Assignment:
Question 1
Gunst and Mason (1980) reported a study about the anthropometric and physical fitness measurements based on 50 white-male applicants to the police department of a major metropolitan city. The data are available in the file "police.xls". The variables included in this data set are:
REACT = reaction time in seconds
HEIGHT = applicant's height in centimeters
WEIGHT = applicant's weight in kilograms
SHLDR = applicant's shoulder width in centimeters
PELVIC = applicant's pelvic width in centimeters
CHEST = applicant's minimum chest circumference in centimeters
THIGH = applicant's thigh skinfold thickness in millimeters
PULSE = applicant's resting pulse rate
DIAST = applicant's diastolic blood pressure
CHNUP = number of chin-ups an applicant can complete
BREATH = applicant's maximum breathing capacity in liters
RECVR = applicant's pulse rate after 5 minutes of recovery from treadmill running
SPEED = applicant's maximum treadmill speed
ENDUR = applicant's treadmill endurance time in minutes
FAT = applicant's total body fat measurement
In answering the following questions, you are not required to check the distributional assumptions.
(a) Suppose we wish to develop an appropriate factor analysis model for these variables, how many common factors do you recommend? Why?
(b) Test the hypothesis that the number of common factors recommended in (a) is sufficient. State any assumption(s) needed.
Note: for (c), (d) and (e) below you are required to compute the factor loadings and residual matrices using the relevant formulae given in the lecture notes; you cannot use the build in factor analysis procedure in R nor other similar programs/software.
(c) Based on the result of (a), compute the principal component factor loading values using the sample covariance matrix and the method of principal components (without iterations).
(d) Now compute the principal component factor loadings with iterations (perform 3 iterations).
(e) Compute the residual matrices of (c) and (d) and comment on the results.
(f) Perform the varimax rotation to the loading matrix obtained in (d). Give physical interpretations to the rotated factors.
(g) Conduct a principal component analysis to this data set. Briefly report your findings.