Author: Stefanie Muff (stefanie.muff@uzh.ch), Johannes Signer, John Fieberg

# 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.