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R代写-MTHM506

时间：2021-01-16

MTHM506 Statistical Data Modelling

Problem Sheet 2

(Covers Topics 3-4)

You should attempt all questions on this sheet. The questions constitute both summative (indicated by marks) and

formative assessment. Marks achieved in this assignment will contribute 25% of the final module mark. Solutions

are expected to be clearly explained, concise, well structured and well presented. Give R input commands

for each model fitted (e.g. ‘model <- gam(...)’). Do not display too much raw R output as part of your

solutions (e.g. don’t display the full output of ‘summary(model)’), but edit this down to the essentials. All

plots should have titles and appropriately labelled axes. Hand written solutions will be accepted, but a more

professional word processed submission is preferred (as is a mixture of the two).

Topic 3

1. The data set globalMeanTemp contains global mean temperature monthly “anomalies” for 1880–2013.

The term “anomaly” means an overall mean value was substracted from the original data points. Interest

lies in quantifying the temporal trend (change in the mean over time) in the data.

(a) Why does an additive model make sense for this data set?

(b) Fit an additive model to this data, ensuring this has adequate flexibility and that it fits the data well.

(c) Produce a plot of the data and estimated mean (trend) along with 95% confidence as well as

prediction intervals.

(d) Investigate (using additive models) whether there is a significant seasonal cycle in the data.

2. In this question we return to question 2 from sheet 1 where the following non-linear Gaussian model

was implemented:

yi ∼ N

(

θ1xi

θ2 + xi

, σ2

)

.

The relevant data are stored in the dataframe nlmodel.

(a) Fit a Gaussian GAM with identity link using the function gam() in R package mgcv. Use a cubic

spline basis and assume a basis dimension of q = 9 by denoting the smooth function of x as

s(x,k=9). (1)

(b) First check whether the rank of 9 is enough by running the function gam.check() on the fitted

model and checking whether the effective degrees of freedom is close to the maximum possible

(q− 1). This function also produces residual plots so also comment on those (but note that it plots

deviance residuals and not standardised deviance residuals). (6)

(c) Use the function predict() with se.fit=T to produce the fitted line along with 95% confidence

intervals. Given that the true relationship between x and the mean is the one given above, i.e.

(θ1x)/(θ2 + x), state why you would want to reduce the rank of this model. (3)

(10)

3. In this question we return to the AIDS data from sheet 1: the number of quarterly aids cases in the UK,

yi, from January 1983 to March 1994.

(a) Fit a Poisson GAM (using a cubic spline) with a log link, where the response is the number of cases

and date is the predictor. Plot the counts against date and add the predicted line with associated

95% confidence intervals, and comment on the fit. Make sure to use an appropriate rank and

perform relevant model checking with respect to residual plots and the deviance. (8)

(b) Suggest two alternative models that would improve the fit. Implement one of these, and perform the

same model checking as in 3a. Also produce a plot of the predicted line with 95% CIs. Comment

on any differences between the predicted smooth lines of the two models and the possible reason

behind this. (8)

(16)

1

4. The dataframe munichrent which contains data on net rent per month in Euro (‘rent’) for more 3,000

randomly sampled similar sized apartments in Munich Germany. Other variables in the data set are

the year of construction (‘yearc’) the quality of the location according to an expert assessment (a three

level factor (‘location’) — average location (1), good location (2), top location (3)) and an additional

factor (‘district’) giving the actual district of munich in which the apartment is situated. Interest lies in the

relationship (if any) between net rent and the year of construction and also the expert assessment of

the location and possibly the interaction between year of construction and assessed quality of location.

(a) Fit a Normal GLM to the relationship between net rent and year of construction, location assess-

ment and the interaction between them. Produce a density plot of the distribution of the residuals

and a Q-Q plot of the residuals and also plot the residuals against the fitted values from the model.

On the basis of these plots explain with reasons why the linear model assumptions of a Normal

distribution with constant variance are not suitable for this data set.

(b) Suggest an alternative GLM for this data set using the same predictors and choosing a suitable

distribution from the exponential family with an appropriate link function. Justify your choices. Fit

your proposed model, report and interpret the results and carry out appropriate model checking.

(c) Extend your GLM to an equivalent GAM where the linear relationship between net rent and year

of construction and it’s interaction with assessed quality of location are replaced by appropriate

smooth functions. Fit this model, report and interpret the results, plot the associated estimated

smooth functions, and carry out appropriate model checking. Say (with reasons) whether you

prefer the GAM formulation to that of the GLM.

