Instructions for assignment: You will need to submit two documents for this assignment. The first document is a pdf document named Assign3 StNo.pdf which will provide all your analysis and solutions for this assignment. To produce this pdf document you will need to use LaTeX. The LaTeX document which was used to produce this assignment is named Assign3 StNo.tex and is located in the Assignment 3 folder in the Topic 9 section of LMS. You can use LaTeX online via Overleaf which is a website dedicated to producing documents from LaTeX. To use LaTeX, follow the instructions in the Overleaf.pdf document located in the Assignment 3 folder. The second document that you will need to submit is an R document named Assign3 R StNo.R which is located in the Assignment 3 folder. This document should provide the R code you used to perform all your data manipulation and analysis.
Assessment information for assignment: There are a total of 80 marks for this assignment.
Description of assignment: The data presented in this assignment is adapted from Ocampo (2005)1 and is stored in the file named Veneer.csv located in the Assignment 3 folder in the Topic 9 section of LMS. A study was conducted on fifty-five adult teeth to examine the impact of veneer placement treatment on gingival (gum) health. The variables of interest for Assignment 3 are:
• Tooth: This is a factor variable that identifies the tooth.
MA: This is a continuous variable that measures the age of the tooth less the mean tooth age (where the mean tooth age is calculated by averaging over the teeth in the sample).
MBG: This is a continuous variable that measures the pre-treatment gingival crevicular fluid of a tooth less the mean pre-treatment gingival crevicular fluid of teeth (where the mean pre-treatment gingival crevicular fluid of teeth is calculated by averaging over the teeth in the sample).
MC: This is a continuous variable that measures the post-treatment contour difference of a tooth less the mean post-treatment contour difference of teeth (where the mean post-treatment contour difference of teeth is calculated by averaging over the teeth in the sample).
T : This is a factor variable that measures two post-treatment time points. It has two levels (0 =
3-months post-treatment, 1 = 6-months post-treatment ).
G: This is the response variable. It’s a continuous variable that measures the gingival crevicular fluid of a tooth at 3-months and 6-months post-treatment.
2 marks are allocated for each question that requires the use of the R computer package. These marks are awarded using the following criterion:
1. R code that accurately produces the analysis/output required in the question.
1 Graphical analysis
1. Use the R computer package to produce a plot of G vs MA grouped by T . The scale of the vertical and horizontal axes of your figure should be identical to Figure 1 below. (Note that the categories of the T variable need to be represented by colours in Figure 1. Also, for each category of T , you must include a line of best fit which represents the linear relationship between G and MA. Please refer to Figure 3 in the Topics 7 and 8 lectures as a guide.) (2 marks). Do you think that there is an interaction effect between MA and T ? Explain. (2 marks).
2. Use the R computer package to produce a plot of the values of G vs T for each Tooth. The scale of the vertical and horizontal axes of your figure should be identical to Figure 2 below (2 marks). Do you think that the random intercept should be included in the linear mixed model? Explain. (2 marks) Do you think random effect of T on G should be included in the linear mixed model? Explain. (2 marks)
2 Describing the model
The researchers in the study set up the following linear mixed model to analyze their research questions.
Gti = β0 + β1 MAi + β2 MBGi + β3 MCi + β4 Tti
+ β5 MAi × Tti + β6 MBGi × Tti + β7 MCi × Tti
+ µ0i + µ1i Tti + εti, (1)
• where Gti is the gingival crevicular fluid of for tooth i (i = 1, 2, . . . , 55) at occasion t (t = 1, 2),
• MAi is the age of tooth i less the mean tooth age,
MBGi is the pre-treatment gingival crevicular fluid of tooth i less the mean pre-treatment gingival crevicular fluid of teeth,
MCi is the post-treatment contour difference of tooth i less the mean post-treatment contour difference of teeth,
Tti = 1 if the post-treatment time at occasion t for tooth i is 6 months, and Tti = 0 if the post-treatment time at occasion t for tooth i is 3 months,
• β0 is the fixed intercept,
• β1, β2, β3 and β4 are the fixed simple effects of MA, MBG, MC and T respectively,
• β5, β6 and β7 are the fixed two-way interaction effects of MA× T , MBG× T and MC × T respectively,
• µ0i is the random intercept specific to tooth i,
• µ1i is the random effect of T on G specific to tooth i,
• εti is the random error associated with measuring G for tooth i, on occasion t.
