Consider the data found in Table B.15 detailing mortality and air pollution on a by city/demographic basis. Following Example 9.3, use principle-components regression in comparison with standard ordinary least squares. Highlight what relationships can be elucidated from the principle components.

table B.15

City Mort Precip Educ Nonwhite Nox SO2

Akron-OH 921.87 36 11.4 8.8 15 59

Albany-NY 997.88 35 11 3.5 10 39

Allentown-PA 962.35 44 9.8 0.8 6 33

Atlanta-GA 982.29 47 11.1 27.1 8 24

Baltimore-MD 1071.29 43 9.6 24.4 38 206

Birmingham-AL 1030.38 53 10.2 38.5 32 72

Boston-MA 934.7 43 12.1 3.5 32 62

Bridgeport-CT 899.53 45 10.6 5.3 4 4

Buffalo-NY 1001.9 36 10.5 8.1 12 37

Canton-OH 912.35 36 10.7 6.7 7 20

Chattanooga-TN 1017.61 52 9.6 22.2 8 27

Chicago-IL 1024.89 33 10.9 16.3 63 278

Cincinnati-OH 970.47 40 10.2 13 26 146

Cleveland-OH 985.95 35 11.1 14.7 21 64

Columbus-OH 958.84 37 11.9 13.1 9 15

Dallas-TX 860.1 35 11.8 14.8 1 1

Dayton-OH 936.23 36 11.4 12.4 4 16

Denver-CO 871.77 15 12.2 4.7 8 28

Detroit-MI 959.22 31 10.8 15.8 35 124

Flint-MI 941.18 30 10.8 13.1 4 11

Fort Worth-TX 891.71 31 11.4 11.5 1 1

Grand Rapids-MI 871.34 31 10.9 5.1 3 10

Greensboro-NC 971.12 42 10.4 22.7 3 5

Hartford-CT 887.47 43 11.5 7.2 3 10

Houston-TX 952.53 46 11.4 21 5 1

Indianapolis-IN 968.66 39 11.4 15.6 7 33

Kansas City-MO 919.73 35 12 12.6 4 4

Lancaster-PA 844.05 43 9.5 2.9 7 32

Los Angeles-CA 861.83 11 12.1 7.8 319 130

Louisville-KY 989.27 30 9.9 13.1 37 193

Memphis-TE 1006.49 50 10.4 36.7 18 34

Miami-FL 861.44 60 11.5 11.5 1 1

Milwasukee-WI 929.15 30 11.1 5.8 23 125

Minneapolis-MN 857.62 25 12.1 3 11 26

Nashville-TN 961.01 45 10.1 21 14 78

New Haven-CT 923.23 46 11.3 8.8 3 8

New Orleans-LA 1113.06 54 9.7 31.4 17 1

New York-NY 994.65 42 10.7 11.3 26 108

Philadelphia-PA 1015.02 42 10.5 17.5 32 161

Pittsburgh-PA 991.29 36 10.6 8.1 59 263

Portland-OR 893.99 37 12 3.6 21 44

Providence-RI 938.5 42 10.1 2.2 4 18

Reading-PA 946.18 41 9.6 2.7 11 89

Richmond-VA 1025.5 44 11 28.6 9 48

Rochester-NY 874.28 32 11.1 5 4 18

Saint Louis-MO 953.56 34 9.7 17.2 15 68

San Diego-CA 839.71 10 12.1 5.9 66 20

San Francisco-CA 911.7 18 12.2 13.7 171 86

San Jose-CA 790.73 13 12.2 3 32 3

Seattle-WA 899.26 35 12.2 5.7 7 20

Springfield-MA 904.16 45 11.1 3.4 4 20

Syracuse-NY 950.67 38 11.4 3.8 5 25

Toledo-OH 972.46 31 10.7 9.5 7 25

Utica-NY 912.2 40 10.3 2.5 2 11

Washington-DC 967.8 41 12.3 25.9 28 102

Wichita-KS 823.76 28 12.1 7.5 2 1

Wilmington-DE 1003.5 45 11.3 12.1 11 42

Worcester-MA 895.7 45 11.1 1 3 8

York-PA 911.82 42 9 4.8 8 49

Youngstown-OH 954.44 38 10.7 11.7 13 39

Example 9.3 Principal-Component Regression for the Acetylene Data

We illustrate the use of principal-component regression for the acetylene data. We

begin with the linear transformation Z = XT that transforms the original standardized

