STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
Final Exam
Instructions:
• You must work independently. You may not discuss this exam with anyone except the
course instructor.
• Any submissions made after the submission deadline will not be accepted.
• You may use R to perform any calculations or data analysis.
• Do not submit pages of raw computer outputs, but do include important outputs or attach
graphs in appropriate places. Answer each question and clearly state your answer.
• For handwritten submissions, please write as clearly and neatly as possible. If I cannot
read your answers, I cannot give you any credit.
Please sign or type, acknowledging that it is your own work.
I have neither given nor received any
unauthorized aid in completing this exam
Signed
Department of Statistics, University of Pittsburgh 1
STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
1. In a study of the effects on future development of the size of apple trees at the time of transplanting, four measurements of size were made on each of n = 50 trees when they were transplanted.
The four variables are:
X1 = tree weight (kg),
X2 = the square of the trunk circumference (cm2),
X3 = the length of the laterals (cm) (the total length of the branches),
X4 = the length of the central leader (cm) (a measure of the height of the tree).
The sample correlation matrix is
R =
2664
1 0:75 0:78 0:50
0:75 1 0:65 0:60
0:78 0:65 1 0:30
0:50 0:60 0:30 1
3775
for the p = 4 standardized random variables Z1, Z2, Z3, and Z4.
(a) Find the eigenvalues and eigenvectors associated with the correlation matrix R. If the
elements of the first eigenvector, e^1, are all negative, then multiply by -1 to obtain the
vector with all positive components (use this e^1 for the remaining questions).
(b) Using the results in Part (a), write down the formulas for the first two (sample) principal
components (using Y^i’s) and find their variances.
(c) What is the proportion of total (standardized) sample variance explained by the first principal component?
(d) What interpretation, if any, can you give to the first principal component?
(e) Estimate the correlation between the second principal component and the standardized
length of the central leader.
(f) Estimate the correlation between the scores for the first and second principal components.
(g) Obtain the sample principal component scores on the first two components (^ y1; y^2) for the
vector of standardized observations, z = [-0:2; -0:7; 1:4; 0:5]0.
Department of Statistics, University of Pittsburgh 2
STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
2. Refer to Question #1, use the four variables and the sample correlation matrix R to answer
questions.
X1 = tree weight (kg),
X2 = the square of the trunk circumference (cm2),
X3 = the length of the laterals (cm) (the total length of the branches),
X4 = the length of the central leader (cm) (a measure of the height of the tree).
R =
2664
1 0:75 0:78 0:50
0:75 1 0:65 0:60
0:78 0:65 1 0:30
0:50 0:60 0:30 1
3775
(a) Obtain the varimax rotated principal component estimates of factor loadings (Le∗) for an
m = 2 factor model.
(b) Using the result from Part (a), determine the specific variances and communalities.
(c) Using the result from Part (a), report the proportion of total (standardized) sample variation explained by each rotated factor.
(d) Using the result from Part (a), determine the residual matrix.
(e) Give a brief interpretation of each of those two rotated factors.
Department of Statistics, University of Pittsburgh 3
STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
3. The U.S. crime data set (crime.txt) consists of the reported number of crimes in the 50 U.S.
states in 1984. The crimes were classified according to 7 categories (X3 ∼ X9): murder (X3),
rape (X4), robbery (X5), assault (X6), burglary (X7), larceny (X8), and auto theft (X9). The
variables X1 and X2 represent the land area and the population in 1984, respectively. The
dataset also contains identifications of the region (X10): Northeast(1), Midwest(2), South(3),
and West(4), and identifications of the division (X11): New England(1), Mid-Atlantic(2), E-N
Central(3), W-N Central(4), S Atlantic(5), E-S Central(6), W-S Central(7), Mountain(8), Pacific(9).
Conduct a principal component analysis of the data (7 categories, X3 ∼ X9) using the correlation matrix R.
(a) Evaluate the sample correlation matrix, and interpret the pairwise correlations.
(b) Determine the appropriate number of components to effectively summarize the sample
variability. How many principal components should be considered? Give a justification.
(c) Interpret the first two sample principal components.
