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

Step 3: FormQpercentile group-
ings.
Typically,Q¼10, i.e., deciles of
risk e.g.,n¼ 200

Decile No. of subjects

1  20

2  20

... ...

9  20

10  20

Total 200

Ties)#of subjects 6 ¼exactly

20 (¼n/Q) in all deciles

Must keep subjects with identical

values of^PðXiÞin the same decile

Step 4: Form table of observed
and expected cases and
noncases
Deciles
of risk
1
2
3

10

Oc3

Oc2

Oc1

Onc3

Onc10

Onc2

Onc1

Ec3

Ec10

Ec2

Ec1

Enc3

Enc10

Enc2

Enc1

Oc10

Obs.

cases

Obs. non

cases

Exp. non

cases

Exp.

cases

Observed cases and noncases:

Ocqcounts#of cases (Yi¼1) in
qth decile
Oncqcounts#of noncases
(Yi¼1) inqth decile

Note:Oncq¼nqOcq

Expected cases and noncases:

Ecq¼~

nq

i¼ 1

P^ðXiqÞandEncq¼nqEcq,

where
Xiq¼covariate values forith subj
inqth decile

At the third step, we divide the ordered pre-

dicted risks intoQ percentile groupings. The

typical grouping procedure involves Q¼ 10

deciles. Thus, if the sample size is 200, each

decile will contain approximately 20 subjects.

Henceforth, we will assume that Q¼10.

Note, however, because some subjects may

have identical predicted risks (i.e., ties), the

number of subjects per decile may vary some-

what to keep subjects with identical predicted

risks in the same decile.

At the fourth step, we form (typically using

a convenient computer program) the table,

shown at the left, that contains observed and

expected cases and noncases within each dec-

ile. In this table, the valuesOcq,Ecq,Oncq, and

Encq,q¼1, 2,..., 10 are defined as follows:

Ocq¼#of observed cases in theqth decile

Ecq¼#of expected cases in theqth decile

Oncq¼#of observed noncases in theqth

decile

Encq¼#of expected noncases in theqth

decile

The observed cases (Ocq) and noncases (Oncq)

in each decile are obtained by simply counting

the numbers of subjects in that decile who are

cases (i.e.,Yi¼1) and noncases (i.e.,Yi¼0),

respectively. Note that once we countOcq,we

can obtainOncqby subtraction fromnq, the

total number of subjects in theqth decile.

The expected cases (Ecq) in each decile are

obtained by summing the predicted risks

P^ðXiÞ for all subjects in that decile. The

expected number of noncases (Encq) are

obtained by subtraction fromnq.

Presentation: IV. The Hosmer–Lemeshow (HL) Statistic 319