regressors into an orthogonal set of variables (the principal components). The

TABLE 9.10 Matrix T of Eigenvectors and Eigenvalnes j for the Acetylene Data

Eigenvectors

Eigenvalues

j

.3387 .1057 .6495 .0073 .1428 .2488 .2077 .5436 .1768 4.20480

.1324 .3391 .0068 .7243 5843 .0205 .0102 .0295 .0035 2.16261

.4137 .0978 .4696 .0718 .0182 .0160 .1468 .7172 .2390 1.13839

.2191 .5403 .0897 .3612 .1661 .3733 .5885 .0909 .0003 1.04130

.4493 .0860 .2863 .1912 .0943 .0333 .0575 .1543 .7969 0.38453

.2524 .5172 .0570 .3447 .2007 .3232 .6209 .1280 .0061 0.04951

.4056 .0742 .4404 .2230 .1443 .5393 .3233 .0565 .4087 0.01363

.0258 .5316 .2240 .3417 .7342 .0705 .0057 .0761 .0050 0.00513

.4667 .0969 .1421 .1337 .0350 .6299 .3089 .3631 .3309 0.00010

eigenvalues j and the T matrix for the acetylene data are shown in Table 9.10. This

matrix indicates that the relationship between z1 (for example) and the standardized

regressors is

z1 0 3387x1 0 1324x2 0 4137x3 0 2191x1x2 0 4493x1x3

0 2524

. . . . .

. x2x3 x1 x x

2

2

2

3

0.4056 0.0258 0.4667 2

The relationships between the remaining principal components z2, z3, . . . , z9 and

the standardized regressors are determined similarly. Table 9.11 shows the elements

of the Z matrix (sometimes called the principal-component scores).

The principal-component estimator reduces the effects of multicollinearity by

using a subset of the principal components in the model. Since there are four small

eigenvalues for the acetylene data, this implies that there are four principal components

that should be deleted. We will exclude z6, z7, z8, and z9 and consider regressions

involving only the first five principal components.

Suppose we consider a regression model involving only the first principal component,

as in

y 1z1

The fitted model is

y 0.35225z1

or PC 0.35225, 0, 0, 0, 0, 0, 0, 0, 0 . The coefficients in terms of the standardized

regressors are found from PC T PC. Panel A of Table 9.11 shows the resulting

standardized regression coefficients as well as the regression coefficients in terms

of the original centered regressors. Note that even though only one principal component

is included, the model produces estimates for all nine standardized regression

coefficients.

The results of adding the other principal components z2, z3, z4, and z5 to the model

one at a time are displayed in panels B, C, D, and E, respectively, of Table 9.12. We

see that using different numbers of principal components in the model produces

TABLE 9.12 Principal Components Regression for the Acetylene Data

Principal Components in Model

Parameter

A B C D E

z1 z1, z2 z1, z2, z3 z1, z2, z3, z4 z1, z2, z3, z4, z5

Standarized

Estimate

Original

Estimate

Standardized

Estimate

Original

Estimate

Standardized

Estimate

Original

Estimate

Standardized

Estimate

Original

Estimate

Standardized

Estimate

Original

Estimate

0 .0000 42.1943 .0000 42.2219 .0000 36.6275 .0000 34.6688 .0000 34.7517

1 .1193 1.4194 .1188 1.4141 .5087 6.0508 .5070 6.0324 .5056 6.0139

2 .0466 .5530 .0450 .5346 .0409 .4885 .2139 2.5438 .2195 2.6129

3 .1457 1.7327 .1453 1.7281 .4272 5.0830 .4100 4.8803 .4099 4.8757

12 .0772 1.0369 .0798 1.0738 .0260 .3502 .1123 1.5115 .1107 1.4885

13 .1583 2.0968 .1578 2.0922 .0143 .1843 .0597 .7926 .0588 .7788

23 .0889 1.2627 .0914 1.2950 .0572 .8111 .1396 1.9816 .1377 1.9493

11 .1429 2.1429 .1425 2.1383 .1219 1.8295 .1751 2.6268 .1738 2.6083

22 .0091 .0968 .0065 .0691 .1280 1.3779 .0460 .4977 .0533 .5760

33 .1644 1.9033 .1639 1.8986 .0786 .9125 .0467 .5392 .0463 .5346

R2 .5217 .5218 .9320 .9914 .9915

MSRes .079713 .079705 .011333 .001427 .00142