Introductory Biostatistics

otherX’s with theX’s under investigation deleted to obtain the regression sum
of squares (SSR 2 ). Define the mean square due toH 0 as

MSR¼

SSR 1
SSR 2

m

ThenH 0 can be tested using

F¼

MSR

MSE

anFtest atðm;n
k
1 Þdegrees of freedom. Thismultiple contribution pro-
cedureis very useful for assessing the importance of potential explanatory vari-
ables. In particular, it is often used to test whether a similar group of variables,
such asdemographic characteristics, is important for prediction of the response;
these variables have some trait in common. Another application would be a
collection of powersand=orproduct terms (referred to asinteraction variables).
It is often of interest to assess the interaction e¤ects collectively before trying to
consider individual interaction terms in a model as suggested previously. In
fact, such use reduces the total number of tests to be performed, and this, in
turn, helps to provide better control of overall type I error rates, which may be
inflated due to multiple testing.

Example 8.9 Refer to the data on liver transplants of Example 8.7 consisting
of three independent variables: hepatic systolic pressure at transplant time
(called pressure 1 ), age (at the second measurement time), and gender of the
child. Let us consider all five quadratic terms and products of these three orig-
inal factors (x 1 ¼pressure 12 ,x 2 ¼age^2 ,x 3 ¼pressure 1 age,x 4 ¼pressure 1 
gender, andx 5 ¼agegender). Using the second measurement of the systolic
pressure of the hepatic artery as our dependent variable and fitting the multiple
regression model with all eight independent variables (three original plus five
newly defined terms), we have

SSR¼ 1944 :70 with 8 df

SSE¼ 710 :25 with 12 df; or

MSE¼ 19 : 19

TABLE 8.12

Variable Coe‰cient Standard Error tStatistic pValue

Smoke
220.324 58.143
3.789 0.0026
Sulfur 1051.816 212.596 4.947 0.0003

302 CORRELATION AND REGRESSION