Topic 4

5. A tyre manufacturer wants to investigate amount of tyre wear, and whether this differs substantially

among the four positions on a car that each tyre can occupy (front offside, front nearside, rear offside

and rear nearside), and also whether the car type affects the amount of wear (i.e. whether there is an

interaction). An experiment was conducted in which tyres were fitted to a car, the car was driven at

a fixed speed and distance and the reduction in depth of tread caused by the test was measured (in

hundredths of a mm) for each tyre. The process was repeated 3 times with a new set of tyres each time,

for 3 different types of car (A, B and C). Results obtained are given in the data frame tyres.

(a) Fit a Normal GLM to test the effects of tyre position and car type.

(b) Briefly explain why it would be sensible to consider the effects for car_type as random instead

of assuming they are fixed, and fit a model with wheel position as a fixed effect and car type as

a random effect. Write down the mathematical formulation of this model and the estimates of the

conditional variance of the response and the variance of the random effects.

(c) Test for the significance of the fixed effects, using both the likelihood ratio test and a bootstrap test,

stating which is preferable and why.

(d) Repeat this to test for the significance of the random effects.

(e) Use the standard errors produced by R to conduce crude significance tests for the parameters of

car type. State any assumptions you are making.

(f) Conduct more accurate significance tests using bootstapping to construct confidence intervals.

(g) Predict the mean wear for the front offside wheel for car A but also the mean wear for an entirely

new car.

(h) Produce the residuals vs fitted values but also the QQ plot of the residuals and comment of the

validity of the model.

6. An experiment was conducted to compare 4 different treatments (A,B,C, and D) on the production of

penicillin. The material used for producing the penicillin is quite variable and it can only be made in

blends sufficient for 4 runs. So the data consists of 5 blends and the 4 treatments were applied to each

blend. The data can be found in dataframe penicillin. Clearly, the treatment can be considered as

fixed effects, however blend should be random (many more blends could have been chosen).

2

(a) Fit a normal mixed model with a fixed effect for treatment and a random effect for blend, making

sure you define the model mathematically. By fitting appropriate reduced models, test for the

significance of both the fixed and the random effects using likelihood ratio tests. (10)

(b) Comment on the validity of using these tests in mixed effects models, suggest an alternative way

of implementing these tests, and use it to compare with results in (a). (5)

(15)

7. The dataframe pupils which involves language scores in Dutch schools. This is an example of a

two level situation. Specifically, the data considers 131 schools (but only 1 class per school) for

i = 1, 2, . . . , 2287 students in grades 7 and 8. The nesting therefore occurs within each school

j = 1, . . . , 131.

Interest lies in assessing the impact on language scores of pupil factors such as IQ (IQ) and pupil social

status (ses). The response variable is denoted as test. The categorical variable (factor) Class refers

to the class that each pupil belongs to (so Class= 1, . . . , 131).

(a) First fit a (Normal) linear model using glm() with test as the response and IQ, ses and factor

Class as the covariates. Comment on the significance of the two continuous variables and perform

a likelihood ratio test to test on the overall significance of the factor Class. (2)

(b) i. State two reasons why one might want to treat the class effects as random. (4)

ii. Write down the mathematical formulation of a Normal random effect model (IQ and ses as

fixed effects and Class as a random effect). (2)

iii. Fit this model and comment on the significance of the fixed effects based on the t-tests. State

any assumptions you are making. (3)

iv. What is the estimate of the “within-class” variance and the “between-class” variance. What

is the estimate of marginal variance of the response, it is different to the (marginal) variance

from the model in (a) and if so, why? (5)

v. Test whether the variance of the random effects is zero, using a likelihood ratio test and com-

ment on its validity. (6)

vi. Plot a density estimate of the predicted random effects and superimpose their theoretical

Normal distribution using the estimate of their variance. Use functions qqnorm() and qqline

to produce a QQ plot of the random effects and comment on the validity of a Gaussian model

for the random effects. (4)

vii. Note that the functions fitted() and resid() in lme4, will produce the fitted values yˆ and

raw residuals y − yˆ. Use these functions, in conjunction with the two functions in (iii) to

produce a QQ plot of the residuals and a residuals vs fitted values plot. Comment on the

model assumptions using the two plots. (4)

(c) One of the student-level covariates is IQ which may affect the test results per student. However,

there may be class level (latent) variables, such as teacher competence, which may have an effect

on how IQ relates to the test result in each class. Such a scenario may be accommodated by

considering the parameter of IQ to be random rather than it being fixed (and constant across

classes).

i. Extend the model in (b) to make the parameter of IQ vary with Class. Compare (qualitatively)

the overall effect of IQ on test between this model and the model in (b). (4)

ii. Test for the significance of the random slope for IQ using a likelihood ratio test. (3)

(37)

8. In this question we return to the dataframe sexr which relates to the ratio of male to female births

across various countries. This can of course be modelled as a Binomial situation with number of male

births (Male) out a total number of births (Total). The hypothesis is that the male to female birth ratio

increases when “things are bad”. As such, two deprivation measures are included: totleatbirth (life

expectancy at birth) and tinmort (infant mortality) – higher value of which indicated more deprivation.