• For model (1), the researchers choose an unstructured structure for the variance-covariance matrix of the random effect vector, µi. That is, the variance-covariance matrix of the random effect vector, µi, is
ñ ô
D = θ0 θ01 ,
θ01 θ1
– where θ0 and θ1 denotes the variance of the random effects µ0i and µ1i, respectively,
– θ01 denotes the covariance between the random effects µ0i and µ1i.
• Also for model (1), the researchers choose the following structure for the variance-covariance matrix of the random error vector, εi,
ñ ô
τ 0
R = 0 τ
– where τ = V ar(ε1i) = V ar(ε2i).
3. The researchers would like to express model (1) in matrix form, Yi = Xi β + Zi µi + εi, where Yi represents the response vector for tooth i, Xi represents a matrix, for tooth i, that contains the values of the predictors associated with the fixed effects of model (1), β is the fixed effect vector, Zi is a matrix, for tooth i, that contains the values of the predictors associated with the random effects of model (1), µi is the random effect vector for tooth i and εi is the random error vector for tooth i. Answer the following questions.
(a) Write down the observed response vector, Yi, of model (1), for tooth i = 7 (the 7th tooth in the data set). (2 marks)
(b) Write down the observed matrix, Xi, of model (1), for tooth i = 7 (the 7th tooth in the data set). (4 marks)
(c) Write down the fixed effect vector, β, of model (1). (1 mark)
(d) Write down the matrix, Zi, of model (1), for tooth i. (2 marks)
(e) Write down the random effect vector, µi, of model (1), for tooth i. (2 marks)
(f) Write down the random error vector, εi, of model (1), for tooth i. (2 marks)
(g) For model (1), derive the variance-covariance matrix of the response vector, Yi, for tooth i. Show all workings. (5 marks)
4. Interpret the intercept β0. (3 marks)
5. Interpret the simple effects β1, β2, β3 and β4. (12 marks)
6. Interpret the interaction effects β5, β6 and β7. (9 marks)
3 Variance-covariance estimates of the final linear mixed model
As their final linear mixed model the researchers choose model (1) which has variance-covariance matrices,
D and R, defined in section 2. Use this model to answer the questions in this section and in section 4.
7. Use the R computer package to calculate the estimate of the R matrix of the final linear mixed model. Present this estimate below. Note, round all the elements in the matrix to two decimal places. (2 marks)
8. Use the R computer package to calculate the estimate of the variance-covariance matrix of the response vector of the final linear mixed model. Note, round all the elements in the matrix to two decimal places. (2 marks)
9. Use your solutions for questions 3(g), 7 and 8 to calculate the estimates of θ0, θ01 and θ1, respectively. Show all your workings. (4 marks)
4 Predicted values and residuals of the final linear mixed model
10. Use the R computer package to produce a table that lists the estimates of the fixed effects in the final linear mixed model, together with their corresponding standard errors, degrees of freedom, observed test statistics and p-values. Present this table below. Note that each value in the table needs to be rounded to two decimal places. (2 marks)
11. Figure 3 presents the random effect predictions of the final linear mixed model, for the first 10 teeth in the study. These predictions were obtained by using the ranef() command in R.
Calculate by hand the predicted conditional value of G for tooth i = 3 at 6-months post-treatment. Show all your workings. (Note: To answer this question, you will need to use the appropriate information presented in Figure 3, the fixed effect estimates you computed in question 10 and the raw data presented in the Veneer.csv file). (4 marks)
12. Calculate by hand the marginal residual for tooth i = 9 at 3-months post-treatment. Show all your workings. (Note: To answer this question, you will need to use the appropriate fixed effect estimates you computed in question 10 and the raw data presented in the Veneer.csv file). (4 marks)
CS 340 Milestone One Guidelines and Rubric Overview: For this assignment, you will implement the fundamental operations of create, read, update,
Retail Transaction Programming Project Project Requirements: Develop a program to emulate a purchase transaction at a retail store. This
7COM1028 Secure Systems Programming Referral Coursework: Secure
Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip
CS 340 Final Project Guidelines and Rubric Overview The final project will encompass developing a web service using a software stack and impleme
