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

II. The Alternating
Logistic Regressions
Algorithm

Modeling associations:

GEE approach ALR approach

correlationsðrsÞ odds ratiosðORsÞ

ORijk¼
PðYij¼ 1 ;Yik¼ 1 ÞPðYij¼ 0 ;Yik¼ 0 Þ
PðYij¼ 1 ;Yik¼ 0 ÞPðYij¼ 0 ;Yik¼ 1 Þ

GEE:as andbs estimated by alter-
nately updating estimates until
convergence

ALR:as andbs estimated similarly

BUT

ALR:aare log ORs

ðGEE:aarersÞ

ALR ORjk> 1 ,GEErjk> 1

ALR ORjk< 1 ,GEErjk< 1

Same OR can correspond to differ-
entrs

ORjk

ra

jk

rb

jk

The alternating logistic regressions (ALR) algo-

rithm is an analytic approach that can be used

to model correlated data with dichotomous out-

comes (Carey et al., 1993; Lipsitz et al., 1991).

This approach is very similar to that of GEE.

What distinguishes the two approaches is that

with the GEE approach, associations between

pairs of outcome measures are modeled with

correlations, whereas with ALR, they are mod-

eled with odds ratios. The odds ratio (ORijk)

between thejth andkth responses for theith

subject can be expressed as shown on the left.

Recall that in a GEE model, the correlation

parameters (a) are estimated using estimates

of the regression parameters (b). The regres-

sion parameter estimates are, in turn, updated

using estimates of the correlation parameters.

The computational process alternately updates

the estimates of the alphas and then the betas

until convergence is achieved.

The ALR approach works in a similar manner,

except that the alpha parameters are log odds

ratio parameters rather than correlation para-

meters. Moreover, for the same data, an odds

ratio between thejth andkth responses that is

greater than 1 using an ALR model corre-

sponds to a positive correlation between the

jth andkth responses using a GEE model. Sim-

ilarly, an odds ratio less than 1 using an ALR

model corresponds to a negative correlation

between responses.

However, the correspondence is not one-to-

one, and examples can be constructed in

which the same odds ratio corresponds to dif-

ferent correlations (see Practice Exercises 1–3).

Presentation: II. The Alternating Logistic Regressions Algorithm 571