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6 Testing the Utility of a Model: The Analysis of Variance F-Test

176 Chapter 4 Multiple Regression Models

Note that the denominator of the F statistic, MSE, represents the unexplained

(or error) variability in the model. The numerator, MS(Model), represents the

variability in y explained (or accounted for) by the model. (For this reason,

the test is often called the ‘‘analysis-of-variance’’ F -test.) Since F is the ratio of

the explained variability to the unexplained variability, the larger the proportion

of the total variability accounted for by the model, the larger the F statistic.

To determine when the ratio becomes large enough that we can conﬁdently

reject the null hypothesis and conclude that the model is more useful than no model

at all for predicting y, we compare the calculated F statistic to a tabulated F -value

with k df in the numerator and [n − (k + 1)] df in the denominator. Recall that

tabulations of the F -distribution for various values of α are given in Tables 3, 4, 5,

and 6 of Appendix D.

Rejection region: F > Fα , where F is based on k numerator and n − (k + 1)

denominator degrees of freedom (see Figure 4.4).

However, since statistical software printouts report the observed signiﬁcance level

(p-value) of the test, most researchers simply compare the selected α value to the

p-value to make the decision.

The analysis of variance F -test for testing the usefulness of the model is

summarized in the next box.

Testing Global Usefulness of the Model: The Analysis of Variance

F-Test

H0 : β1 = β2 = · · · = βk = 0 (All model terms are unimportant for predicting y)

Ha : At least one βi = 0

(At least one model term is useful for predicting y)

Test statistic: F =

=

(SSyy − SSE)/k

R 2 /k

=

SSE/[n − (k + 1)]

(1 − R 2 )/[n − (k + 1)]

Mean square (Model)

Mean square (Error)

where n is the sample size and k is the number of terms in the model.

Rejection region: F > Fα , with k numerator degrees of freedom and [n − (k + 1)]

denominator degrees of freedom.

or

α > p-value, where p-value = P (F > Fc ), Fc is the computed value of the test

statistic.

Assumptions: The standard regression assumptions about the random error

component (Section 4.2).

Example

4.3

Refer to Example 4.2, in which an antique collector modeled the auction price y

of grandfather clocks as a function of the age of the clock, x1 , and the number of

bidders, x2 . The hypothesized ﬁrst-order model is

y = β0 + β1 x1 + β2 x2 + ε

Testing the Utility of a Model: The Analysis of Variance F-Test

177

Figure 4.4 Rejection

region for the global F -test

a

0

F

Fa

Rejection region

A sample of 32 observations is obtained, with the results summarized in the

MINITAB printout in Figure 4.5. Conduct the global F -test of model usefulness at

the α = .05 level of signiﬁcance.

Figure 4.5 MINITAB

regression printout for

grandfather clock model

Solution

The elements of the global test of the model follow:

H0 : β1 = β2 = 0

[Note: k = 2]

Ha : At least one of the two model coefﬁcients is nonzero

Test statistic: F = 120.19 (shaded in Figure 4.5)

p-value = .000 (shaded in Figure 4.5)

Conclusion: Since α = .05 exceeds the observed signiﬁcance level, p = .000, the data

provide strong evidence that at least one of the model coefﬁcients is nonzero. The

overall model appears to be statistically useful for predicting auction prices.

178 Chapter 4 Multiple Regression Models

Can we be sure that the best prediction model has been found if the global

F -test indicates that a model is useful? Unfortunately, we cannot. The addition

of other independent variables may improve the usefulness of the model. (See

the accompanying box.) We consider more complex multiple regression models in

Sections 4.10–4.12.

Caution

A rejection of the null hypothesis H0 : β1 = β2 = · · · = βk in the global

F -test leads to the conclusion [with 100(1 − α)% conﬁdence] that the model

is statistically useful. However, statistically ‘‘useful’’ does not necessarily mean

‘‘best.’’ Another model may prove even more useful in terms of providing more

reliable estimates and predictions. This global F -test is usually regarded as a

test that the model must pass to merit further consideration.

