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

F. Using DevETðb^Þto test for GOF:

i. WhenG<<n, can assume DevETðb^Þis

approximatelyw^2 np 1 underH 0 : good fit

ii. However, whenGn,cannotassume

DevETðb^Þis approximatelyw^2 np 1 under

H 0 : good fit

iii. Xicontinuous, e.g.,Xi¼AGE)Gn,

so cannot test for GOF

G. Why DevSSð^bÞcannot be used to test for GOF:

i. Alternative formula for

DevSSð^bÞ:DevSSðb^Þ

¼ 2 ~

n

i¼ 1

P^ðXiÞln P^ðXiÞ

1 ^PðXiÞ



þln 1 P^ðXiÞ

hi

ii. The above formula contains only the

predictedvaluesP^ðXiÞfor each subject;

tells nothing about the agreement

betweenobserved(0, 1) outcomes and

their corresponding predicted

probabilities

IV. The HL Statistic (pages 318–320)

A. Used to provide a significance test for

assessing GOF:

i. Avoids questionable use of the deviance

whenGn

ii. Available in most computer procedures

for logistic regression

iii. Requires that the model considers at

least three covariate patterns, rarely

results in significance whenGis less

than 6, and works best whenGis close to

n, e.g., with continuous predictors

B. Steps for computation:

1. ComputeP^ðXiÞfor allnsubjects


2. OrderP^ðXiÞfrom largest to smallest
   values


3. Divide ordered values intoQpercentile
   groupings (usuallyQ¼10, i.e., deciles)


4. Form table of observed and expected
   counts


5. Calculate HL statistic from table


6. Compare computed HLtow^2 withQ2df

Detailed Outline 331