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

Presentation

I. Overview

Focus

Does estimated

logistic model predict

observed outcomes

in data?

 Considers a given model

 Does not consider comparing
models

Primary analysis goal:
Assess E–D relationship to derive
“best” model

GOF goal:
Determine how well final (“best”)
model fits the data

Assume:Yis binary (0,1)
Units of analysis: individual
subjects

GOF: Summary measure that com-
pares

YitoY^i; where

Yi¼observed response for

subjecti

Y^i¼^PðXiÞ¼predicted response

for subjecti

i¼1, 2,...,n

Good fit: GOF measure “small” or “n.s.”

Lack of fit: Otherwise

Not sufficient evidence to conclude

a bad fit

This presentation describes methods for asses-

sing the extent to which a logistic model esti-

mated from a dataset predicts the observed

outcomes in the dataset. The classical term

for this topic isgoodness of fit (GOF).

GOF is an issue that considers how well a given

model, considered by itself, fits the data, rather

than whether or not the model is more appro-

priate than another model.

In most epidemiologic analyses, the primary

goal is to assess an exposure–disease relation-

ship, so that we are usually more interested in

deriving the “best” model for the relationship

(which typically involves a strategy requiring

the comparison of various models) than in

using a GOF procedure. Nevertheless, once

we have obtained a final (i.e., best”) model,

we would also like this model to fit the data

well, thus justifying a GOF procedure.

Assuming that the outcome (Y) is binary, say

coded as 0 or 1, and the unit of analysis is an

individual subject, GOFtypically requires a

summary measure over all subjects that com-

pares the observed outcome (Yi) for subjectito

the predicted outcome (Y^i) for this subject

obtained from the fitted model, i.e.,Y^i¼P^ðXiÞ.

If when determined collectively over all sub-

jects, the GOF measure is “small” or “nonsig-

nificant,” we say the model has good fit,

although, technically, we mean there is not

sufficient evidence to conclude a bad fit.

Otherwise, we say that the model has evidence

oflack of fit.

304 9. Assessing Goodness of Fit for Logistic Regression