4.7 Inferences About the Individual β Parameters

Inferences about the individual β parameters in a model are obtained using either a

conﬁdence interval or a test of hypothesis, as outlined in the following two boxes.∗

Test of an Individual Parameter Coefﬁcient in the Multiple Regression

Model

ONE-TAILED TESTS

Test statistic:

Rejection region:

H0 : βi = 0

Ha : βi < 0

βˆi

t=

sβˆt

t < −tα

H0 : βi = 0

Ha : βi > 0

TWO-TAILED TEST

H0 : βi = 0

Ha : βi = 0

|t| > tα/2

t > tα

where tα and tα/2 are based on n − (k + 1) degrees of freedom and

n = Number of observations

k + 1 = Number of β parameters in the model

Note: Most statistical software programs report two-tailed p-values on their

output. To ﬁnd the appropriate p-value for a one-tailed test, make the following

adjustment to P = two-tailed p-value:

For Ha : βi > 0, p-value =

P /2

if t > 0

1 − P /2 if t < 0

For Ha : βi < 0, p-value =

1 − P /2 if t > 0

P /2

if t < 0

Assumptions: See Section 4.2 for assumptions about the probability distribution

for the random error component ε.

∗ The formulas for computing βˆ and its standard error are so complex, the only reasonable way to present them

i

is by using matrix algebra. We do not assume a prerequisite of matrix algebra for this text and, in any case, we

think the formulas can be omitted in an introductory course without serious loss. They are programmed into all

statistical software packages with multiple regression routines and are presented in some of the texts listed in

the references.

Inferences About the Individual β Parameters

179

A 100 (1 − α)% Conﬁdence Interval for a β Parameter

βˆi ± (tα/2 )sβˆt

where tα/2 is based on n − (k + 1) degrees of freedom and

n = Number of observations

k + 1 = Number of β parameters in the model

We illustrate these methods with another example.

Example

4.4

Refer to Examples 4.1–4.3. A collector of antique grandfather clocks knows that the

price (y) received for the clocks increases linearly with the age (x1 ) of the clocks.

Moreover, the collector hypothesizes that the auction price (y) of the clocks will

increase linearly as the number of bidders (x2 ) increases. Use the information on

the SAS printout, Figure 4.6, to:

(a) Test the hypothesis that the mean auction price of a clock increases as the

number of bidders increases when age is held constant, that is, β2 > 0. Use

α = .05.

(b) Form a 95% conﬁdence interval for β1 and interpret the result.

Solution

(a) The hypotheses of interest concern the parameter β2 . Speciﬁcally,

H0 : β2 = 0

Ha : β2 > 0

Figure 4.6 SAS regression output for the auction price model, Example 4.4

180 Chapter 4 Multiple Regression Models

The test statistic is a t statistic formed by dividing the sample estimate βˆ2 of

the parameter β2 by the estimated standard error of βˆ2 (denoted sβˆ2 ). These

estimates, βˆ2 = 85.953 and sβˆ2 = 8.729, as well as the calculated t-value, are

highlighted on the SAS printout, Figure 4.6

βˆ2

85.953

= 9.85

Test statistic: t =

=

sβˆ2

8.729

The p-value for the two-tailed test of hypothesis, Ha : β2 ± 0, is also shown

on the printout under Pr > |t|. This value (highlighted) is less than .0001. To

obtain the p-value for the one-tailed test, Ha : β2 > 0, we divide this p-value in

half. Consequently, the observed signiﬁcance level for our upper-tailed test is

.0001

= .00005.

p-value =

2

Since α = .05 exceeds p-value = .0005, we have sufﬁcient evidence to

reject H0 . Thus, the collector can conclude that the mean auction price of a

clock increases as the number of bidders increases, when age is held constant.

(b) A 95% conﬁdence interval for β1 is (from the box):

βˆ1 ± (tα/2 )sβˆ1 = βˆ1 ± (t.05 )sβˆ1

Substituting βˆ1 = 12.74, sβˆi = .905 (both obtained from the SAS printout,

Figure 4.6) and t.025 = 2.045 (from Table C.2) into the equation, we obtain

12.74 ± (2.045)(.905) = 12.74 ± 1.85

or (10.89, 14.59). This interval is also shown (highlighted) on the SAS printout.

Thus, we are 95% conﬁdent that β1 falls between 10.89 and 14.59. Since β1

is the slope of the line relating auction price (y) to age of the clock (x1 ),

we conclude that price increases between $10.89 and $14.59 for every 1-year

increase in age, holding number of bidders (x2 ) constant.

