Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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10.3.6 The following data were collected on a simple random sample of 20 patients with hypertension.

The variables are

Patient YX

1 105 47 85.4 1.75 5.1 63 33

2 115 49 94.2 2.10 3.8 70 14

3 116 49 95.3 1.98 8.2 72 10

4 117 50 94.7 2.01 5.8 73 99

5 112 51 89.4 1.89 7.0 72 95

6 121 48 99.5 2.25 9.3 71 10

7 121 49 99.8 2.25 2.5 69 42

8 110 47 90.9 1.90 6.2 66 8

9 110 49 89.2 1.83 7.1 69 62

10 114 48 92.7 2.07 5.6 64 35

11 114 47 94.4 2.07 5.3 74 90

12 115 49 94.1 1.98 5.6 71 21

13 114 50 91.6 2.05 10.2 68 47

14 106 45 87.1 1.92 5.6 67 80

15 125 52 101.3 2.19 10.0 76 98

16 114 46 94.5 1.98 7.4 69 95

17 106 46 87.0 1.87 3.6 62 18

18 113 46 94.5 1.90 4.3 70 12

19 110 48 90.5 1.88 9.0 71 99

20 122 56 95.7 2.09 7.0 75 99

10.4 EVALUATING THE MULTIPLE

REGRESSION EQUATION

Before one uses a multiple regression equation to predict and estimate, it is desirable to

determine ﬁrst whether it is, in fact, worth using. In our study of simple linear regression

we have learned that the usefulness of a regression equation may be evaluated by a con-

sideration of the sample coefﬁcient of determination and estimated slope. In evaluating a

multiple regression equation we focus our attention on the coefﬁcient of multiple deter-

mination and the partial regression coefﬁcients.

The Coefﬁcient of Multiple Determination In Chapter 9 the coefﬁ-

cient of determination is discussed in considerable detail. The concept extends logically

= measure of stress

= basal pulse 1beatsthn/min2

= duration of hypertension 1years2

= body surface area 1sq m2

= weight 1kg2

= age 1years2

Y = mean arterial blood pressure 1mm Hg2

10.4 EVALUATING THE MULTIPLE REGRESSION EQUATION 497

to the multiple regression case. The total variation present in the Y values may be par-

titioned into two components—the explained variation, which measures the amount of

the total variation that is explained by the ﬁtted regression surface, and the unexplained

variation, which is that part of the total variation not explained by ﬁtting the regression

surface. The measure of variation in each case is a sum of squared deviations. The total

variation is the sum of squared deviations of each observation of Y from the mean of the

observations and is designated by or SST. The explained variation, desig-

nated is the sum of squared deviations of the calculated values from the

mean of the observed Y values. This sum of squared deviations is called the sum of

squares due to regression (SSR). The unexplained variation, written as is

the sum of squared deviations of the original observations from the calculated values.

This quantity is referred to as the sum of squares about regression or the error sum of

squares (SSE). We may summarize the relationship among the three sums of squares with

the following equation:

(10.4.1)

The coefﬁcient of multiple determination, is obtained by dividing the

explained sum of squares by the total sum of squares. That is,

(10.4.2)

The subscript indicates that in the analysis Y is treated as the dependent vari-

able and the X variables from through are treated as the independent variables. The

value of indicates what proportion of the total variation in the observed Y values

is explained by the regression of Y on In other words, we may say that

is a measure of the goodness of ﬁt of the regression surface. This quantity is anal-

ogous to which was computed in Chapter 9.

EXAMPLE 10.4.1

Refer to Example 10.3.1. Compute

Solution: For our illustrative example we have in Figure 10.3.1

y.12

393.39

1061.36

= .3706 L .371

SSE = 667.97

SSR = 393.39

SST = 1061.36

y .12

y.12 . . . k

, X

,...,X

y.12 . . . k

g1y

- y2

g1y

- y2

SSR

SST

y.12 . . . k

+ unexplained 1error2sum of squares

total sum of squares = explained 1regression2sum of squares

SST = SSR + SSE

g1y

- y2

= g1y

- y2

+ g1y

- y

g1y

- y

g1y

- y2

g1y

- y2

498 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION

We say that about 37.1 percent of the total variation in the Y values is

explained by the ﬁtted regression plane, that is, by the linear relationship

with age and education level. ■

Testing the Regression Hypothesis To determine whether the overall

regression is signiﬁcant (that is, to determine whether is signiﬁcant), we may per-

form a hypothesis test as follows.