(d) Create a biplot showing that 50 states (PC scores) plotted according to the first two principal components. Discuss and interpret any important features (at least five features) in
the plot including correlations among the seven variables and regional/state crime trends.
(e) Create a biplot of the first two PCs considering the regions variable, X10. Can you see
any difference between the four regions? Discuss.
Department of Statistics, University of Pittsburgh 4
STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
4. Refer to Question #3, perform a factor analysis with a varimax rotation using the maximum
likelihood (ML) procedure on the U.S. crime data set (for the seven variables, X3 ∼ X9) and
estimate the factor scores.
(a) Obtain the estimates of factor loadings for m = 2 and m = 3 common factors, by the
maximum likelihood method and varimax rotation.
(b) Using the results from Part (a), determine the specific variances and communalities for
m = 2 and m = 3.
(c) Using the results from Part (a), report the proportion of total sample variation explained
by each rotated factor for m = 2 and m = 3.
(d) Determine the residual matrix for m = 2 and m = 3.
(e) Determine how many factors are needed to explain the pattern of relationships among
these variables. Give some justification for your answer.
(f) Using the result from the m = 3 factor solutions, give interpretations of the rotated factors. Make comparison with the results obtained from the PC analysis in Question #3.
(g) Create three biplots (for all pairs of factor scores and factor loadings) of the first three
common factors. Discuss and interpret any important features (at least five features) in
the plots.
Department of Statistics, University of Pittsburgh 5
STAT 1311/2310 (Spring 2020) Due: Wed, Apr 22, Online submission by 1:00 PM
# Criminal Data ( f o r Qu estions 3 & 4)
l i b r a r y ( psych )
l i b r a r y ( ggplot2 )
l i b r a r y ( d e v t o ol s )
l i b r a r y ( g g b i p l o t )
df <- read . t a b l e (” crime . txt ” , header = T)
crime <- df [ , 3 : 1 0 ]
colnames ( crime ) <- c (“mur” , ” rap ” , ” rob ” , ” ass ” , “bur ” , ” l a r ” , ” aut ” , ” r eg “)
# Add S t a t e s
c rim e $ s t a t e <- c (“ME” , “NH” , “VT” , “MA” , “RI ” , “CT” , “NY” , “NJ” , “PA” , “OH” , “IN ” ,
” IL ” , “MI” , “WI” , “MN” , “IA ” , “MO” , “ND” , “SD” , “NE” , “KS” , “DE” , “MD” , “VA” , “VW” ,
“NC” , “SC” , “GA” , “FL” , “KY” , “TN” , “AL” , “MS” , “AR” , “LA” , “OK” , “TX” , “MT” ,
“ID ” , “WY” , “CO” , “NM” , “AZ” , “UT” , “NV” , “WA” , “OR” , “CA” , “AK” , “HI “)
head ( crime )
# S c a t t e r pl o t
p a i r s . pan els ( crime [ , – c ( 8 , 9 ) ] , show . p oin t s=T, gap=0, d en si t y=T, e l l i p s e s=F, pch=19)
# For Question 4 ( Factor Analysis with ML Method )
df2 <- crime [ , – c ( 8 , 9 ) ]
df2 . f a <- f a c t a n a l ( df2 , f a c t o r s =3, r o t a t i o n = ” varimax ” , s c o r e s=”r e g r e s s i o n “)
par ( mfrow=c ( 1 , 3 ) )
b i p l o t ( df2 . f a $ s c o r e s [ , 1 : 2 ] , l o a d in g s ( df2 . f a ) [ , 1 : 2 ] , cex=c ( 0 . 9 , 0 . 9 ) )
b i p l o t ( df2 . f a $ s c o r e s [ , c ( 1 , 3 ) ] , l o a d in g s ( df2 . f a ) [ , c ( 1 , 3 ) ] , cex=c ( 0 . 9 , 0 . 9 ) )
b i p l o t ( df2 . f a $ s c o r e s [ , c ( 2 , 3 ) ] , l o a d in g s ( df2 . f a ) [ , c ( 2 , 3 ) ] , cex=c ( 0 . 9 , 0 . 9 ) )
Department of Statistics, University of Pittsburgh 6
The post Any submissions made after the submission appeared first on My Assignment Online.