After we have determined that the overall model is useful for predicting y

using the F -test (Section 4.6), we may elect to conduct one or more t-tests on

the individual β parameters (as in Example 4.4). However, the test (or tests) to

be conducted should be decided a priori, that is, prior to ﬁtting the model. Also,

we should limit the number of t-tests conducted to avoid the potential problem

of making too many Type I errors. Generally, the regression analyst will conduct

t-tests only on the ‘‘most important’’ β’s. We provide insight in identifying the most

important β’s in a linear model in the next several sections.

Recommendation for Checking the Utility of a Multiple Regression

Model

1. First, conduct a test of overall model adequacy using the F -test, that is, test

H0 : β1 = β2 = · · · = βk = 0

If the model is deemed adequate (i.e., if you reject H0 ), then proceed to

step 2. Otherwise, you should hypothesize and ﬁt another model. The new

model may include more independent variables or higher-order terms.

2. Conduct t-tests on those β parameters in which you are particularly interested

(i.e., the ‘‘most important’’ β’s). These usually involve only the β’s associated

with higher-order terms (x 2 , x1 x2 , etc.). However, it is a safe practice to limit

the number of β’s that are tested. Conducting a series of t-tests leads to a

high overall Type I error rate α.

Multiple Coefﬁcients of Determination: R2 and R2a

181

We conclude this section with a ﬁnal caution about conducting t-tests on

individual β parameters in a model.

Caution

Extreme care should be exercised when conducting t-tests on the individual β

parameters in a ﬁrst-order linear model for the purpose of determining which

independent variables are useful for predicting y and which are not. If you fail

to reject H0 : βi = 0, several conclusions are possible:

1. There is no relationship between y and xi .

2. A straight-line relationship between y and x exists (holding the other x’s

in the model ﬁxed), but a Type II error occurred.

3. A relationship between y and xi (holding the other x’s in the model

ﬁxed) exists, but is more complex than a straight-line relationship (e.g.,

a curvilinear relationship may be appropriate). The most you can say

about a β parameter test is that there is either sufﬁcient (if you reject

H0 : βi = 0) or insufﬁcient (if you do not reject H0 : βi = 0) evidence of a

linear (straight-line) relationship between y and xi .

4.8 Multiple Coefﬁcients of Determination:

R2 and R2a

Recall from Chapter 3 that the coefﬁcient of determination, r 2 , is a measure of how

well a straight-line model ﬁts a data set. To measure how well a multiple regression

model ﬁts a set of data, we compute the multiple regression equivalent of r 2 , called

the multiple coefﬁcient of determination and denoted by the symbol R2 .

Deﬁnition 4.1 The multiple coefﬁcient of determination, R2 , is deﬁned as

SSE

0 ≤ R2 ≤ 1

R2 = 1 −

SSyy

where SSE = (yi − yˆ i )2 , SSyy = (yi − y)

¯ 2 , and yˆ i is the predicted value of

yi for the multiple regression model.

Just as for the simple linear model, R 2 represents the fraction of the sample

variation of the y-values (measured by SSyy ) that is explained by the least squares

regression model. Thus, R 2 = 0 implies a complete lack of ﬁt of the model to the

data, and R 2 = 1 implies a perfect ﬁt, with the model passing through every data

point. In general, the closer the value of R 2 is to 1, the better the model ﬁts the data.

To illustrate, consider the ﬁrst-order model for the grandfather clock auction

price presented in Examples 4.1–4.4. A portion of the SPSS printout of the analysis

is shown in Figure 4.7. The value R 2 = .892 is highlighted on the printout. This

relatively high value of R 2 implies that using the independent variables age and

number of bidders in a ﬁrst-order model explains 89.2% of the total sample variation

(measured by SSyy ) in auction price y. Thus, R 2 is a sample statistic that tells how

well the model ﬁts the data and thereby represents a measure of the usefulness of

the entire model.

182 Chapter 4 Multiple Regression Models

Figure 4.7 A portion of

the SPSS regression output

for the auction price model

A large value of R 2 computed from the sample data does not necessarily mean

that the model provides a good ﬁt to all of the data points in the population. For

example, a ﬁrst-order linear model that contains three parameters will provide a

perfect ﬁt to a sample of three data points and R 2 will equal 1. Likewise, you will

always obtain a perfect ﬁt (R 2 = 1) to a set of n data points if the model contains

exactly n parameters. Consequently, if you want to use the value of R 2 as a measure

of how useful the model will be for predicting y, it should be based on a sample

that contains substantially more data points than the number of parameters in the

model.