1. Data. The research situation and the data generated by the research are examined

to determine if multiple regression is an appropriate technique for analysis.

2. Assumptions. We assume that the multiple regression model and its underlying

assumptions as presented in Section 10.2 are applicable.

3. Hypotheses. In general, the null hypothesis is

and the alternative is not all In words, the null hypothesis

states that all the independent variables are of no value in explaining the variation

in the Y values.

4. Test statistic. The appropriate test statistic is V.R., which is computed as part

of an analysis of variance. The general ANOVA table is shown as Table 10.4.1.

In Table 10.4.1, MSR stands for mean square due to regression and MSE stands

for mean square about regression or, as it is sometimes called, the error mean

square.

5. Distribution of test statistic. When is true and the assumptions are met, V.R.

is distributed as F with k and degrees of freedom.

6. Decision rule. Reject if the computed value of V.R. is equal to or greater than

the critical value of F.

7. Calculation of test statistic. See Table 10.4.1.

8. Statistical decision. Reject or fail to reject in accordance with the decision

rule.

9. Conclusion. If we reject we conclude that, in the population from which the

sample was drawn, the dependent variable is linearly related to the independent vari-

ables as a group. If we fail to reject we conclude that, in the population from

which our sample was drawn, there may be no linear relationship between the

dependent variable and the independent variables as a group.

10. p value. We obtain the p value from the table of the F distribution.

We illustrate the hypothesis testing procedure by means of the following example.

n - k - 1

= 0.H

= 0

: b

= b

. . .

y.12

10.4 EVALUATING THE MULTIPLE REGRESSION EQUATION

499

TABLE 10.4.1 ANOVA Table for Multiple Regression

Source SS

d.f.

MS V.R.

Due to regression

SSR k

About regression

SSE

Total

SST

n - 1

MSE = SSE>1n - k - 12n - k - 1

MSR>MSEMSR = SSR>k

EXAMPLE 10.4.2

We wish to test the null hypothesis of no linear relationship among the three variables

discussed in Example 10.3.1: CDA score, age, and education level.

Solution:

1. Data. See the description of the data given in Example 10.3.1.

2. Assumptions. We assume that the assumptions discussed in Section

10.2 are met.

3. Hypotheses.

4. Test statistic. The test statistic is V.R.

5. Distribution of test statistic. If is true and the assumptions are met,

the test statistic is distributed as F with 2 numerator and 68 denomina-

tor degrees of freedom.

6. Decision rule. Let us use a signiﬁcance level of The decision

rule, then, is reject if the computed value of V.R. is equal to or greater

than 4.95 (obtained by interpolation).

7. Calculation of test statistic. The ANOVA for the example is shown

in Figure 10.3.1, where we see that the computed value of V.R. is

20.02.

8. Statistical decision. Since 20.02 is greater than 4.95, we reject

9. Conclusion. We conclude that, in the population from which the sam-

ple came, there is a linear relationship among the three variables.

10. p value. Since 20.02 is greater than 5.76, the p value for the test is less

than .005.

■

Inferences Regarding Individual Frequently, we wish to evaluate the

strength of the linear relationship between Y and the independent variables individually.

That is, we may want to test the null hypothesis that against the alternative

The validity of this procedure rests on the assumptions stated earlier:

that for each combination of values there is a normally distributed subpopulation of Y

values with variance

Hypothesis Tests for the To test the null hypothesis that is equal to

some particular value, say, the following t statistic may be computed:

(10.4.3)

where the degrees of freedom are equal to and is the standard deviation

of the b

n - k - 1,

t =

- b

1i = 1, 2, . . . , k2.

Z 0b

= 0

B’s

a = .01.

: = not all b

= 0

: = b

= b

= 0

500 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION

The standard deviations of the are given as part of the output from most com-

puter software packages that do regression analysis.