There is also information on each country (Country) to allow for country variability of the ratio.

(a) Fit a Binomial GLM with totleatbirth, tinmort and Country as covariates. Check whether the

model fits (w.r.t. to the saturated model) and also produce the QQ plot of the residuals.

3

(b) Now fit the model as a Binomial GLMM with a random effect for each observation. State why we

would not have to check whether this model fits (w.r.t. to the saturated model), and also produce

the QQ plot of the residuals and compare with the GLM.

(c) Check for the significance of the random effects using the likelihood ratio test, commenting on its

reliability.

(d) Comment on the significance of the fixed effects and they say about male to female birth ratio.

9. The dataframe hip contains information on the number of hip fractures (Nfract) in elderly population

per municipality (municipality), in Portugal. The data are stratified by gender (sex, 1=males and 2=fe-

males) and the socio-economic status, ses, of each municipality (factor 1=poor, 2=average, 3=good).

Interest here lies on the effect of ses on the rate of hip fractures per 1000 population, after allowing for

gender. The data frame also contains information on the number of people (Npop) in 1000s, in each

sex and ses combination per municipality.

(a) Give two reasons why we would want to treat the municipality effects as random. (4)

(b) Write down (mathematically) an appropriate GLMM with an appropriate offset to assess the effect

of ses on the hip incidence rate. (6)

(c) Fit this model in R and comment on parameter significance using the z-tests. (3)

(d) Produce a QQ plot of the residuals and comment appropriately. Also produce a QQ plot of the

random effects and comment on the appropriateness of the Normal distribution. (4)

(e) The function glmer fits the model using maximum likelihood. Use this fact to perform a likelihood

ratio test to assess the significance of the random effects, commenting appropriately on the validity

of the test. (5)

(22)

(Total=100)

4

Problem Sheet 2

(Covers Topics 3-4)

You should attempt all questions on this sheet. The questions constitute both summative (indicated by marks) and

formative assessment. Marks achieved in this assignment will contribute 25% of the final module mark. Solutions

are expected to be clearly explained, concise, well structured and well presented. Give R input commands

for each model fitted (e.g. ‘model <- gam(...)’). Do not display too much raw R output as part of your

solutions (e.g. don’t display the full output of ‘summary(model)’), but edit this down to the essentials. All

plots should have titles and appropriately labelled axes. Hand written solutions will be accepted, but a more

professional word processed submission is preferred (as is a mixture of the two).

Topic 3

1. The data set globalMeanTemp contains global mean temperature monthly “anomalies” for 1880–2013.

The term “anomaly” means an overall mean value was substracted from the original data points. Interest

lies in quantifying the temporal trend (change in the mean over time) in the data.

(a) Why does an additive model make sense for this data set?

(b) Fit an additive model to this data, ensuring this has adequate flexibility and that it fits the data well.

(c) Produce a plot of the data and estimated mean (trend) along with 95% confidence as well as

prediction intervals.

(d) Investigate (using additive models) whether there is a significant seasonal cycle in the data.

2. In this question we return to question 2 from sheet 1 where the following non-linear Gaussian model

was implemented:

yi ∼ N

(

θ1xi

θ2 + xi

, σ2

)

.

The relevant data are stored in the dataframe nlmodel.

(a) Fit a Gaussian GAM with identity link using the function gam() in R package mgcv. Use a cubic

spline basis and assume a basis dimension of q = 9 by denoting the smooth function of x as

s(x,k=9). (1)

(b) First check whether the rank of 9 is enough by running the function gam.check() on the fitted

model and checking whether the effective degrees of freedom is close to the maximum possible

(q− 1). This function also produces residual plots so also comment on those (but note that it plots

deviance residuals and not standardised deviance residuals). (6)

(c) Use the function predict() with se.fit=T to produce the fitted line along with 95% confidence

intervals. Given that the true relationship between x and the mean is the one given above, i.e.

(θ1x)/(θ2 + x), state why you would want to reduce the rank of this model. (3)

(10)

3. In this question we return to the AIDS data from sheet 1: the number of quarterly aids cases in the UK,

yi, from January 1983 to March 1994.

