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

dcovð^y 1 ;^y 2 Þ¼r 12 s 1 s 2

Importance ofV^ð^uÞ:

Inferences require variances and

covariances

(3) Variable listing:

Variable ML Coefficient S.E.

Intercept ^a s^a

X 1 b^ 1 s^b 1







Xk b^k s^bK

III. Models for Inference-

Making

Model 1: logitP 1 ðXÞ¼aþb 1 X 1 þb 2 X 2

Model 2: logitP 2 ðXÞ¼aþb 1 X 1 þb 2 X 2

þb 3 X 3

Model 3: logitP 3 ðXÞ¼aþb 1 X 1 þb 2 X 2

þb 3 X 3 þb 4 X 1 X 3

þb 5 X 2 X 3

L^ 1 ,L^ 2 ,L^ 3 areL^s for models 1–3

L^ 1 L^ 2 L^ 3

The reader may recall that the covariance

between two estimates is the correlation times

the standard errors of each estimate.

The variance–covariance matrix is important

because hypothesis testing and confidence inter-

val estimation require variances and sometimes

covariances for computation.

In addition to the maximized likelihood value

and the variance–covariance matrix, other

information is also provided as part of the out-

put. This typically includes, as shown here,a

listing of each variable followed by its ML esti-

mate and standard error. This information pro-

vides another way of carrying out hypothesis

testing and confidence interval estimation, as

we will describe shortly. Moreover, this listing

gives the primary information used for calcu-

lating odds ratio estimates and predicted risks.

The latter can only be done, however, provided

the study has a follow-up type of design.

To illustrate how statistical inferences are

made using the above information, we con-

sider the following three models, each written

in logit form. Model 1 involves two variablesX 1

andX 2. Model 2 contains these same two vari-

ables and a third variableX 3. Model 3 contains

the same threeX’s as in model 2 plus two addi-

tional variables, which are the product terms

X 1 X 3 andX 2 X 3.

LetL^ 1 ,L^ 2 , andL^ 3 denote the maximized likeli-

hood values based on fitting Models 1, 2, and 3,

respectively. Note that the fitting may be done

either by unconditional or conditional meth-

ods, depending on which method is more

appropriate for the model and data set being

considered.

Because the more parameters a model has, the

better it fits the data, it follows thatL^ 1 must be

less than or equal toL^ 2 , which, in turn, must be

less than or equal toL^ 3.

Presentation: III. Models for Inference-Making 133