EXAMPLE 10.4.3

Let us refer to Example 10.3.1 and test the null hypothesis that age (years) is irrelevant

in predicting the capacity to direct attention (CDA).

Solution:

1. Data. See Example 10.3.1.

2. Assumptions. See Section 10.2.

3. Hypotheses.

4. Test statistic. See Equation 10.4.3.

5. Distribution of test statistic. When is true and the assumptions are

met, the test statistic is distributed as Student’s t with 68 degrees of

freedom.

6. Decision rule. Reject if the computed t is either greater than or equal

to 1.9957 (obtained by interpolation) or less than or equal to

7. Calculation of test statistic. By Equation 10.4.3 and data from Figure

10.3.1 we compute

8. Statistical decision. The null hypothesis is rejected since the computed

value of is less than

9. Conclusion. We conclude, then, that there is a linear relationship

between age and CDA in the presence of education level.

10. p value. For this test, because

(obtained by interpolation).

■

Now, let us perform a similar test for the second partial regression coefﬁcient,

t =

- 0

.6108

.1357

= 4.50

a = .05

: b

Z 0

: b

= 0

-3.80 6-2.6505p 6 21.0052= .01

-1.9957.t, -3.80,

t =

- 0

-.18412

.04851

=-3.80

-1.9957.

Let a = .05

: b

Z 0

: b

= 0

10.4 EVALUATING THE MULTIPLE REGRESSION EQUATION

501

In this case also the null hypothesis is rejected, since 4.50 is greater than 1.9957. We

conclude that there is a linear relationship between education level and CDA in the pres-

ence age, and that education level, used in this manner, is a useful variable for predict-

ing CDA. [For this test, ]

Conﬁdence Intervals for the

When the researcher has been led to conclude

that a partial regression coefﬁcient is not 0, he or she may be interested in obtaining a

conﬁdence interval for this Conﬁdence intervals for the may be constructed in the

usual way by using a value from the t distribution for the reliability factor and standard

errors given above.

A percent conﬁdence interval for is given by

For our illustrative example we may compute the following 95 percent conﬁdence

intervals for and

The 95 percent conﬁdence interval for is

We may give these intervals the usual probabilistic and practical interpretations. We are

95 percent conﬁdent, for example, that is contained in the interval from .3400 to .8816

since, in repeated sampling, 95 percent of the intervals that may be constructed in this

manner will include the true parameter.

Some Precautions

One should be aware of the problems involved in carrying out

multiple hypothesis tests and constructing multiple conﬁdence intervals from the same

sample data. The effect on of performing multiple hypothesis tests from the same data

is discussed in Section 8.2. A similar problem arises when one wishes to construct

conﬁdence intervals for two or more partial regression coefﬁcients. The intervals will not

be independent, so that the tabulated conﬁdence coefﬁcient does not, in general, apply.

In other words, all such intervals would not be percent conﬁdence intervals.

In order to maintain approximate confidence intervals for partial

regression coefﬁcients, adjustments must be made to the calculation of errors in the pre-

vious equations. These adjustments are sometimes called family-wise error rates, and

can be found in many computer software packages. The topic is discussed in detail by

Kutner, et al. (4).

10011 - a2

.3400, .8816

.6108 ; .2708

.6108 ; 11.995721.13572

-.28092, -.08732

-.18412 ; .0968

-.18412 ; 1.99571.048512

; t

1-1a>22,n-k-1

10011 - a2

p 6 21.0052= .01.

502

CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION

Another problem sometimes encountered in the application of multiple regression

is an apparent incompatibility in the results of the various tests of signiﬁcance that one

may perform. In a given problem for a given level of signiﬁcance, one or the other of

the following situations may be observed.

1. and all signiﬁcant

2. and some but not all signiﬁcant

3. signiﬁcant but none of the signiﬁcant

4. All signiﬁcant but not

5. Some signiﬁcant, but not all nor

6. Neither nor any signiﬁcant

Notice that situation 1 exists in our illustrative example, where we have a signiﬁcant

and two signiﬁcant regression coefﬁcients. This situation does not occur in all cases.

In fact, situation 2 is very common, especially when a large number of independent vari-

ables have been included in the regression equation.

