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

EXAMPLE

Covariate

pattern

Obs.

risk

Pred.

risk

X 1 :E¼1,

V¼ 1

^p 1 ¼ 0 : 6 P^ðX 1 Þ¼ 0 : 6

X 2 :E¼0,

V¼ 1

P^ 2 ¼ 0 : 4 P^ðX 2 Þ¼ 0 : 4

X 3 :E¼1,

V¼ 0

^p 3 ¼ 0 : 3 P^ðX 3 Þ¼ 0 : 3

X 4 :E¼0,

V¼ 0

^p 4 ¼ 0 : 7 P^ðX 4 Þ¼ 0 : 7

DevSSðb^Þ

¼ 2 ð 10 Þ½ 0 : 6 lnð 0 : 6 = 0 : 4 Þþlnð 0 : 4 Þ

þ 0 : 4 lnð 0 : 4 = 0 : 6 Þþlnð 0 : 6 Þ

þ 0 : 3 lnð 0 : 3 = 0 : 7 Þþlnð 0 : 7 Þ

þ 0 : 7 lnð 0 : 7 = 0 : 3 Þþlnð 0 : 3 ފ

¼ 51 : 3552

DevETð^bÞ¼ 0 : 06 ¼DevSSðb^Þ¼ 51 : 3552

DevETðb^Þ¼ 0 : 0 because

 2 lnL^ET saturated¼ 2 lnL^Model 3

¼ 2 lnL^C

so

DevETðb^Þ¼ 2 lnL^C

ð 2 lnL^ET saturatedÞ

¼ 2 lnL^Model 3

ð 2 lnL^Model 3 Þ

¼ 0 : 0

DevSSð^bÞ 6 ¼ 0 : 0 because

 2 lnL^SS saturated¼ 0 : 0

so

DevSSðb^Þ¼ 2 lnL^C

ð 2 lnL^SS saturatedÞ

¼ 2 lnL^Model 3  0

¼ 51 : 3552

Note: 2 lnL^C;ET 6 ¼ 2 lnL^C;SS

–2 ln LˆC,ET = –2 ln LˆC,SS −2K, where

ˆ

∑

G

g=1 dg!(ng – dg)!

ng!

K = ln

Model 3: –2 ln LC,ET = 10.8168

K does not involve β, so

ˆβ is same for SS or ET

 2 lnL^C;SS¼ 51 : 3552 ;K¼ 20 : 2692

To compute this formula for Model 3, we need

to provide values of^PðXiÞfor each of the 40

subjects in the data set. Nevertheless, this cal-

culation can be simplified since there are only

four distinct values ofP^ðXiÞover all 40 sub-

jects. These correspond to the four covariate

patterns of Model 3.

The calculation now shown at the left, where

we have substituted each of the four distinct

values of^PðXiÞ10 times in the above formula.

We have thus seen that the events–trial and

subject-specific deviance values obtained for

Model 3 are numerically quite different.

The reason why DevETðb^Þis zero is because the

ET formula assumes that the (group-) saturated

model is the fully parameterized Model 3 and

the current model being considered is also

Model 3. So the values of their corresponding

log likelihood statistics ( 2 lnL^) are equal and

their difference is zero.

In contrast, the reason why DevSSðb^Þis differ-

ent from zero is because the SS formula

assumes that the saturated model is the “clas-

sical” (SS) saturated model that perfectly pre-

dicts each subject’s 0 or 1 outcome. As

mentioned earlier,  2 lnL^is always zero for

the SS saturated model. Thus, DevSSðb^Þsimpli-

fies to  2 lnL^C for the current model (i.e.,

Model 3), whose value is 51.3552.

Mathematically, the formula for 2 lnL^Cdiffers

with the (ET or SS) formatbeing used to specify

the data. However, these formulae differ by a

constantK, as we show on the left and illustrate

for Model 3. The formula forK, however, does

not involveb. Thus, the ML estimateb^will

be the same for either format. Consequently,

some computer packages (e.g., SAS) present

the same value (i.e.,  2 lnL^C;SS) regardless of

the data layout used.

316 9. Assessing Goodness of Fit for Logistic Regression