Caution

In a multiple regression analysis, use the value of R 2 as a measure of how useful

a linear model will be for predicting y only if the sample contains substantially

more data points than the number of β parameters in the model.

As an alternative to using R 2 as a measure of model adequacy, the adjusted

multiple coefﬁcient of determination, denoted Ra2 , is often reported. The formula

for Ra2 is shown in the box.

Deﬁnition 4.2 The adjusted multiple coefﬁcient of determination is given by

Ra2 = 1 −

=1−

(n − 1)

n − (k + 1)

SSE

SSyy

(n − 1)

(1 − R 2 )

n − (k + 1)

Note: Ra2 ≤ R 2 and, for poor-ﬁtting models Ra2 may be negative.

R 2 and Ra2 have similar interpretations. However, unlike R 2 , Ra2 takes into account

(‘‘adjusts’’ for) both the sample size n and the number of β parameters in the model.

Ra2 will always be smaller than R 2 , and more importantly, cannot be ‘‘forced’’ to 1 by

simply adding more and more independent variables to the model. Consequently,

analysts prefer the more conservative Ra2 when choosing a measure of model

adequacy. The value of Ra2 is also highlighted in Figure 4.7. Note that Ra2 = .885, a

value only slightly smaller than R 2 .

Despite their utility, R 2 and Ra2 are only sample statistics. Consequently, it is

dangerous to judge the usefulness of the model based solely on these values. A

prudent analyst will use the analysis-of-variance F -test for testing the global utility

of the multiple regression model. Once the model has been deemed ‘‘statistically’’

useful with the F -test, the more conservative value of Ra2 is used to describe the

proportion of variation in y explained by the model.

Multiple Coefﬁcients of Determination: R2 and R2a

183

4.8 Exercises

4.1 Degrees of freedom.

How is the number of

degrees of freedom available for estimating σ 2 , the

variance of ε, related to the number of independent

variables in a regression model?

4.2 Accounting and Machiavellianism. Refer to the

Behavioral Research in Accounting (January 2008)

study of Machiavellian traits (e.g., manipulation,

cunning, duplicity, deception, and bad faith) in

accountants, Exercise 1.47 (p. 41). Recall that a

Machiavellian (‘‘Mach’’) rating score was determined for each in a sample of accounting alumni

of a large southwestern university. For one portion

of the study, the researcher modeled an accountant’s Mach score (y) as a function of age (x1 ),

gender (x2 ), education (x3 ), and income (x4 ). Data

on n = 198 accountants yielded the results shown

in the table.

INDEPENDENT

VARIABLE

t-VALUE FOR H0 : βi = 0

p-VALUE

Age (x1 )

0.10

> .10

−0.55

> .10

Gender (x2 )

Education (x3 )

1.95

< .01

Income (x4 )

0.52

> .10

Overall model: R 2 = .13, F = 4.74 (p-value < .01)

(a) Write the equation of the hypothesized model

relating y to x1 , x2 , x3 , and x4 .

(b) Conduct a test of overall model utility. Use

α = .05.

(c) Interpret the coefﬁcient of determination, R 2 .

(d) Is there sufﬁcient evidence (at α = .05) to say

that income is a statistically useful predictor of

Mach score?

4.3 Study of adolescents with ADHD. Children with

attention-deﬁcit/hyperactivity disorder (ADHD)

were monitored to evaluate their risk for substance

(e.g., alcohol, tobacco, illegal drug) use (Journal

of Abnormal Psychology, August 2003). The following data were collected on 142 adolescents

diagnosed with ADHD:

y = frequency of marijuana use the past 6 months

x1 = severity of inattention (5-point scale)

x2 = severity of impulsivity–hyperactivity

(5-point scale)

x3 = level of oppositional–deﬁant and conduct

disorder (5-point scale)

(a) Write the equation of a ﬁrst-order model for

E(y).

(b) The coefﬁcient of determination for the model

is R 2 = .08. Interpret this value.

(c) The global F -test for the model yielded a

p-value less than .01. Interpret this result.

(d) The t-test for H0 : β1 = 0 resulted in a p-value

less than .01. Interpret this result.

(e) The t-test for H0 : β2 = 0 resulted in a p-value

greater than .05. Interpret this result.

(f) The t-test for H0 : β3 = 0 resulted in a p-value

greater than .05. Interpret this result.

