  Biased sampling designs can be highly efficient when studying rare (binary)
or low variability (continuous) endpoints. We consider longitudinal data
settings in which the probability of being sampled depends on a repeatedly
measured response through an outcome-related, auxiliary variable. Such
auxiliary variable- or outcome-dependent sampling improves observed response
and possibly exposure variability over random sampling, {even though} the
auxiliary variable is not of scientific interest. {For analysis,} we propose a
generalized linear model based approach using a sequence of two offsetted
regressions. The first estimates the relationship of the auxiliary variable to
response and covariate data using an offsetted logistic regression model. The
offset hinges on the (assumed) known ratio of sampling probabilities for
different values of the auxiliary variable. Results from the auxiliary model
are used to estimate observation-specific probabilities of being sampled
conditional on the response and covariates, and these probabilities are then
used to account for bias in the second, target population model. We provide
asymptotic standard errors accounting for uncertainty in the estimation of the
auxiliary model, and perform simulation studies demonstrating substantial bias
reduction, correct coverage probability, and improved design efficiency over
simple random sampling designs. We illustrate the approaches with two examples.
