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

A widely used GOF measure for many mathe-

matical models is called thedeviance. However,

as we describe later, for a binary logistic

regression model, the use of the deviance for

assessing GOF is problematic.

A popular alternative is the Hosmer–Lemeshow

(HL) statistic. Both the deviance and the HL

statistic will be defined and illustrated in this

chapter.

Widely used GOF measure:devi-
ance

But: deviance problematic for
binary logistic regression

Popular alternative:
Hosmer–Lemeshow (HL)
statistic

II. Saturated vs. Fully
Parameterized Models

GOF: Overall comparison between
observed and predicted out-
comes

Perfect fit:YiY^i¼ 0 for alli.

 Rarely happens

 Typically 0 <Y^i< 1 for mosti

Perfect fit:Not practical goal
Conceptual ideal
Saturated model
(a reference point)

As stated briefly in the previous overview sec-

tion, a measure of goodness of fit (GOF) pro-

vides an overall comparison between observed

(Yi) and predicted valuesðY^iÞof the outcome

variable.

We say there isperfect fitifYiY^i¼ 0 for alli.

Although the mathematical characteristics of

any logistic model require that the predicted

value for any subject must lie between or

including 0 or 1, it rarely happens that pre-

dicted values are either 0 or 1 forallsubjects

in a given dataset. In fact, the predicted values

for most, if not all, subjects will lie above 0 and

below 1.

Thus, achieving “perfect fit” is typically not a

practical goal, but rather is a conceptual ideal

when fitting a model to one’s data. Neverthe-

less, since we typically want to use this “ideal”

model as a reference point for assessing the fit

of any specific model of interest, it is conve-

nient to identify such a model as asaturated

model.

A trivial example of a saturated regression

model is obtained if we have a dataset contain-

ing onlyn¼2 subjects, as shown on the left.

Here, the outcome variable, SBP, is continu-

ous, as is the (nonsense) predictor variable foot

length (FOOT). A “perfect” straight line fits the

data.

EXAMPLE

n= 2

FOOT

SBP

9 11

• 
  
  
  
  • 115
  
  
  
  

170

Subject # SBP FOOT

(^11159)
2 170 11
Presentation: II. Saturated vs. Fully Parameterized Models 305