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

Recall that the previously shown output gave

Wald statistics for HEAD, AGECAT, and FLEX

that were nonsignificant. A backward elimina-

tion approach that begins by dropping the least

significant of these variables (i.e., HEAD), refit-

ting the model, and dropping additional non-

significant variables results in a model that

contains only two predictor variables, WEIGHT

and PATELLAR. (Note that we are treating all

predictor variables as exposure variables.)

The fitted logistic model that involves only

WEIGHT and PATELLAR is shown at the left.

We also show the discrimination measures

that result for this model, including the c

(¼AUC) statistic. Thecstatistic here is 0.731,

which is slightly smaller than thecstatistic of

0.745 obtained for the full model. The reduced

model has slightly less discriminatory power

than the full model. (See Hanley (1983) for a

statistical test of significance between two or

more AUCs.)

The ROC plot for the reduced model is shown

here.

Notice that there are fewer cut-pts plotted on

this graph than on the ROC plot for the full

model (previous page). The reason is that the

number of possible cut-pts for a given model is

always equal to or less than the number of

covariate patterns (i.e., distinct combinations

of predictors) defined by the model.

The reduced model (with only two binary pre-

dictors) contains four (¼ 22 ) covariate patterns

whereas the full model (with 5 binary predic-

tors) contains 32 (¼ 25 ) covariate patterns.

Moreover, because the reduced model is nested

within the full model, the AUC for the reduced

model will always be smaller than the AUC for

the full model, i.e., similar to the characteris-

tics ofR^2 in linear regression. That’s the case

here, since the AUC is 0.731 for the reduced

model compared to 0.745 for the full model.

EXAMPLE (continued)

Backward elimination:

Step 1: Drop HEAD

(highestP-value 0.5617)

Step 2: Drop AGECAT

(highestP-value 0.2219)

Step 3: Drop FLEX

(highestP-value 0.1207)

Step 4: Keep WEIGHT or

PATELLAR

(highestP-value 0.0563)

Reduced Model After BW
Elimination
Parameter DF Estimate Std
Err

Wald

ChiSq

Pr>
ChiSq
Intercept 1 3.1790 0.3553 80.0692<.0001
WEIGHT 1 1.7743 0.3781 22.0214 <.0001
PATELLAR 1 0.6504 0.3407 3.6437 0.0563

Association of Predicted Probabil-
ities and Observed Responses
Percent Concordant 61.4 Somers’ D 0.463
Percent Discordant 15.2 Gamma 0.604
Percent Tied 23.4 Tau-a 0.105
Pairs 14,214 c 0.731

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.1 0.2 0.3 0.4 0.5 0.6

1 – specificity

Sensitivity

AUC = 0.731

0.7 0.8 0.90.10

ROC Plot for the Reduced Model

Reduced model (2Xs) Full model (5Xs)

22 ¼4 covariate
patterns

25 ¼32 covariate
patterns
4 cut-pts  28 cut-pts

AUCReduced¼0.731  AUCFull¼0.745

In general:

Model 1 isnestedwithin Model 2

+

AUCModel 1 AUCModel 2

Presentation: V. Example from Study on Screening for Knee Fracture 369