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

EXAMPLE

Edited Output (Model EC2):

(Variables – CAT, AGE, ECG, AGE×ECG, CAT×AGE,

AGE×ECG, and CAT×AGE×ECG)

0.0000 4 1.0000

0.0000 4

0.0000 4

• 2 Log L 417.226

1

2 0.12295 0.12295

3

4 0.15625 0.15625

5 .012500 0.12500

6

7 0.17647 0.17647

8

1 274 17 17.00 257 257.00

2 59 7 7.00 52 52.00

3 122 15 15.00 107 107.00

4 57 9 9.00 48 48.00

5 39 9 9.00 30 30.00

644.00 58 14 14.00 44

0.06205 0.06205

0.11864 0.11864

0.23079 0.23079

0.24138 0.24138

We now show edited ouput for the fully para-

meterized Model EC2.

As with Model EC1, Model EC2 has eight covar-

iate patterns, and onlyQ¼6 percentile groups

are obtained in the table of observed and

expected cases and noncases. However, since

Model EC2 is fully parameterized (kþ 1 ¼ 8 ¼

G), corresponding observed and expected cases

and noncases are identical throughout the table.

Consequently, the HL test statistic is zero, as

are both the Deviance and Pearson statistics.

The log likelihood statistic of 417.226 is equiv-

alent to the SS deviance (i.e., DevSSðb^Þ) for

Model EC2. Since this deviance value is differ-

ent from 0, we know that Model EC2 is not the

(SS) saturated model that perfectly predicts

the 0 or 1 outcome for each of the 609 subjects

in the dataset.

EXAMPLE (continued)

SinceG¼ 8 <<n¼ 609 ,

Pearson statistic and Dev statistic

approxw^2 underH 0

 2 lnL^C;SS¼ 418 : 181

¼DevSSðb^Þfor Model EC1

418,181

417,226

DevET(βˆ) = 0.9544

= –2 ln LˆEC1,SS

–(–2 ln LˆEC2,SS)

Table of Probabilitiesðphatvs:pÞ

+

No perfect group prediction

e.g., group 3: phat¼0.09497

p¼0.11864

ThePearson statisticis another GOF statistic

that is similar to the Deviance in that it is not

recommended when the number of covariate

patterns is close to the sample size. However,

since the number of covariate patterns (G¼8)

for Model EC1 is much less than the sample

size (n¼609), both statistics can be assumed

to be approximately chi square under H 0.

Notice also that the Pearson and Deviance

values are very close to the HL value (0.9474).

The2 logLvalue (418.181) in the output is

the SS statistic  2 lnL^C;SS, where C is Model

EC1. This statistic is equivalent to DevSSðb^Þfor

Model EC1, sinceL^max;SSis always one.

The Deviance in the output (0.9544) is computed

using the DevETðb^Þformula based on the ET data

layout. This formula is also equivalent to the

difference between SS log-likelihood statistics

for Model EC1 (418.181) and the (fully parame-

terized) Model EC2 (417.226, in output below).

Also, from the table of probabilities (above left),

the observed and predicted probabilities are dif-

ferent, so Model EC1 does not provide perfect

group (i.e., covariate pattern) prediction.

322 9. Assessing Goodness of Fit for Logistic Regression