EXERCISES

10.4.1 Refer to Exercise 10.3.1. (a) Calculate the coefﬁcient of multiple determination; (b) perform an

analysis of variance; (c) test the signiﬁcance of each Let for all tests of sig-

niﬁcance and determine the p value for all tests; (d) construct a 95 percent conﬁdence interval for

each signiﬁcant sample slope.

10.4.2 Refer to Exercise 10.3.2. Do the analysis suggested in Exercise 10.4.1.

10.4.3 Refer to Exercise 10.3.3. Do the analysis suggested in Exercise 10.4.1.

10.4.4 Refer to Exercise 10.3.4. Do the analysis suggested in Exercise 10.4.1.

10.4.5 Refer to Exercise 10.3.5. Do the analysis suggested in Exercise 10.4.1.

10.4.6 Refer to Exercise 10.3.6. Do the analysis suggested in Exercise 10.4.1.

10.5 USING THE MULTIPLE

REGRESSION EQUATION

As we learned in the previous chapter, a regression equation may be used to obtain a

computed value of when a particular value of X is given. Similarly, we may use our

multiple regression equation to obtain a value when we are given particular values of

the two or more X variables present in the equation.

Just as was the case in simple linear regression, we may, in multiple regression,

interpret a value in one of two ways. First we may interpret as an estimate of the

mean of the subpopulation of Y values assumed to exist for particular combinations of

values. Under this interpretation is called an estimate, and when it is used for this pur-

pose, the equation is thought of as an estimating equation. The second interpretation of

a = .05b

1i 7 02.

10.5 USING THE MULTIPLE REGRESSION EQUATION

503

is that it is the value Y is most likely to assume for given values of the In this case

is called the predicted value of Y, and the equation is called a prediction equation. In

both cases, intervals may be constructed about the value when the normality assump-

tion of Section 10.2 holds true. When is interpreted as an estimate of a population mean,

the interval is called a conﬁdence interval, and when is interpreted as a predicted value

of Y, the interval is called a prediction interval. Now let us see how each of these inter-

vals is constructed.

The Conﬁdence Interval for the Mean of a Subpopulation of

Values Given Particular Values of the We have seen that a

percent conﬁdence interval for a parameter may be constructed by the gen-

eral procedure of adding to and subtracting from the estimator a quantity equal to the reli-

ability factor corresponding to multiplied by the standard error of the estimator.

We have also seen that in multiple regression the estimator is

(10.5.1)

If we designate the standard error of this estimator by the percent con-

ﬁdence interval for the mean of Y, given speciﬁed is as follows:

(10.5.2)

The Prediction Interval for a Particular Value of

Given

Particular Values of the

When we interpret as the value Y is most

likely to assume when particular values of the are observed, we may construct a

prediction interval in the same way in which the confidence interval was constructed.

The only difference in the two is the standard error. The standard error of the predic-

tion is slightly larger than the standard error of the estimate, which causes the pre-

diction interval to be wider than the confidence interval.

If we designate the standard error of the prediction by the percent

prediction interval is

(10.5.3)

The calculations of and in the multiple regression case are complicated and will

not be covered in this text. The reader who wishes to see how these statistics are calcu-

lated may consult the book by Anderson and Bancroft (3), other references listed at the

end of this chapter and Chapter 9, and previous editions of this text. The following exam-

ple illustrates how MINITAB may be used to obtain conﬁdence intervals for the mean

of Y and prediction intervals for a particular value of Y.

EXAMPLE 10.5.1

We refer to Example 10.3.1. First, we wish to construct a 95 percent conﬁdence inter-

val for the mean CDA score (Y) in a population of 68-year-old subjects who com-

pleted 12 years of education Second, suppose we have a subject who is 68 years1X

; t

11-a>22,n-k-1

s¿

10011 - a2s

; t

11-a>22,n-k -1

10011 - a2s

= b

+ b

. . .

+ b

1 - a

10011 - a2

504

CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION

of age and has an education level of 12 years. What do we predict to be this subject’s

CDA score?

