating the effects of all other terms in mod1. Provide your conclusion
with clear explanation. [4]
(d) Based on the displayed R output in (a), test the adequacy of model Cnt+Smo+Can+Cnt:Smo+Cnt:Can
at significance level 0.05. Provide your conclusion with clear explanation. [4]
(e) Are your conclusions in (c) and (d) contradictory? You must give an explanation to get any score.
[5]
3. A variable Y taking values in {0, 1, 2, . . . } has a Negative Binomial (NB) distribution if its probability
mass function has the form
p(Y = y;µ, κ) =
Γ(κ+ y)
Γ(κ)y!
κκµy
(µ+ κ)κ+y
,
for y = 0, 1, . . . , where µ is the mean of Y .
(a) When κ is considered as fixed (or known), the NB distribution belongs to the exponential dispersion
model (EDM) discussed in class. Write out its form as an EDM explicitly. In particular, you have
to identify the natural parameter θ and the dispersion parameter φ in terms of µ and κ whenever
appropriate, and identify b(·) (as a function of θ). You can simply take the weight ω to be 1. [5]
(b) Let σ2 be the variance of Y . From your answer above, derive the formula for σ2 as a function of µ.
Why do we say that the NB distribution can be used as a likelihood model to handle “overdispersion”
compared to the Poisson distribution? [4]
2
MAST90084 Statistical Modelling Assignment 1 Semester 1, 2021
Total marks = 40

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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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