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 $j^\text{th}$ observed binary response $y_{nj}$ in stratum $n$, the model is given as $$\mathsf{Prob}(y_{nj}=1 \,|\, \eta_{n\cdot}) = p_{nj} = \frac{\exp(\eta_{nj})}{\sum_i \exp( \eta_{ni})} \ , \quad y_{nj} \sim \mathsf{Bern}(p_{nj}) \ ,$$ with linear predictor $\eta_{nj}$ and success probability $p_{nj}$. 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 by using a likelihood-equivalent reformulation as a Poisson model $$\mathsf{E}(y_{nj}) = \mu_{nj} = \exp(\alpha_{n} + \eta_{nj}) \ , \quad \quad y_{nj} \sim \mathsf{Po}(\mu_{nj}) \ ,$$ with stratum-specific intercepts $\alpha_{n}$. If the number of strata is large, the explicit estimation of these intercepts can be circumvented by $\alpha_{n} \sim \mathsf{N}(0,\tau_\alpha)$ and fixing the precision $\tau_\alpha$ 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 $\alpha_{n}$ 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 $\alpha_{n}$ use
  • model="iid"
  • hyper=list(theta = list(initial=log(1e-6),fixed=T))

Example

The 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 (indicated by 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.

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, and real corridors. Landscape Ecology, 28, 1615–1630.