4.4 Characteristics of lead users. During new product development, companies often involve ‘‘lead

users’’ (i.e., creative individuals who are on the

leading edge of an important market trend).

Creativity and Innovation Management (February

2008) published an article on identifying the social

network characteristics of lead users of children’s

computer games. Data were collected for n = 326

children and the following variables measured:

lead-user rating (y, measured on a 5-point scale),

gender (x1 = 1 if female, 0 if male), age (x2 , years),

degree of centrality (x3 , measured as the number of direct ties to other peers in the network),

and betweenness centrality (x4 , measured as the

number of shortest paths between peers). A ﬁrstorder model for y was ﬁt to the data, yielding the

following least squares prediction equation:

yˆ = 3.58 + .01x1 − .06x2 − .01x3 + .42x4

(a) Give two properties of the errors of prediction that result from using the method of least

squares to obtain the parameter estimates.

(b) Give a practical interpretation the estimate of

β4 in the model.

(c) A test of H0 : β4 = 0 resulted in a two-tailed

p-value of .002. Make the appropriate conclusion at α = .05.

4.5 Runs scored in baseball. In Chance (Fall 2000),

statistician Scott Berry built a multiple regression

model for predicting total number of runs scored

by a Major League Baseball team during a season.

Using data on all teams over a 9-year period (a

sample of n = 234), the results in the next table

(p. 184) were obtained.

(a) Write the least squares prediction equation for

y = total number of runs scored by a team in

a season.

(b) Conduct a test of H0 : β7 = 0 against Ha : β7 < 0

at α = .05. Interpret the results.

(c) Form a 95% conﬁdence interval for β5 . Interpret the results.

(d) Predict the number of runs scored by your

favorite Major League Baseball team last

184 Chapter 4 Multiple Regression Models

year. How close is the predicted value to the

actual number of runs scored by your team?

(Note: You can ﬁnd data on your favorite team

on the Internet at www.mlb.com.)

INDEPENDENT

VARIABLE

Intercept

Walks (x1 )

Singles (x2 )

Doubles (x3 )

Triples (x4 )

Home Runs (x5 )

Stolen Bases (x6 )

Caught Stealing (x7 )

Strikeouts (x8 )

Outs (x9 )

β ESTIMATE

3.70

.34

.49

.72

1.14

1.51

.26

−.14

−.10

−.10

STANDARD ERROR

15.00

.02

.03

.05

.19

.05

.05

.14

.01

.01

Source: Berry, S. M. ‘‘A statistician reads the sports pages:

Modeling offensive ability in baseball,’’ Chance, Vol. 13,

No. 4, Fall 2000 (Table 2).

4.6 Earnings of Mexican street vendors. Detailed

interviews were conducted with over 1,000 street

vendors in the city of Puebla, Mexico, in order

to study the factors inﬂuencing vendors’ incomes

(World Development, February 1998). Vendors

were deﬁned as individuals working in the street,

and included vendors with carts and stands on

wheels and excluded beggars, drug dealers, and

prostitutes. The researchers collected data on gender, age, hours worked per day, annual earnings,

and education level. A subset of these data appears

in the accompanying table.

(a) Write a ﬁrst-order model for mean annual

earnings, E(y), as a function of age (x1 ) and

hours worked (x2 ).

SAS output for Exercise 4.6

STREETVEN

VENDOR

NUMBER

ANNUAL

EARNINGS y

AGE x1

HOURS WORKED

PER DAY x2

21

53

60

184

263

281

354

401

515

633

677

710

800

914

997

$2841

1876

2934

1552

3065

3670

2005

3215

1930

2010

3111

2882

1683

1817

4066

29

21

62

18

40

50

65

44

17

70

20

29

15

14

33

12

8

10

10

11

11

5

8

8

6

9

9

5

7

12

Source: Adapted from Smith, P. A., and Metzger, M. R.

‘‘The return to education: Street vendors in Mexico,’’

World Development, Vol. 26, No. 2, Feb. 1998, pp.

289–296.

(b) The model was ﬁt to the data using SAS. Find

the least squares prediction equation on the

printout shown below.

(c) Interpret the estimated β coefﬁcients in your

model.

(d) Conduct a test of the global utility of the model

(at α = .01). Interpret the result.

(e) Find and interpret the value of Ra2 .

(f) Find and interpret s, the estimated standard

deviation of the error term.

