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

IV. The Hosmer–
Lemeshow (HL)
Statistic

 Alternative to questionable use
of deviance

 Available in most computer
packages

HL widely used regardless of
whetherG<<norGn:

 RequiresG> 3

 Rarely significant whenG< 6
 Works best whenGn(e.g.,
someXs are continuous)

 HL0 for fully parameterized
model

 “Saturated” model is fully
parameterized in ET format

Steps for computing HL statistic:

1. ComputeP^ðXiÞfor allnsubjects


2. OrderP^ðXiÞfrom largest to
   smallest values


3. Divide ordered values intoQ
   percentile groupings (usually
   Q¼ 10 )deciles of risk)


4. Form table of observed and
   expected counts


5. Calculate HL statistic from table


6. Compare computed HL to
   w^2 withQ2df

Step 1: ComputeP^ðXiÞ;i¼ 1 ; 2 ;...;n
n¼ 200 ) 200 values forP^ðXiÞ
although some values are identi-
cal ifXiXjfor subjectsiandj.

Step 2: Order values ofP^ðXiÞ:
e.g.,n¼ 200
Order # P^ðXiÞ
1 0.934 (largest)
2 0.901 tie
3 0.901
... ...
199 0.123
200 0.045 (smallest)

Toavoid questionable useofthe deviance to pro-

vide a significance test for assessing GOF, the

Hosmer–Lemeshow (HL) statistic has been deve-

loped and is available in most computer packages.

The HL statistic is widely used regardless of

whetherornotthenumberofcovariatepatterns

(G) is close to the number of observations. Nev-

ertheless, this statistic requires that the model

considers at least three covariate patterns, rarely

results in significance whenGis less than 6, and

works best whenGis close ton(the latter occurs

when some of the predictors are continuous).

Moreover, the HL statistic has the property

that it will always be zero for a fully parame-

terized model. In other words, the “saturated”

model for the HL statistic is essentially a fully

parameterized (group-saturated) model coded

in events–trials format.

The steps involved in computing the HL statis-

tic are summarized at the left. Each step will

then be described and illustrated below, fol-

lowing which we will show examples obtained

from different models with differing numbers

of covariate patterns.

At the first step, wecompute the predicted risks

P^ðXiÞfor all subjectsin the dataset. If there are,

say,n¼200 subjects, there will be 200 pre-

dicted risks, although some predicted risks

will be identical for those subjects with the

same covariate pattern.

At the second step, weorder the predicted risks

from largest to smallest (or smallest to largest).

Again, there may be several ties when doing

this if some subjects have the same covariate

pattern.

o

318 9. Assessing Goodness of Fit for Logistic Regression