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

Interaction: variance calculation
difficult

No interaction: variance directly
from printout

var^l



¼var ~b^iðÞX 1 iX 0 i

hi

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
linear sum
^biare correlated for differenti

Must use var ^bi



and cov b^i;^bj



The difficult part in computing the confidence

interval for an odds ratio involving interaction

effects is the calculation for the estimated vari-

ance or corresponding square root, the stan-

dard error. When there isno interaction,so

that the parameter of interest is a single regres-

sion coefficient, this variance is obtained

directly from the variance–covariance output

or from the listing of estimated coefficients

and corresponding standard errors.

However, when the odds ratio involvesinterac-

tioneffects, the estimated variance considers a

linear sum of estimated regression coefficients.

The difficulty here is that, because the coeffi-

cients in the linear sum are estimated from the

same data set, these coefficients are correlated

with one another. Consequently, the calcula-

tion of the estimated variance must consider

both the variances and the covariances of the

estimated coefficients, which makes computa-

tions somewhat cumbersome.

Returning to the interaction example, recall

that the confidence interval formula is given

by exponentiating the quantityl^plus or minus

aZpercentage point times the square root of

the estimated variance ofl^, where^lis given by

^b 3 plus^b 4 timesX 1 plus^b 5 timesX 2.

It can be shown that the estimated variance of

this linear function is given by the formula

shown here.

The estimated variances and covariances in

this formula are obtained from the estimated

variance–covariance matrix provided by the

computer output. However, the calculation of

bothl^and the estimated variance ofl^requires

additional specification of values for the effect

modifiers in the model, which in this case are

X 1 andX 2.

EXAMPLE (model 3)

exp^lZ 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvar^l

q 

;

wherel^¼^b 3 þ^b 4 X 1 þ^b 5 X 2

vard^l



¼dvar^b 3



þðÞX 12 dvar^b 4



þðÞX 22 dvar^b 5



þ 2 X 1 dcov^b 3 ;^b 4



þ 2 X 2 dcov^b 3 ;^b 5



þ 2 X 1 X 2 dcov^b 4 ;^b 5



var(bi) and covb^i;^bj



obtained from

printout BUT must specifyX 1 andX 2

144 5. Statistical Inferences Using Maximum Likelihood Techniques