STA5ARM Assignment 2 Assignment 2 is due no later than 5pm Friday, 13th of May, 2022. In submitting your work, you are consenting that it may be copied and transmitted by the University for the detection of plagiarism. Submission is your guarantee that the below statement of originality is correct. \"This is my own work. I have not copied any of it from anyone else.\" NAME: Replace this text with your name. STUDENT NUMBER: Replace this text with your student number. Instructions for assignment: You will need to submit two documents for this assignment. The first document is a pdf document named Assign2_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 Assign2_StNo .tex and is located in the Assignment 2 folder in the Topic 7 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 Dverleaf .pdf document located in the Assignment 2 folder. The second document that you will need to submit is an R document named Assign2JLStNo .R which is located in the Assignment 2 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 50 marks for this assignment. Description of assignment: The data presented in this assignment is stored in the le named Veneer. csv located in the Assignment 2 folder in the Topic 7 section of LMS. A study was conducted on the teeth of 25 adults to examine gingival (gum) health. The variables of interest for Assignment 2 are: 0 Adult: This is a factor variable that identies the adult in the study. 0 A: This is a continuous variable that measures the age of the adult less the mean age (where the mean age is equal to the mean age of the adults in the sample). 0 W: This is a continuous variable that measures the weight of the adult less the mean weight (where the mean weight is equal to the mean weight of the adults in the sample). 0 G: This is a factor variable that identies the gender of the adult (0 = female? 1 = male). 0 C: This is the response variable. It's a continuous variable that measures the gingival crevicular uid of a tooth. 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. Answer the following questions. \fc 5,,- is the random error associated with measuring C for tooth i, belonging to adult 3'. o For model (1), the researchers choose an unstructured structure for the variance-covariance matrix of the random effect vector, pg. That is, the variance-covariance matrix of the random effect vector, #1: is D=[6], where 6' denotes the variance of the random effect #0,. 0 Also for model (1), the researchers choose the following structure for the variance-covariance matrix of the random error vector, 83-, T Tp 7,9 TP Tl\" T H) Tp if gender for adult j is male Tp 79 T TP Tp T9 T9 T 10 up up up 11))\" ff' if\") 19,0 if gender for adult j 13 female 7,129 up a W 7,09 up ip w where 'r denotes the constant variance of the random errors associated with adult j where the gender of adult j is male, 1;; denotes the constant variance of the random errors associated with adult 3' where the gender of adult 3." is female, p denotes the correlation between any two random errors belonging to adult 3". 1. The researchers would like to express model (1) in matrix form, Y3; = X, ,6 + Z5.- pj + 53-, where Y,- represents the response vector for adult j, X ,- represents a matrix, for adult j, that contains the values of the predictors associated with the xed effects of model (1), )3 is the xed effect vector, Z ,- is a matrix, for adult 3', that contains the values of the predictors associated with the random eifects of model (1), #5,: is the random effect vector for adult 3' and ej- is the random error vector for adult 3'. Answer the following questions. (a) Write down the observed response vector, Y,.-, of model (1), for adult j = 5 (the 5th adult in the data set). (2 marks) (b) Write down the observed matrix, X j, of model (1), for adult 3." = 15 (the 15th adult in the data set). (5 marks) (c) Write down the xed eifect vector, 5, of model (1). (1 mark) (d) Write down the matrix, Z,, of model (1), for adult 3'. (2 marks) (e) Write down the random effect vector, :15, of model (1), for adult j. (2 marks) (f) Write down the random error vector, 8,, of model (1), for adult 3'. (2 marks) (g) For model (1), derive the variance-covariance matrix of the response vector, Y,, for a male adult in this study. Show all workings. (5 marks) 2. Interpret the fixed effects g, .61 and BS. (9 marks)