Conditional Logistic Regression Model
Stefanie Muff (stefanie.muff@uzh.ch), Johannes Signer, John Fieberg
July 16th 2018
Parametrization
Some binomial sampling schemes in Biostatistics or Biology may result in what is called matched case-control
data, which require a conditional logistic regression model. For the j th observed binary response ynj instratum n, the model is given asexp(ηnj )Prob(ynj = 1 | ηn· ) = pnj = P,i exp(ηni )ynj ∼ Bern(pnj ) ,with linear predictor ηnj and success probability pnj . The sum in the denominator is over all observations in
the respective stratum. This model is a special case of a multinomial model, and as such it can be fitted byusing a likelihood-equivalent reformulation as a Poisson modelE(ynj ) = µnj = exp(αn + ηnj ) ,ynj ∼ Po(µnj ) ,with stratum-specific intercepts αn . If the number of strata is large, the explicit estimation of these intercepts
can be circumvented by αn ∼ N(0, τα ) and fixing the precision τα at a very small value, e.g. 10−6 or 10−12 ,which corresponds to a large variance. This mimicks a uniform distribution and ensures that the αn can be
estimated freely instead of being shrunken towards 0.

Hyperparameters
None.

Specification• family =Poisson• To fix the variance at a large value the stratum-specific intercept αn use– model="iid"– hyper=list(theta = list(initial=log(1e-6),fixed=T))

ExampleThe following example stems from a habitat selection study of 6 radio collared fishers (Pekania pennanti)(LaPoint et al. 2013), and was adapted from Signer et al. (2018). Outcomes with y = 1 represent locations
that were visited by fishers, and y = 0 represents nearby locations that were not visited. Each visited location
was matched to 2 nearby available locations, and together these 3 observations form a stratum (indicatedby stratum). By design, only exactly one location can be visited in each stratum, thus these data need to
be analyzed by a conditional logistic regression model. Covariates include sex (sex), land use (landuse,categorical covariate) and distance to the center of the habitat (dist_center), with individual-dependent
random slopes for dist_cent. The 6 individuals are represented using id and id1. Shown is a reduced
dataset with only 100 steps per individual and a sampling ratio of 1:2.

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fisher.dat <- readRDS(system.file("demodata/data_fisher2.rds", package= "INLA"))fisher.dat$id1 <- fisher.dat$id
fisher.dat$dist_cent <- scale(fisher.dat$dist_cent)formula.inla <- y ~ sex + landuse + dist_cent +f(stratum,model="iid",hyper=list(theta = list(initial=log(1e-6),fixed=T))) +f(id1,dist_cent, model="iid")
r.inla <- inla(formula.inla, family ="Poisson", data=fisher.dat)

References
Muff, S., Signer, J. and Fieberg, J. (preprint) Accounting for individual-specific variation in habitat selection
studies: Efficient estimation using integrated nested Laplace approximations
Signer, J., Fieberg, J. and Avgar, T. In press. Animal Movement Tools (amt): R-Package for Managing
Tracking Data and Conducting Habitat Selection Analyses. Ecology and Evolution.
LaPoint, S., Gallery, P., Wikelski, M. and Kays, R. (2013) Animal behavior, cost-based corridor models, andreal corridors. Landscape Ecology, 28, 1615–1630.

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