(g) Is age (x1 ) a statistically useful predictor of

annual earnings? Test using α = .01.

(h) Find a 95% conﬁdence interval for β2 . Interpret the interval in the words of the problem.

Multiple Coefﬁcients of Determination: R2 and R2a

4.7 Urban population estimation using satellite

images. Can the population of an urban area be

estimated without taking a census? In Geographical Analysis (January 2007) geography professors

at the University of Wisconsin–Milwaukee and

Ohio State University demonstrated the use of

satellite image maps for estimating urban population. A portion of Columbus, Ohio, was partitioned

into n = 125 census block groups and satellite

imagery was obtained. For each census block,

the following variables were measured: population

density (y), proportion of block with low-density

residential areas (x1 ), and proportion of block with

high-density residential areas (x2 ). A ﬁrst-order

model for y was ﬁt to the data with the following

results:

yˆ = −.0304 + 2.006x1 + 5.006x2 , R 2 = .686

(a) Give a practical interpretation of each

β-estimate in the model.

(b) Give a practical interpretation of the coefﬁcient of determination, R 2 .

(c) State H0 and Ha for a test of overall model

adequacy.

(d) Refer to part c. Compute the value of the test

statistic.

(e) Refer to parts c and d. Make the appropriate

conclusion at α = .01.

4.8 Novelty of a vacation destination. Many tourists

choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the

itinerary. Texas A&M University professor J. Petrick investigated the relationship between novelty

and vacationing golfers’ demographics (Annals

of Tourism Research, Vol. 29, 2002). Data were

obtained from a mail survey of 393 golf vacationers to a large coastal resort in southeastern United States. Several measures of novelty

level (on a numerical scale) were obtained for

each vacationer, including ‘‘change from routine,’’

‘‘thrill,’’ ‘‘boredom-alleviation,’’ and ‘‘surprise.’’

The researcher employed four independent variables in a regression model to predict each of the

novelty measures. The independent variables were

x1 = number of rounds of golf per year, x2 = total

number of golf vacations taken, x3 = number of

years played golf, and x4 = average golf score.

(a) Give the hypothesized equation of a ﬁrst-order

model for y = change from routine.

(b) A test of H0 : β3 = 0 versus Ha : β3 < 0 yielded a

p-value of .005. Interpret this result if α = .01.

(c) The estimate of β3 was found to be negative.

Based on this result (and the result of part

b), the researcher concluded that ‘‘those who

have played golf for more years are less apt

to seek change from their normal routine in

185

their golf vacations.’’ Do you agree with this

statement? Explain.

(d) The regression results for the three other

dependent novelty measures are summarized

in the table below. Give the null hypothesis for

testing the overall adequacy of each ﬁrst-order

regression model.

DEPENDENT VARIABLE

F -VALUE

Thrill

Boredom-alleviation

Surprise

5.56

3.02

3.33

p-VALUE

< .001

.018

.011

R2

.055

.030

.023

Source: Reprinted from Annals of Tourism Research,

Vol. 29, Issue 2, J. F. Petrick, ‘‘An examination of golf

vacationers’ novelty,” Copyright © 2002, with permission

from Elsevier.

(e) Give the rejection region for the test, part d,

using α = .01.

(f) Use the test statistics reported in the table and

the rejection region from part e to conduct

the test for each of the dependent measures of

novelty.

(g) Verify that the p-values in the table support

your conclusions in part f.

(h) Interpret the values of R 2 reported in the table.

4.9 Highway crash data analysis.

Researchers at

Montana State University have written a tutorial

on an empirical method for analyzing before and

after highway crash data (Montana Department

of Transportation, Research Report, May 2004).

The initial step in the methodology is to develop

a Safety Performance Function (SPF)—a mathematical model that estimates crash occurrence for

a given roadway segment. Using data collected

for over 100 roadway segments, the researchers

ﬁt the model, E(y) = β0 + β1 x1 + β2 x2 , where

y = number of crashes per 3 years, x1 = roadway

length (miles), and x2 = AADT (average annual

daily trafﬁc) (number of vehicles). The results are

shown in the following tables.

