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STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
Note: there is no submission required for lab 7. This worksheet doesn’t include everything
you need to review for the midterm. Please see the midterm study guide posted on bCourses
for a more comprehensive list of concepts, examples and exercises.
1
Data transformation
Problem 1 Conceptual Review
(a) Why do we transform data?
(b) What is Box-Cox transformation on
X
?
(c) What
p
do you use to correct positive skewness (right skew)? What
p
do you use to
correct negative skewness (left skew)?
(d) A good transformation will make this ratio
UQ
−
M
M
−
LQ
close to 1.
(e) What is Tukey and Mosteller’s bulging rule and how to use it to correct monotone
non-linearity?
Problem 2 Excercise 4.1 - Fox
Creat a graph for the ordinary power transformations
X
→
X
p
for
p
=
−
1
,
0
,
1
,
2
,
3. (When
p
= 0, however, use the log transformation.) Compare the graph to Figure 4.1, and comment
on the similarities and differences between the two families of transformations
x
p
and (
x
p
−
1)
/p
.
1
STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
2
Simple linear regression
Problem 3 SLR review
Consider simple linear regression
y
i
=
β
0
+
β
1
x
i
+
ϵ
i
.
(a) what are the assumptions?
(b) Derive the least squares estimates of
β
0
and
β
1
.
(c) Show that
ˆ
β
0
and
ˆ
β
1
are unbiased. What assumptions are used?
(d) Derive
var
(
ˆ
β
0
),
var
(
ˆ
β
1
) and
cov
(
ˆ
β
0
,
ˆ
β
1
). What assumptions are used?
(e) What is an unbiased estimator for
σ
2
?
Problem 4 TSS, RSS and
R
2
review
Consider simple linear regression
y
i
=
β
0
+
β
1
x
i
+
ϵ
i
under standard linear model assumptions:
(a) What is residual standard error and how to interpret it?
(b) What are total sum of squares, regression sum of squares, and residual sum of squares?
(c) Definition of R-squared and what does it represent?
Problem 5 (SP23 HW)
Consider simple linear regression where there is one response variable
y
and an explanatory
variable
x
and there are
n
subjects with values
y
1
,
·
, y
n
and
x
1
,
· · ·
, x
n
.
(a) What are the estimates for
α
0
and
α
1
if we regress
x
on
y
?
(b) Let
ˆ
β
0
and
ˆ
β
1
be the estimate from regressing
y
on
x
.
Intuition might suggest that
ˆ
α
1
= 1
/
ˆ
β
1
. Is this true?
Problem 6 Excercise 5.9
Show that in simple-regression analysis, the standardized slope coefficient
B
is equal to
the correlation coefficient
r
.
(In general, however, standardized slope coefficients are not
correlations and can be outside of the range [0, 1].)
2
STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
3
Multiple regression
Problem 7 MR Review
Consider multiple regression
⃗
y
=
Xβ
+
⃗
ϵ
.
(a) what are the assumptions?
(b) Derive the least squares estimates of
β
.
(c) Show that
ˆ
β
is unbiased. What assumptions are used?
(d) Derive
cov
(
ˆ
β
). What assumptions are used?
(e) What is an unbiased estimator for
σ
2
?
Problem 8 Other concepts of MR
(a) what is adjusted R-squared? Why
R
2
can only rise?
(b) How do correlated variables impact the regression coefficient?
(c) What are the standardized coefficient and how to interpret them?
Problem 9 True/False (Past midterm)
(a)
R
2
is an effective model selection criterion for deciding the best size for a linear model.
(b) If I assume the data-generating process is
⃗
y
=
Xβ
+
⃗
ϵ
with full rank matrix
X
treated
as fixed, then the following is true:
arg min
||
Xβ
−
⃗
y
||
2
2
= (
X
T
X
)
−
1
X
T
⃗
y
regardless of the distribution of
ϵ
.
(c) The R-squared summary output will always increase if I add more covariates to the
regression.
Problem 10 SP23 midterm
In many data analyses,
⃗
y
observations are collected from various sensors with different mea-
surement variabilities. Let’s say that I know the variability of each sensor such that I can
safely assume the following model:
3
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STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
⃗
y
=
Xβ
+
⃗
ϵ,
⃗
ϵ
∼
N
(0
, σ
2
w
2
1
0
· · ·
0
0
w
2
2
· · ·
0
.
.
.
.
.
.
0
0
0
· · ·
w
2
n
)
(a) What is the solution (call this
ˆ
β
OLS
) to the following optimization problem:
arg min
||
Xβ
−
⃗
y
||
2
2
(b) Let
⃗w
=
w
1
w
2
.
.
.
w
n
.
Show the following
V ar
(
ˆ
β
OLS
) =
σ
2
(
X
T
X
)
−
1
X
T
⃗w ⃗w
T
X
(
X
T
X
)
−
1
(c) Let
⃗w ⃗w
T
=
W
, and note
W
−
1
=
1
/w
1
1
/w
2
· · ·
1
/w
n
1
/w
1
1
/w
2
.
.
.
1
/w
n
.
