Statistics

Maths 428/528 Fall 2016

Midterm (Takehome) Total Points: 60

Due: October 18 by 2:00PM

Instructions: Please be as rigorous as possible in all of your answers and show

all your work. I prefer that you use a word processing utility to prepare your

answers. However, be as neat as possible if you are simply writing your answers.

Attach SAS or R codes if you used these software packages in your exam.

You may not consult with anyone except the instructor for clarification of questions. The

work you present should be your work alone. Violation of the honor code will be prosecuted

according to Ball State honor policy. Please sign the honor code pledge and submit it with

your report.

MATH 528 students: You should present your answers as a report format

Honor Code Pledge: On my honor, I have neither given nor received aid on

this examination

Name:

Signature:

Date:

1. (16 pts) Let us consider the general linear model, where y = X� + ✏, ✏ ⇠ N(O, �2In⇥n

and Xn⇥p is full rank. Principal Components Regression (PCR) is a technique defined

by the model y = Z� + ✏, where Z = XV is the n ⇥ p matrix of principal component

scores and � is the set of regression coe�cients corresponding to Z. V is the p ⇥ p

matrix obtained from the spectral decomposition of X0X such that VDV0 = X0X, and

D is the p ⇥ p diagonal matrix of eigenvalues corresponding to V. That is, in PCR, we

regress our responses y on the set of principal component scores Z. Here, ✏ and y are

defined as previously in the general linear model and the columns of Z are uncorrelated.

(a) (3 pts) For the general linear model, state the model assumptions. Would these

assumptions also extend to the model defined for PCR? Why or why not?

(b) (4 pts) Derive the form of �̂. Reduce your expression to the simplest form possible.

(c) (4 pts) Derive the form of cov(�̂). Reduce your expression to the simplest form

possible.

(d) (5 pts) Let �̂ be the least squares estimate for � from the general linear model.

Show how �̂ can be written in terms of �̂. Use this to describe the relationship

between the predicted values from the general linear model and the predicted

values from PCR?

2. (10 pts) A researcher has obtained data on n = 6 subjects measuring the body weight

(in kg) of each subject in addition to their resting heart rate (in beats/minute). Let Xi

and Yi represent the body weight and resting heart rate, respectively, for subject i. It is

of interest to determine whether there is a relationship between an individual’s weight

and resting heart rate. Unfortunately, we are only given the following quantities by the

researcher:

6X

i=1

Xi = 488, 6X

i=1

Yi = 319, 6X

i=1

X2i = 40, 092, 6X

i=1

Y 2i = 17, 399, 6X

i=1

XiYi = 26, 184

(a) (2 pts) Write the general linear model for modeling our response, resting heart

rate, with respect to our predictor, body weight, in matrix notation assuming an

intercept, writing the dimensions of y, X, �, and ✏.

(b) (4 pts) Compute �̂. Interpret each of the parameter estimates in a manner a

non-statistician can understand.

(c) (4 pts) What is the SSE for this model? What are the number of degrees of

freedom pertaining to the SSE?

3. (10 pts) The following X0X�1, �̂, and SSE was obtained from the regression of tumor

weight in grams on blood T-cell concentration in million cells/mL and age in years

pertaining to n = 7 mice, i = 1, . . . , 7.

(X0X)�1 =

2

6666 4

1.7995972 �.0685472 �.2531648

�.0685472 .0100774 �.0010661

�.2531648 �.0010661 .0570789

3

7777 5

�̂ =

0

BBBB @

51.5697

1.4974

6.7233

1

CCCC A

SSE = 27.5808

(a) (2 pts) How many degrees of freedom does the SSE have? Compute �̂2

(b) (4 pts) Compute the variance of �̂1 and use this value to get the 95% CI for �̂1.

(c) (4 pts) A previous study has determined that the increase in tumor weight per 1

million cells/mL increase in blood T-cell concentration was 0.34. We wish to test

the null hypothesis that the parameter corresponding to blood T-cell concentration

in our study is equal to this previously observed value. Write the test in terms

of the general linear hypothesis testing framework, specifying C, ✓, and ✓0. Is this

hypothesis testable? Write the expression of the F statistic associated with this

test in addition to its degrees of freedom.

4. (24 pts) Reducing saturated fatty acids in the diet is believed to lead to lower blood

cholesterol levels. High levels of cholesterol, a soft, waxy substance found among the

lipids (fats) in the bloodstream and in all the body’s cells, have been linked to increased

risk of heart disease. Investigators measured a variety of covariates and confounders on

n = 103 subjects and are interested in predictors of the level of high-density lipoprotein

(mg/dl), also called HDL or “good cholesterol”, which is associated with reduced risk

of heart disease. These predictors included triglycerides (mg/dl) (TRIG), which may

lead to hardening of the arteries and increased heart attack risk; low-density lipoprotein

(mg/dl) (LDL), also known as “bad cholesterol” and associated with increased risk of

heart disease; Lp(a) (mg/dl) (LPA), a lipoprotein that is a genetic variation of LDL that

may be important in heart disease, although its exact e↵ect has not been determined;

Apo-A1 (mg/dl) (APOA1) and Apo-B (mg/dl) (APOB), proteins that carry cholesterol

in the blood;

brinogen (mg/dl) (FIBRIN), factor VII (%) (FACTOR7), and PAI-1 (ng/ml) (PAI),

factors involved in blood clotting; the calorie level (CALORIES) of the diet; age (years)

(AGEYRS); birth date (BIRTHDAT); height (cm) (HEIGHT); weight (kg) (WEIGHT);

and body mass index (kg/m2) (BMI), a height-adjusted measure of weight calculated as

weight in kg divided by the square of height in m. The data are in the file chol.sas7bdat

on blackboard.

To report a test for any of the following questions, provide H0, the test

statistic, the degrees of freedom, the p-value, the decision (accept/reject

H0), and an interpretation of the result in terms of the subject matter.

(a) Write the predicted general linear model to study the relationship between hdl and

the group of predictors (trig, ldl, lpa, apoa1, apob, fibrin, factor7, pai, calories,

height, weight, BMI, and age) base on the trig dataset.

(b) Report the test of whether the group of predictors (trig, ldl, lpa, apoa1, apob,

fibrin, factor7, pai, calories, height, weight, BMI, and age) is important.

(c) Report a test of the hypothesis that trig, ldl, lpa, apoa1, apob, has no a↵ect on

hdl after adjusting for all the other predictors in the model.

(d) Describe the relationship between calories, height, weight, BMI, and age and hdl

in these data.

(e) Based on the original model, which characteristics are associated with the best

(highest) hdl?

(f) Give the two models being compared in testing the following hypothesis for these

data, and report each test.

“The group of protein variables (ldl, lpa, apoa1, apob) pro- vides no additional

information about FVC compared to a model for only the mean level of hdl”.