ST30005 - Multivariate Analysis - Statistics Assignment Help

Assignment Task

Instructions:

Please prepare your submission in a document (Word or PDF) and clearly label all answers and output with their corresponding question number and part.

You should aim to answer questions concisely and focus only on the relevant aspects. Only include a figure if it is relevant to answering the question. A strict maximum of one figure per question part may be included. For example, a separate figure may be included for both Q1a and Q1b, but neither part can have more than one figure.

Any software output, such as figure, should be included in the document (either at the relevant question, or in an appendix). All such output MUST be referred to in your answer. For example, “The scatterplot in figure 1a shows a strong negative relationship between …”. Any software output that is not referred to in your answer should not be included.

Take care to write any equations clearly and using a suitably sized font. You may consider using the equation editor in Word, but it is not required.

Any relevant R code should be included in an appendix. It MUST have documentation to indicate the purpose of the code, and be labelled with its corresponding question number and part.

Question 1

A naturalist for the Alaska Fish and Game Department studies grizzly bears with the goal of maintaining a healthy population. An analysis of measurements on N = 61 bears produced the following covariance matrix called bears:

The variables are X1: weight (kg), X2: body length (cm), X3: neck circumference (cm), X4: girth (cm), X5: head length (cm) and X6: head width (cm).

a. Read the data into R and explain why the matrix elements bears[2,3] and bears[3,2] are equal. (2 Marks)

b. Verify that bears is a covariance matrix. (2 Marks)

c. Conduct a principal component analysis on the covariance matrix, bears, and show your R code. Can the data be effectively summarised in fewer than six dimensions? Justify your answer. (2 Marks)

d. Based on the information provided in the question, and your results to part c, interpret the meaning of the first principal component. (2 Marks)

e. Which two variables have, respectively, the most and least influence on the value of first principal component? Justify your answer. (2 Marks)

Question2

The file wine.csv contains data on concentrations of 13 different chemicals in wines grown in the same region in Italy. They were derived from three different cultivars: a grape variety which has been selectively cultivated. The data were collected by the Italian Wine Association (IWA) and is available on Canvas.

You are contacted by the IWA and asked to analyse the data. They would like you to study the chemicals related to smell (chem6, chem7, chem9), taste (chem3, chem5) and alcohol content (chem8). The IWA inform you that each of these variables is as important as any other, and ask you to focus on summarising the information in this set of 6 variables by a smaller number of components.

Answer the following questions or complete the task.

a. Prepare the dataset for the analysis by extracting the relevant variables and check for any missing values. (2 Marks)

b. Examine the mean, median, standard deviation, and range of the data. Which aspect of these descriptive statistics is particularly relevant for the analysis asked for by the IWA? Justify your answer. (2 Marks)

c. Produce a plot to visualise potential univariate outliers in the data that will be used as input to a principal component analysis (PCA) and comment on the key features seen in the results. (3 Marks)

d. Use the relevant Mahalanobis distance measure to identify potential multivariate outliers (do not plot them). How many are there? Based on the number of these potential outliers, should they be investigated further? Justify your answers. (5 Marks)

e. Generate a scatterplot matrix of the relevant data for the PCA such that there are scatterplots in one set of off-diaganol elements, correlation coefficients in the other set of off-diaganol elements, and histograms on the diagonal of the matrix. Briefly comment on what the plot shows for each of these three aspects and what, if any, their relevance is for a PCA. (8 Marks)

f. Assuming that the data have been checked and there is no need to remove outliers or make any other changes, conduct a PCA and decide how many PCs to retain in your results. Justify your answer. (2 Marks)

g. State the value of the second largest eigenvalue found in your analysis of part f. Interpret its meaning in a PCA. (2 Marks)

h. Write the retained PCs from part f in equation form as linear combinations of the original variables. (2 Marks)

i. To what extent do the first two PCs measure different aspects in the data? Justify your answer. (2 Marks)

Question 3

The dataset household2.csv contains responses to a questionnaire on different aspects of housing. In the nine questions, Q1,…,Q9, participants were asked to rate their satisfaction with their living conditions on a scale from 1 to 7, with 7 indicating most satisfaction and 1 indicating most dissatisfaction. There was no missing data in the satisfaction data that was collected. The dataset is available on Canvas.

The primary goal of the analysis is to identify any dimensions that underly satisfaction with living conditions. A secondary goal is to use these results to measure satisfaction with living conditions in other similar populations or on other future occasions.

Answer the following questions or complete the task.

a. Download the dataset and select the variables needed for an exploratory factor analysis (EFA) of satisfaction with living conditions. Explain why the variable Gender should not be included in the EFA. (2 Marks)

b. Based only on the problem and data description given above in the question notes (and not on any results), why would an EFA initially be preferred to a PCA to achieve the analysis goals? (2 Marks)

c. Show for all variables to be included in the EFA that their recorded values are valid based on the data description given above in the question notes. (2 Marks)

d. Assuming that there is no need to remove outliers, show that the data are suitable for an EFA, and decide how many factors should be investigated in your EFA. Justify your answers. (2 Marks)

e. Conduct an EFA on the correlation matrix with as many factors as you think are justified. Is simple structure achieved in the unrotated solution? Justify your answer. 

f. What is the value of the second largest eigenvalue for the analysis done in part e? Interpret its meaning in an EFA. (2 Marks)

g. Does a varimax rotation of the results found in part e improve the solution? Justify your answer. (2 Marks)

h. Compare the unrotated solution of part e with an oblimin rotation of the results. Which of these two would be the preferred solution to answer the analysis goals? 

i. Based on your chosen solution to part h, what would you recommend as a next step (you don’t need to conduct this action) to achieve a solution with simple structure? Justify your answer. (3 Marks)

j. By inspecting the communalities seen with the results for part e, what is there to indicate that an EFA is a preferred analysis strategy to a PCA with these data? Justify your answer. (2 Marks)

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