Consider the
transformed model:
W
−
1
/
2
⃗
y
=
W
−
1
/
2
Xβ
+
W
−
1
/
2
⃗
ϵ
Show that the least square estimator (call this
ˆ
β
WLS
) for model above is
ˆ
β
WLS
= (
X
T
W
−
1
X
)
−
1
X
T
W
−
1
⃗
y
(d) Show that
ˆ
β
WLS
is unbiased for
β
.
(e) Compute variance of
ˆ
β
WLS
as an expression involving one instance of each of the fol-
lowing:
X, X
T
, W
−
1
, σ
2
.
Problem 11 Partial coefficient - FWL theorem
(a) How to compute partial coefficient and its standard error?
(b) What is the variance-inflation factor and how does it relate to the coefficient variance?
4
STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
4
Geometry and matrix form of linear models
Problem 12 Gram-Schmidt (SP23 practice midterm)
Consider running a multiple linear regression of ˆ
y
on
X
= [
⃗
1
⃗x
⃗x
2
]
,
where
⃗x
2
is a vector of the squared corresponding elements of
⃗x
, and the elements of
⃗x
are
larger than 10.
(a) Is there a unique solution to the following optimization problem? Explain why or why
not.
(b) Find an orthogonal basis for
X
.
(c) Write the OLS predictions, ˆ
y
, as a function of the orthogonal basis in (b). No need to
fully simplify.
Problem 13 Hat matrix (SP23 practice midterm)
In this problem we will analyze some properties of the ”hat matrix” from the linear model.
Specifically, consider the multiple linear regression model
⃗
y
=
X
⃗
β
+
⃗
ϵ
, with
⃗
ϵ
∼
N
(0
, σ
2
I
).
Recall the hat matrix is defined as:
H
=
X
(
X
T
X
)
−
1
X
T
,
where
X
∈
R
n
×
(
p
+1)
is full column rank.
(a) Consider the predicted values ˆ
y
=
X
ˆ
β
. Show that ˆ
y
has variance
σ
2
H
.
(b) Let
⃗e
=
⃗
y
−
ˆ
y
. Show that
⃗e
= (
I
−
H
)
⃗
y
.
(c) Show that (
I
−
H
) is symmetric and idempotent.
(d) Show that
var
(
⃗e
) =
σ
2
(
I
−
H
).
(e) Show that
ˆ
β
and
⃗e
are independent.
5
Sample dataset questions
Problem 14 Modeling Sugar Cane Production (SP23 midterm)
Suppose you have been hired as a consultant by the sugar company that operates these
sugarcane fields. Your job is to build a linear model to predict the sugarcane production in
tons per hectare. You are provided a dataset with columns:
5
STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
•
Region: region (defined by physical position and average rainfall) in which each paddock
is located.
•
Position:
geographic position of each paddock in the general area according to the
compass directions (E = east, W = west, N = north, S = south, C = central).
•
Area: size of the paddock in hectares.
•
Age: years elapsed since the paddock was plowed out and planted with new sugarcane
seeds.
•
HarvestMonth: month of the year in which the harvest took place (1 = January, 2 =
February, etc.).
•
HarvestDuration: time taken to harvest the sugarcane in days.
•
Tonn.Hect: tons per hectare of sugarcane produced by this paddock.
•
Rainfall.96: Total rainfall for the district from July 1996 through December 1996 (mil-
limeters).
(a) Which variable is your response variable. Which variables are continuous/categorical?
(b) You plot the distribution for Tonn.Hect.
How would you describe the spread of the
variable. For the purposes of your model, would you transform this variable, and if yes,
how so?
6
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STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
(c) You create the following correlation matrix.
Based on the following correlations, in a linear regression model where Tonn.Hect is
the response variable, how would the covariates Age and HarvestDuration impact the
coefficient of Area?
(d) HarvestMonth can be used as a continuous or categorical variable for our model. What
are some drawbacks of using HarvestMonth as continuous over categorical?
This question is a little beyond the scope of this midterm.
So here is the solution if
you’re curious:
Categorical variable captures non-linear relationship between variable and response (i.e
if there is a difference in re- sponse variable between various months).
Additionally,
treating Month as continuous us uninterpretable, as our model will con- sider non-
integer values for Month.
(e) You fit the following linear model, and the R summary is as follows:
Tonn.Hect
∼
Area
+
HarvestMonth
+
Position
+
Region
+
Rainfall.
96
7
STAT 151A
Lab 7: Midterm Review Session
October 6, 2023
Based on this output, is it safe to assume that Region is unimportant to our model due
to most of the categories having a low t-value? Explain your reasoning.
This question is a little beyond the scope of this midterm.
So here is the solution if
you’re curious:
No, we cannot look at the individual t-values of each category to make an overall claim
about the entire variable. We would need to use an F-test where we test the significance
of all categories together.
Problem 15 Model diagnostic (SP23 practice exam)
Below are the residual vs. fitted plot and the Q-Q plot from a model. Describe what
problems you see, if any, in the assumptions of the model. If you see problems in these
diagnostic plots, describe what you might suggest to get an improved regression model.
8
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