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

EXAMPLE

4 predictors: E, AGE, RACE, SEX)

4 Dbvalues for each subject (i)

n¼ 100 )Compute 400 Dbjvalues

DbE;i;DbAGE;i;DbRACE;i;DbSEX;i

iðsubjectÞ¼ 1 ; 2 ;...; 100

DbE,i¼Ais “large” or “significant”:

+

removing subject A from analysis

changes conclusions about effect of E

Summary measure for linear
regression:

Cook’s distance (CD)

combinesDbj,iinformationfor
allXjpredictors for subjecti,
e.g., a weighted average of the
form
CDi¼~jwjDbj;i=~jwj;
i¼ 1 ; 2 ;...;n

Logistic regression:

 Cook’s distance-type index

 Uses approximation to change
in logit values

 Similar to combining Delta-
betas

Suggested alternative for logistic

regression:

CD*i¼~

j

wjDðexp½bŠÞj;i=~

j

wj

but not available in computer

packages.

For example, a model containing four predic-

tors, say E, AGE, RACE, and SEX, would pro-

duce four Delta-betas for each subject.

If the dataset contained 100 subjects, 400

Delta-betas would be computed, 4 for each

subject.

If subject A, say, has a “large” or “significant”

Delta-beta for the variableE, then one may

conclude that removal of this subject from the

analysis may change the conclusions drawn

about the effect ofE.

Also, a summary measure that combines the

Delta-beta information from all variables is

typically computed. For linear regression, one

such measure is calledCook’s distance, which

is a form of weighted average of the Delta-betas

over all predictor variablesXjin one’s model.

For logistic regression, a similar measure

(Pregibon, 1981) is used and often referred to

as aCook’s distance-type index. This measure is

derived using an approximation to the change

in logit values when a subject is dropped from

the data and, in essence, combines Delta-beta

values using a logistic model.

However, since the effect measure in logistic

regression is typically an odds ratio, which

exponentiates regression coefficients, a modi-

fied Cook’s distance-type index that computes

a weighted average of changes in exp[b], i.e.,

Dexp[b], might be preferable, but is not avail-

able in most computer packages.

276 8. Additional Modeling Strategy Issues