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

For the specific interaction model (B) we have

been considering, the variance of^lis given by

the formula shown here.

In computing this variance, there is an issue

concerning round-off error. Computer packages

typically maintain 16 decimal places for calcu-

lations, with final answers rounded to a set

number of decimals (e.g., 4) on the printout.

The variance results we show here were

obtained with such a program (see Computer

Appendix) rather than using the rounded values

from the variance–covariance matrix presented

at left.

For this model,W 1 is CHL andW 2 is HPT.

As an example of a confidence interval calcula-

tion, we consider the values CHL equal to 220

and HPT equal to 1. Substituting^b;^d 1 , and^d 2

into the formula for^l, we obtain the estimatel^

equals 0.1960.

The corresponding estimated variance is

obtained by substituting into the above vari-

ance formula the estimated variances and co-

variances from the variance–covariance matrix

Vb. The resulting estimate of the variance of^lis

equal to 0.2279. The numerical values used in

this calculation are shown at left.

We can combine the estimates of^land its vari-

ance to obtain the 95% confidence interval.

This is given by exponentiating the quantity

0.1960 plus or minus 1.96 times the square

root of 0.2279. The resulting confidence limits

are 0.48 for the lower limit and 3.10 for the

upper limit.

The 95% confidence intervals obtained for

other combinations of CHL and HPT are

shown here. For example, when CHL equals

200 and HPT equals 1, the confidence limits

are 0.10 and 0.91. When CHL equals 240 and

HPT equals 1, the limits are 1.62 and 14.52.

EXAMPLE

dvarl^



¼dvar ^b



þðÞW 12 vard^d 1



þðÞW 22 vard^d 2



þ 2 W 1 Covd ^b;^d 1



þ 2 W 2 Covd ^b;^d 2



þ 2 W 1 W 2 Covd ^d 1 ;^d 2



W 1 ¼CHL,W 2 ¼HPT

CHL¼220, HPT¼1:

^l¼^bþ^d 1 ðÞþ 220 ^d 2 ðÞ 1

¼ 0 : 1960

dvarl^



¼ 0 : 2279

Vb¼

dvar^b

dcov^b;^d 1



dvar^d 1

dcov^b;^d 2



dcov^d 1 ;^d 2



dvar^d 2

2

(^66)
4
3
(^77)
5
¼
9 : 6389
 0 :0437 0: 0002
 0 : 0049  0 :0016 0: 5516
2
4
3
5
95% CI for adjusted OR:
CI limits: (0.48, 3.10)
exp[0.1960±1.96 0.2279]
HPT¼ 0 HPT¼ 1
CHL¼ 200
OR : 3.16d
CI : (0.89, 11.03)
dOR : 0.31
CI : (0.10, 0.91)
CHL¼ 220
dOR : 12.61
CI : (3.65, 42.94)
dOR : 1.22
CI : (0.48, 3.10)
CHL¼ 240
dOR : 50.33
CI : (11.79, 212.23)
dOR : 4.89
CI : (1.62, 14.52)
152 5. Statistical Inferences Using Maximum Likelihood Techniques