Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

dORs¼point estimators

Variability ofdOR considered for

statistical inferences

Two types of inferences:
(1) Testing hypotheses

(2) Interval estimation

Two testing procedures:

(1) Likelihood ratio test: a chi-
square statistic using2lnL^.
(2) Wald test:aZtest using
standard errors listed with
each variable.

Note: Since Z^2 isw^2 1df, the Wald test
can equivalently be considered a
chi-square test.

The estimated model coefficients and the cor-

responding odds ratio estimates that we have

just described are point estimates of unknown

population parameters. Such point estimates

have a certain amount of variability associated

with them, as illustrated, for example, by the

standard errors of each estimated coefficient

provided in the output listing. We consider

the variability of our estimates when we

make statistical inferences about parameters

of interest.

We can use two kinds of inference-making pro-

cedures. One is testing hypotheses about cer-

tain parameters; the other is deriving interval

estimates of certain parameters.

As an example of a test, we may wish to test the

null hypothesis that an odds ratio is equal to

the null value.

Or, as another example, we may wish to test

for evidence of significant interaction, for

instance, whether one or more of the coeffi-

cients of the product terms in the model are

significantly nonzero.

As an example of an interval estimate, we may

wish to obtain a 95% confidence interval for

the adjusted odds ratio for the effect of CAT on

CHD, controlling for the fiveVvariables and

the twoWvariables. Because this model con-

tains interaction terms, we need to specify the

values of theWs to obtain numerical values for

the confidence limits. For instance, we may

want the 95% confidence interval when CHL

equals 220 and HPT equals 1.

When using ML estimation, we can carry out

hypothesis testing by using one of two proce-

dures, thelikelihood ratio testand theWald test.

The likelihood ratio test is a chi-square test that

makes use of maximized likelihood values such

as those shown in the output. The Wald test is a

Ztest; that is, the test statistic is approximately

standard normal. The Wald test makes use of

the standard errors shown in the listing of vari-

ables and associated output information. Each

of these procedures is described in detail in the

next chapter.

EXAMPLES

(1) Test for H 0 :OR¼ 1

(2) Test for significant interaction,

e.g.,d 16 ¼0?

(3) Interval estimate: 95% confidence

interval for ORCAT, CHD

controlling for 5Vsand2Ws

Interaction: must specifyWs

e.g., 95% confidence interval when

CAT¼220 and HPT¼ 1

120 4. Maximum Likelihood Techniques: An Overview