Introductory Biostatistics

knowing if we may conclude thatr 0 0, that is, that the two variables

under investigation are really correlated. The test statistic is

t¼r

ffiffiffiffiffiffiffiffiffiffiffiffiffi

n
 2

1 
r^2

r

The procedure is often performed as two-sided, that is,

HA:r 00

and it is attest withn
2 degrees of freedom, the samettest as used in

the comparisons of population means in Chapter 7.

2. The role of the slopeb 1 can be seen as follows. Since the regression model
   describes the mean of the dependent variableYas a function of the pre-
   dictor or independent variableX,

my¼b 0 þb 1 x

YandXwould be independent ifb 1 ¼0. The test for

H 0 :b 1 ¼ 0

can be performed similar to the method for one-sample problems in Sec-

tions 6.1 and 7.1. In that process the observed/estimated valueb 1 is con-

verted to a standard unit: the number of standard errors away from the

hypothesized value of zero. The formula for a standard error ofb 1 is

rather complicated; fortunately, the resulting test isidenticalto thettest

above. Whenever needed, for example in the computation of confidence

intervals for the slope, we can always obtain the numerical value of its

standard error from computer output.

When thettest for independence above is significant, the value ofXhas a
real e¤ect on the distribution ofY. To be more precise, the square of the
correlation coe‰cient,r^2 , represents the proportion of the variability ofY
accounted for byX. For example,r^2 ¼ 0 :25 indicates that the total variation in
Yis reduced by 25% by the use of information aboutX. In other words, if we
have a sample of the same size, with allnsubjects having the sameXvalue,
the variation inY(say, measured by its variance) is 25% less than the variation
ofYin the current sample. It is interesting to note thatr¼ 0 :5 would give an
impression of greater association betweenXandY, but a 25% reduction in
variation would not. The parameterr^2 is called thecoe‰cient of determination,
an index with a clearer operational interpretation than the coe‰cient of corre-
lationr.

290 CORRELATION AND REGRESSION