wherebb^iis the corresponding estimated regression coe‰cient and SEðbb^iÞis the
estimate of the standard error ofbb^i, both of which are printed by standard
computer-packaged programs. In performing this test, we refer the value of the
zstatistic to percentiles of the standard normal distribution.
Example 9.5 Refer to the data set on prostate cancer of Example 9.1 (Table
9.1). With all five covariates, we have the results shown in Table 9.6. The
e¤ects of x-ray and stage are significant at the 5% level, whereas the e¤ect of
acid is marginally significant (p¼ 0 :0643).
Note: Use the same SAS program as in Example 9.4.
Given a continuous variable of interest, one can fit a polynomial model and
use this type of test to check for linearity. It can also be used to check for a
single product representing an e¤ect modification.
Example 9.6 Refer to the data set on prostate cancer of Example 9.1 (Table
9.1), but this time we investigate only one covariate, the level of acid phospha-
tase (acid). After fitting the second-degree polinomial model,
pi¼1
1 þexp½ðb 0 þb 1 ðacidÞþb 2 ðacidÞ^2 Þi¼ 1 ; 2 ;...;nwe obtained the results shown in Table 9.7, indicating that thecurvature e¤ect
should not be ignoredðp¼ 0 : 0437 Þ.
TABLE 9.6
Variable Coe‰cient
Standard
Error zStatistic pValueIntercept 0.0618 3.4599 0.018 0.9857
X-ray 2.0453 0.8072 2.534 0.0113
Stage 1.5641 0.7740 2.021 0.0433
Grade 0.7614 0.7708 0.988 0.3232
Age 0.0693 0.0579 1.197 0.2314
Acid 0.0243 0.0132 1.850 0.0643
TABLE 9.7
Factor Coe‰cient
Standard
Error zStatistic pValueIntercept 7.3200 2.6229 2.791 0.0053
Acid 0.1489 0.0609 2.445 0.0145
Acid^2 0.0007 0.0003 2.017 0.0437
MULTIPLE REGRESSION ANALYSIS 331