Interstate Highways

VARIABLE

Intercept

Length (x1 )

AADT (x2 )

PARAMETER

ESTIMATE

STANDARD

ERROR

t-VALUE

1.81231

.10875

.00017

.50568

.03166

.00003

3.58

3.44

5.19

PARAMETER

ESTIMATE

STANDARD

ERROR

t-VALUE

1.20785

.06343

.00056

.28075

.01809

.00012

4.30

3.51

4.86

Non-Interstate Highways

VARIABLE

Intercept

Length (x1 )

AADT (x2 )

186 Chapter 4 Multiple Regression Models

(a) Give the least squares prediction equation for

the interstate highway model.

(b) Give practical interpretations of the β estimates, part a.

(c) Refer to part a. Find a 99% conﬁdence interval

for β1 and interpret the result.

(d) Refer to part a. Find a 99% conﬁdence interval

for β2 and interpret the result.

(e) Repeat parts a–d for the non-interstate highway model.

4.10 Snow geese feeding trial. Refer to the Journal

of Applied Ecology (Vol. 32, 1995) study of the

feeding habits of baby snow geese, Exercise 3.46

(p. 127). The data on gosling weight change, digestion efﬁciency, acid-detergent ﬁber (all measured

as percentages) and diet (plants or duck chow) for

42 feeding trials are saved in the SNOWGEESE

ﬁle. (The table shows selected observations.) The

botanists were interested in predicting weight

change (y) as a function of the other variables. The

ﬁrst-order model E(y) = β0 + β1 x1 + β2 x2 , where

x1 is digestion efﬁciency and x2 is acid-detergent

ﬁber, was ﬁt to the data. The MINITAB printout

is given below.

SNOWGEESE (First and last ﬁve trials)

FEEDING

TRIAL DIET

1

2

3

4

5

38

39

40

41

42

Plants

Plants

Plants

Plants

Plants

Duck Chow

Duck Chow

Duck Chow

Duck Chow

Duck Chow

WEIGHT DIGESTION

ACIDCHANGE EFFICIENCY DETERGENT

(%)

(%)

FIBER (%)

−6

−5

−4.5

0

2

9

12

8.5

10.5

14

0

2.5

5

0

0

59

52.5

75

72.5

69

28.5

27.5

27.5

32.5

32

8.5

8

6

6.5

7

Source: Gadallah, F. L., and Jefferies, R. L. ‘‘Forage quality in brood rearing areas of the lesser snow goose and the

growth of captive goslings,’’ Journal of Applied Ecology,

Vol. 32, No. 2, 1995, pp. 281–282 (adapted from Figures 2

and 3).

4.11 Deep space survey of quasars. A quasar is a

(a) Find the least squares prediction equation for

weight change, y.

(b) Interpret the β-estimates in the equation,

part a.

(c) Conduct the F -test for overall model adequacy

using α = .01.

(d) Find and interpret the values of R 2 and Ra2 .

Which is the preferred measure of model ﬁt?

(e) Conduct a test to determine if digestion efﬁciency, x1 , is a useful linear predictor of weight

change. Use α = .01.

(f) Form a 99% conﬁdence interval for β2 . Interpret the result.

MINITAB output for Exercise 4.10

distant celestial object (at least 4 billion lightyears away) that provides a powerful source of

radio energy. The Astronomical Journal (July

1995) reported on a study of 90 quasars detected

by a deep space survey. The survey enabled

astronomers to measure several different quantitative characteristics of each quasar, including

redshift range, line ﬂux (erg/cm2 · s), line luminosity (erg/s), AB1450 magnitude, absolute magnitude,

and rest frame equivalent width. The data for a

sample of 25 large (redshift) quasars is listed in the

table on p. 187.

(a) Hypothesize a ﬁrst-order model for equivalent

width, y, as a function of the ﬁrst four variables

in the table.

## 2011 (7th edition) william mendenhall a second course in statistics regression analysis prentice hall (2011)

## 2 Populations, Samples, and Random Sampling

## 3 Fitting the Model: The Method of Least Squares

## 6 Assessing the Utility of the Model: Making Inferences About the Slope β[sub(1)]

## 4 Fitting the Model: The Method of Least Squares

## 11 A Quadratic (Second-Order) Model with a Quantitative Predictor

## 1 Introduction: Why Model Building Is Important

## 1 Introduction: Why Use a Variable-Screening Method?

## 5 Extrapolation: Predicting Outside the Experimental Region

## 7 Follow-Up Analysis: Tukey’s Multiple Comparisons of Means

## B.7 Standard Errors of Estimators, Test Statistics, and Confidence Intervals for β[sub(0)], β[sub(1)], . . . , β[sub(k)]

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