(a) Fit a Poisson GAM (using a cubic spline) with a log link, where the response is the number of cases

and date is the predictor. Plot the counts against date and add the predicted line with associated

95% confidence intervals, and comment on the fit. Make sure to use an appropriate rank and

perform relevant model checking with respect to residual plots and the deviance. (8)

(b) Suggest two alternative models that would improve the fit. Implement one of these, and perform the

same model checking as in 3a. Also produce a plot of the predicted line with 95% CIs. Comment

on any differences between the predicted smooth lines of the two models and the possible reason

behind this. (8)

(16)

1

4. The dataframe munichrent which contains data on net rent per month in Euro (‘rent’) for more 3,000

randomly sampled similar sized apartments in Munich Germany. Other variables in the data set are

the year of construction (‘yearc’) the quality of the location according to an expert assessment (a three

level factor (‘location’) — average location (1), good location (2), top location (3)) and an additional

factor (‘district’) giving the actual district of munich in which the apartment is situated. Interest lies in the

relationship (if any) between net rent and the year of construction and also the expert assessment of

the location and possibly the interaction between year of construction and assessed quality of location.

(a) Fit a Normal GLM to the relationship between net rent and year of construction, location assess-

ment and the interaction between them. Produce a density plot of the distribution of the residuals

and a Q-Q plot of the residuals and also plot the residuals against the fitted values from the model.

On the basis of these plots explain with reasons why the linear model assumptions of a Normal

distribution with constant variance are not suitable for this data set.

(b) Suggest an alternative GLM for this data set using the same predictors and choosing a suitable

distribution from the exponential family with an appropriate link function. Justify your choices. Fit

your proposed model, report and interpret the results and carry out appropriate model checking.

(c) Extend your GLM to an equivalent GAM where the linear relationship between net rent and year

of construction and it’s interaction with assessed quality of location are replaced by appropriate

smooth functions. Fit this model, report and interpret the results, plot the associated estimated

smooth functions, and carry out appropriate model checking. Say (with reasons) whether you

prefer the GAM formulation to that of the GLM.

Topic 4

5. A tyre manufacturer wants to investigate amount of tyre wear, and whether this differs substantially

among the four positions on a car that each tyre can occupy (front offside, front nearside, rear offside

and rear nearside), and also whether the car type affects the amount of wear (i.e. whether there is an

interaction). An experiment was conducted in which tyres were fitted to a car, the car was driven at

a fixed speed and distance and the reduction in depth of tread caused by the test was measured (in

hundredths of a mm) for each tyre. The process was repeated 3 times with a new set of tyres each time,

for 3 different types of car (A, B and C). Results obtained are given in the data frame tyres.

(a) Fit a Normal GLM to test the effects of tyre position and car type.

(b) Briefly explain why it would be sensible to consider the effects for car_type as random instead

of assuming they are fixed, and fit a model with wheel position as a fixed effect and car type as

a random effect. Write down the mathematical formulation of this model and the estimates of the

conditional variance of the response and the variance of the random effects.

(c) Test for the significance of the fixed effects, using both the likelihood ratio test and a bootstrap test,

stating which is preferable and why.

(d) Repeat this to test for the significance of the random effects.

(e) Use the standard errors produced by R to conduce crude significance tests for the parameters of

car type. State any assumptions you are making.

(f) Conduct more accurate significance tests using bootstapping to construct confidence intervals.

(g) Predict the mean wear for the front offside wheel for car A but also the mean wear for an entirely

new car.

(h) Produce the residuals vs fitted values but also the QQ plot of the residuals and comment of the

validity of the model.

6. An experiment was conducted to compare 4 different treatments (A,B,C, and D) on the production of

penicillin. The material used for producing the penicillin is quite variable and it can only be made in

blends sufficient for 4 runs. So the data consists of 5 blends and the 4 treatments were applied to each

blend. The data can be found in dataframe penicillin. Clearly, the treatment can be considered as

fixed effects, however blend should be random (many more blends could have been chosen).

2

(a) Fit a normal mixed model with a fixed effect for treatment and a random effect for blend, making

sure you define the model mathematically. By fitting appropriate reduced models, test for the

significance of both the fixed and the random effects using likelihood ratio tests. (10)

(b) Comment on the validity of using these tests in mixed effects models, suggest an alternative way

of implementing these tests, and use it to compare with results in (a). (5)

(15)

7. The dataframe pupils which involves language scores in Dutch schools. This is an example of a

two level situation. Specifically, the data considers 131 schools (but only 1 class per school) for

i = 1, 2, . . . , 2287 students in grades 7 and 8. The nesting therefore occurs within each school

j = 1, . . . , 131.