Solution: The point estimate of the mean CDA score is

The point prediction, which is the same as the point estimate obtained pre-

viously, also is

To obtain the confidence interval and the prediction interval for the

parameters for which we have just computed a point estimate and a point

prediction, we use MINITAB as follows. After entering the information

for a regression analysis of our data as shown in Figure 10.3.1, we click

on Options in the dialog box. In the box labeled “Prediction intervals for

new observations,” we type 68 and 12 and click OK twice. In addition

to the regression analysis, we obtain the following output:

New Obs Fit SE Fit 95.0% CI 95.0% PI

1 0.303 0.672 (1.038, 1.644) (6.093, 6.699)

We interpret these intervals in the usual ways. We look ﬁrst at the

confidence interval. We are 95 percent confident that the interval from

to includes the mean of the subpopulation of Y values for the

speciﬁed combination of values, since this parameter would be included

in about 95 percent of the intervals that can be constructed in the manner

shown.

Now consider the subject who is 68 years old and has 12 years of

education. We are 95 percent conﬁdent that this subject would have a CDA

score somewhere between and The fact that the P.I. is wider

than the C.I. should not be surprising. After all, it is easier to estimate the

mean response than it is estimate an individual observation. ■

EXERCISES

For each of the following exercises compute the y value and construct (a) 95 percent conﬁdence

and (b) 95 percent prediction intervals for the speciﬁed values of

10.5.1 Refer to Exercise 10.3.1 and let and

10.5.2 Refer to Exercise 10.3.2 and let and

10.5.3 Refer to Exercise 10.3.3 and let and

10.5.4 Refer to Exercise 10.3.4 and let and x

= 2.x

= 1

= 6.x

= 5

= 22.x

= 50, x

= 20,

= 35.x

= 95

6.699.-6.093

1.644-1.038

yN = 5.494 - .184121682+ .61081122= .3034

EXERCISES 505

10.5.5 Refer to Exercise 10.3.5 and let and

10.5.6 Refer to Exercise 10.3.6 and let and

10.6 THE MULTIPLE CORRELATION MODEL

We pointed out in the preceding chapter that while regression analysis is concerned with

the form of the relationship between variables, the objective of correlation analysis is to

gain insight into the strength of the relationship. This is also true in the multivariable

case, and in this section we investigate methods for measuring the strength of the rela-

tionship among several variables. First, however, let us deﬁne the model and assump-

tions on which our analysis rests.

The Model Equation We may write the correlation model as

(10.6.1)

where is a typical value from the population of values of the variable Y, the are

the regression coefﬁcients deﬁned in Section 10.2, and the are particular (known) val-

ues of the random variables This model is similar to the multiple regression model,

but there is one important distinction. In the multiple regression model, given in Equa-

tion 10.2.1, the are nonrandom variables, but in the multiple correlation model the

are random variables. In other words, in the correlation model there is a joint distribu-

tion of Y and the that we call a multivariate distribution. Under this model, the vari-

ables are no longer thought of as being dependent or independent, since logically they

are interchangeable and either of the may play the role of Y.

Typically, random samples of units of association are drawn from a population of

interest, and measurements of Y and the are made.

A least-squares plane or hyperplane is fitted to the sample data by methods

described in Section 10.3, and the same uses may be made of the resulting equation.

Inferences may be made about the population from which the sample was drawn if it

can be assumed that the underlying distribution is normal, that is, if it can be assumed

that the joint distribution of Y and is a multivariate normal distribution. In addition,

sample measures of the degree of the relationship among the variables may be computed

and, under the assumption that sampling is from a multivariate normal distribution, the

corresponding parameters may be estimated by means of confidence intervals, and

hypothesis tests may be carried out. Speciﬁcally, we may compute an estimate of the

multiple correlation coefﬁcient that measures the dependence between Y and the This

is a straightforward extension of the concept of correlation between two variables that we

discuss in Chapter 9. We may also compute partial correlation coefﬁcients that measure

the intensity of the relationship between any two variables when the inﬂuence of all other

variables has been removed.

The Multiple Correlation Coefficient As a first step in analyzing

the relationships among the variables, we look at the multiple correlation coefficient.

b’sy

= b

+ b

= 70.

= 6.00, x

= 75,x

= 2.00,x

= 50, x

= 95.0,

= 80.x

= 90

506 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION