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

Note, however, that concluding that “none of the

models are saturated” is based on the following

assumption:theunitofanalysisisthesubject.

This assumption is equivalent to listing the

dataset using 40 lines of data, one for each

subject. For Model 2, therefore, each dataline

would contain for a given subject, the values of

the outcome (D) and the predictor variables

(EandV) in the model.

When each subject is the unit of analysis,

we can therefore claim that Model 2 is not

saturated because the number of parameters

(3) in the model is less than total units in the

dataset (n¼40).

However, there is another way to view the

dataset we have been considering:the unit of

analysis is a group of subjects, all of whom have

the same covariate pattern within a group.

This assumption is equivalent to listing the

dataset using only four lines of data, one for

each covariate pattern. Each dataline would

contain for a given group of subjects, the num-

ber of cases (D¼1) in each group (dg), the

number of subjects in each group (ng), and

the values of each predictor variable being

modeled, as shown at the left for our dataset.

This type of data layout is called anevents–trials

format, where there aredgevents andngtrials.

Using events–trials format, we can argue that

the number of observations (n) consists of the

total number of datalines (4 in our example),

rather than the number of subjects (40).

Here, the goal of model prediction no longer is

to predict an individual’s (0 or 1) outcome, but

rather to predict the observed proportion of

persons in a group (the unit of analysis) that

has the outcome, i.e.,^pg¼dg=ng.

Thus, using events–trials format, we can declare

a model to be “group-saturated” if for each covar-

iate pattern listed in the dataset, the model per-

fectly predicts the observed proportionp^g.

EXAMPLE (continued)

Key assumption:

Unit of analysis is the subject

Datalines listed by subject (e.g.,

Model 2)

Subject (i) D E V

1111

2111

... ... ... ...

39 0 0 0

40 0 0 0

If subjects are units of analysis,

then Model 2 is not saturated

(kþ 1 ¼ 3 <n¼40)

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Alternative assumption:

Unit of analysis is a group

(subjects with same covariate

pattern)

Datalines Listed by Group (e.g.,

Model 2)

Group (g) dg ng EV

161011

241001

331010

471000

Events–trials format

(dg)(ng)

n¼#of observations¼ 4

(Model 2)

Goal of model prediction:

p^g¼dg=ng

(group predictionrather than

individual prediction)

Events–trials format:

“Group-saturated” providedn¼kþ 1

forncovariate patterns

(i.e., perfectly predictsp^g)

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Presentation: II. Saturated vs. Fully Parameterized Models 307