---
title: "Conditional Logistic Regression Model"
author: "Stefanie Muff (stefanie.muff@uzh.ch), Johannes Signer, John Fieberg"
date: "July 16th 2018"
output: rmarkdown::pdf_document
vignette: >
  %\VignetteIndexEntry{Conditional Logistic Regression Model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
set.seed(123)
library(INLA)
inla.setOption(num.threads="1:1")
inla.setOption(smtp="taucs")
if (file.exists("myinit.R")) source("myinit.R")
library(knitr)
library(rmarkdown)
knitr::opts_chunk$set(echo=TRUE, cache=FALSE, message=FALSE,warning=FALSE)
knitr::opts_chunk$set(fig.path="figures/conditional-logit/")
```

## 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.


```{r formula, include=TRUE,eval=F}
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_call, include=TRUE,eval=F}
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.

