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

Typical model for random Y:

 Fixed component (fixed effects)

 Random component (error)

Random effects model:

 Fixed component (fixed effects)

 Random components (random
effects)

1. Random effects:b
   VarðbÞ¼G


2. Residual variation:«
   Varð«Þ¼R

Random components layered:

random

effects

residual

variation

Yij =

1

1 +exp –b 0 +Σ bhXhij +boi

+eij

h=1

p

The modeling of any response variable typi-

cally contains a fixed and random component.

The random component, often called the error

term, accounts for the variation in the response

variables that the fixed predictors fail to explain.

A model containing a random effect adds

another layer to the random part of the

model. With a random effects model, there are

at least two random components in the model:

1. The first random component is the
   variation explained by the random effects.
   For the heartburn data set, the random
   effect is designed to account for random
   subject-to-subject variation
   (heterogeneity). The variance–covariance
   matrix of this random component (b)is
   called theGmatrix.


2. The second random component is the
   residual error variation. This is the
   variation unexplained by the rest of the
   model (i.e., unexplained by fixed or
   random effects). For a given subject, this is
   the difference of the observed and expected
   response. The variance–covariance matrix
   of this random component is called theR
   matrix.

For mixed logistic models, the layering of these

random components is tricky. This layering

can be illustrated by presenting the model

(see left side) in terms of the random effect

for theith subject (b 0 i) and the residual varia-

tion (eij) for thejth response of theith subject

(Yij).

582 16. Other Approaches for Analysis of Correlated Data