Interest lies in assessing the impact on language scores of pupil factors such as IQ (IQ) and pupil social

status (ses). The response variable is denoted as test. The categorical variable (factor) Class refers

to the class that each pupil belongs to (so Class= 1, . . . , 131).

(a) First fit a (Normal) linear model using glm() with test as the response and IQ, ses and factor

Class as the covariates. Comment on the significance of the two continuous variables and perform

a likelihood ratio test to test on the overall significance of the factor Class. (2)

(b) i. State two reasons why one might want to treat the class effects as random. (4)

ii. Write down the mathematical formulation of a Normal random effect model (IQ and ses as

fixed effects and Class as a random effect). (2)

iii. Fit this model and comment on the significance of the fixed effects based on the t-tests. State

any assumptions you are making. (3)

iv. What is the estimate of the “within-class” variance and the “between-class” variance. What

is the estimate of marginal variance of the response, it is different to the (marginal) variance

from the model in (a) and if so, why? (5)

v. Test whether the variance of the random effects is zero, using a likelihood ratio test and com-

ment on its validity. (6)

vi. Plot a density estimate of the predicted random effects and superimpose their theoretical

Normal distribution using the estimate of their variance. Use functions qqnorm() and qqline

to produce a QQ plot of the random effects and comment on the validity of a Gaussian model

for the random effects. (4)

vii. Note that the functions fitted() and resid() in lme4, will produce the fitted values yˆ and

raw residuals y − yˆ. Use these functions, in conjunction with the two functions in (iii) to

produce a QQ plot of the residuals and a residuals vs fitted values plot. Comment on the

model assumptions using the two plots. (4)

(c) One of the student-level covariates is IQ which may affect the test results per student. However,

there may be class level (latent) variables, such as teacher competence, which may have an effect

on how IQ relates to the test result in each class. Such a scenario may be accommodated by

considering the parameter of IQ to be random rather than it being fixed (and constant across

classes).

i. Extend the model in (b) to make the parameter of IQ vary with Class. Compare (qualitatively)

the overall effect of IQ on test between this model and the model in (b). (4)

ii. Test for the significance of the random slope for IQ using a likelihood ratio test. (3)

(37)

8. In this question we return to the dataframe sexr which relates to the ratio of male to female births

across various countries. This can of course be modelled as a Binomial situation with number of male

births (Male) out a total number of births (Total). The hypothesis is that the male to female birth ratio

increases when “things are bad”. As such, two deprivation measures are included: totleatbirth (life

expectancy at birth) and tinmort (infant mortality) – higher value of which indicated more deprivation.

There is also information on each country (Country) to allow for country variability of the ratio.

(a) Fit a Binomial GLM with totleatbirth, tinmort and Country as covariates. Check whether the

model fits (w.r.t. to the saturated model) and also produce the QQ plot of the residuals.

3

(b) Now fit the model as a Binomial GLMM with a random effect for each observation. State why we

would not have to check whether this model fits (w.r.t. to the saturated model), and also produce

the QQ plot of the residuals and compare with the GLM.

(c) Check for the significance of the random effects using the likelihood ratio test, commenting on its

reliability.

(d) Comment on the significance of the fixed effects and they say about male to female birth ratio.

9. The dataframe hip contains information on the number of hip fractures (Nfract) in elderly population

per municipality (municipality), in Portugal. The data are stratified by gender (sex, 1=males and 2=fe-

males) and the socio-economic status, ses, of each municipality (factor 1=poor, 2=average, 3=good).

Interest here lies on the effect of ses on the rate of hip fractures per 1000 population, after allowing for

gender. The data frame also contains information on the number of people (Npop) in 1000s, in each

sex and ses combination per municipality.

(a) Give two reasons why we would want to treat the municipality effects as random. (4)

(b) Write down (mathematically) an appropriate GLMM with an appropriate offset to assess the effect

of ses on the hip incidence rate. (6)

(c) Fit this model in R and comment on parameter significance using the z-tests. (3)

(d) Produce a QQ plot of the residuals and comment appropriately. Also produce a QQ plot of the

random effects and comment on the appropriateness of the Normal distribution. (4)

(e) The function glmer fits the model using maximum likelihood. Use this fact to perform a likelihood

ratio test to assess the significance of the random effects, commenting appropriately on the validity

of the test. (5)

(22)

(Total=100)

4