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

Large samples: both procedures give

approximately the same results

Small or moderate samples:different

results possible; likelihood ratio test

preferred

Confidence intervals

 use large sample formulae

 use variance–covariance matrix

No interaction: variance only

Interaction: variances and covar-

iances

5. Statistical Inferences Using
   ML Techniques

Both testing procedures should give appro-

ximately the same answer in large samples

but may give different results in small or

moderate samples. In the latter case, statisti-

cians prefer the likelihood ratio test to the

Wald test.

Confidence intervals are carried out by using

large sample formulae that make use of the

information in the variance–covariance matrix,

which includes the variances of estimated coef-

ficients together with the covariances of pairs of

estimated coefficients.

An example of the estimated variance–covari-

ance matrix is given here. Note, for example,

that the variance of the coefficient of the CAT

variable is 9.6389, the variance for the CC vari-

able is 0.0002, and the covariance of the coeffi-

cients of CAT and CC is0.0437.

If the model being fit contains no interaction

terms and if the exposure variable is a (0, 1)

variable, then only a variance estimate is

required for computing a confidence interval.

If the model contains interaction terms, then

both variance and covariance estimates are

required; in this latter case, the computations

required are much more complex than when

there is no interaction.

We suggest that the reader review the material

covered here by reading the summary outline

that follows. Then you may work the practice

exercises and test.

In the next chapter, we give a detailed descrip-

tion of how to carry out both testing hypoth-

eses and confidence interval estimation for the

logistic model.

EXAMPLEV^ð^uÞ

Intercept

1.5750 –0.6629 –0.0136 0.0034

0.00030.0000–0.0010

–0.0021 –0.0049

–0.0016

0.5516

0.0002

9.6389

Intercept 0.0548

CAT

CAT

AGE

AGE

CC

CC

CH

CH

–0.0437

SUMMARY

Chapters up to this point:

1. Introduction


2. Important Special Cases


3. Computing the Odds Ratio

3 4. ML Techniques: An Overview

This presentation is now complete. In sum-

mary, we have described how ML estimation

works, have distinguished between uncondi-

tional and conditional methods and their

corresponding likelihood functions, and

have given an overview of how to make statis-

tical inferences using ML estimates.

Presentation: V. Overview on Statistical Inferences for Logistic Regression 121