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there is a significant body of literature in statistics , econometrics and ecology dealing with the modeling of discrete responses under biased or preferential sampling designs .
they are particularly popular in the natural sciences when species distributions are studied .
such sample schemes may reduce the survey cost especially when one of the responses is rare .
a large part of statistical literature concerns the case - control design , retrospective , choice - based or response - based sampling @xcite . in the simplest case a sample of cases and a sample of controls are available and for each observation a set of `` attributes / covariates '' is observed in both samples .
then inference is carried out following standard statistical procedures @xcite .
+ a case that has received increasing attention in the literature is the situation where the sample of controls is a random sample from the whole population with information only on the attributes and not on the response @xcite .
this situation is fairly common in ecological studies where only species presence is recorded when field surveys are carried out . in the ecological literature , since the 1990 s such data are called _ presence only data _ ( see * ? ? ?
* and references therein ) .
@xcite define presence - only data as `` consisting only of observations of the organism but with no reliable data where the species was not found '' .
atlases , museum and herbarium records , species lists , incidental observation databases and radio - tracking studies are examples of such data .
+ in recent years we find a considerably growing literature describing approaches to the modeling of this type of data , among the many ecological papers we recall @xcite , @xcite , @xcite , @xcite , @xcite and , most notably , in the statistical literature @xcite , @xcite , @xcite , @xcite and @xcite . while in @xcite and @xcite to model the presence - only data poisson point processes are considered in the likelihood and bayesian framework respectively , in @xcite and @xcite a modified case - control logistic model is adopted in the likelihood and bayesian perspective respectively , in both papers there is no account for possible dependence structure in the observations . in @xcite
the asymptotic relations between the two approaches are discussed .
+ a different approach , maxent , is based on the maximum entropy principle @xcite . in maxent
@xcite the relative entropy between the distribution of covariates at locations where the species is present and the unconditional background distribution of covariates is maximized subject to some constrains concerning empirical statistics ( see philips et al . , 2006 , for details ) . as pointed by @xcite
`` the maxent method requires knowledge of species prevalence for its estimator of occurrence to be consistent '' .
+ in what follows we are going to use the name _ presence - only data _ when referring to the above sketched general problem of having information on the presence and covariates jointly on a sample from a population , while information on only the covariates is available on any sample from the same population .
this work is developed in the same discrete setting as in @xcite and @xcite , i.e. , we have a population of independent units , no dependence structure , such as spatial correlation , is anticipated .
we defer the treatment of this extension to a subsequent work .
+ the main contribution of the paper is a new rigorous formalization of the logistic regression model with presence - only data that allows further insight into the inferential issues .
this leads us to an algorithmic procedure that , among other results , returns a mcmc approximation of the response prevalence under general knowledge of the process generating the data .
we also present a large simulation study involving 24,000 simulated datasets and comparing our approach to other two models representing optimal benchmarks .
+ the paper is organized as follows .
section [ sec : reglog ] introduces a general framework for the presence - only data problems , section [ sec : bayes ] presents our bayesian approach , section [ sec : mcmc ] describes the mcmc algorithm while results related to the simulation study are reported in section [ sec : sim ] .
finally in section [ sec : conc ] some conclusions are drawn and future developments briefly described .
the analysis of a binary response related to a set of explicative covariates is usually carried out through the use of the logistic regression where the logit of the conditional probability of occurrence is modeled as a function of covariates . in this section ,
we first introduce a general framework for the modeling of presence - only data and then consider the case of the linear logistic regression .
the approach proposed is built on two levels and we partially follow the formulation introduced by @xcite but adopting a bayesian scheme as in @xcite .
let @xmath0 be a binary variable informing on the presence ( @xmath1 ) or absence ( @xmath2 ) of a population s attribute and let @xmath3 denote a set of highly informative , on the same attribute , covariates which are available on the same population .
then , the presence - only problem can be formalized by considering a censorship mechanism that acts when observing the response @xmath0 , so that part of the population units are not reachable . in particular , we refer to the situation in which we are able to detect only a partial set of units on which the attribute of interest is present while the information on the covariates @xmath4 is available on the entire population . in this situation
, we have to consider two types of uncertainty : the uncertainty due to the mechanism of censorship and the uncertainty due to the sampling procedure .
moreover , since we are not able to collect a random sample of observable data , we need to adjust for the sampling mechanism through the use of a case - control scheme @xcite .
+ in order to build a bayesian model , in this framework we adopt the following conceptual scheme in two levels .
+          let @xmath9 be the subset of @xmath5 where we observe @xmath1 .
a modified case - control design is applied so that a sample of presences , considered as cases , is selected from @xmath9 and a sample of `` contaminated '' controls @xcite is selected from the whole population @xmath5 , with all the covariates but no information on @xmath0 .      here
, we can not approach the model construction using only a finite population approach @xcite because of the censoring mechanism that `` masks '' distributional information on @xmath0 already at the population level . by the introduction of level 1 we can describe the censored observations as random quantities generated by the model @xmath8 .
hence , the problem of presence - only data can be formalized as a problem of missing data @xcite .      at the first level , we assume that the law @xmath8 is defined in terms of the conditional probability of occurrence @xmath10 , denoted by @xmath11 , when the covariates are @xmath12 .
moreover , we consider that the relation between @xmath0 and @xmath4 is formalized through a regression function @xmath13 on the logit scale @xmath14 that is @xmath15 when the data @xmath7 are independently generated from @xmath8 , we denote by @xmath16 the empirical prevalence of the binary response @xmath0 in @xmath5 , expressed as the ratio of the number of presences @xmath17 to the size of the population , that is @xmath18      at the second level , we adopt a case - control design modified for presence - only data @xcite in order to account for the specific sampling procedure considered .
the use of the case - control scheme is necessary at all times when it is appropriate to select observations in fixed proportions with respect to the values of the response variable .
this can occur when the attribute of interest represents a phenomenon that is rare among the units of the population as for example a rare disease or a rare exposure in epidemiological studies @xcite .
+ now , let @xmath19 be a binary indicator of inclusion into the sample ( @xmath20 denotes that a unit is in the sample ) , let @xmath21 and @xmath22 be the inclusion probability of the absences and the presences , respectively . under the assumption that , given @xmath0 , the sampling mechanism is independent from the covariates @xmath4 , the conditional probability of occurrence is modified through the bayes rule as @xmath23 hence , the corresponding case - control regression function @xmath24 defined as the logit of is given by @xmath25 in particular , if the selection of cases ( @xmath26 ) and controls ( @xmath27 ) is made independently without replacement , the inclusion probabilities are given in terms of the empirical prevalence @xmath16 by @xmath28 and @xmath29 so that the equation becomes @xmath30 in our framework , since the response variable @xmath0 is already censored at the population level , the standard case - control design can not be adopted but it should be modified in such a way that a sample of presences is matched with an independent sample drawn from the entire population , named the _ background sample _ @xcite . remark that in this sample the response variable is unobserved and only the covariates are available .
+ in this way , the complete sample @xmath31 is composed by a set @xmath32 of @xmath33 independent background data , where the response @xmath0 is not observed , drawn from the entire @xmath5 and by a set @xmath34 of @xmath35 independent observations selected from the sub - population of presences @xmath9 .
this procedure implies that the reference population @xmath5 is augmented with its subset @xmath9 so that the total number of observations considered in the sampling scheme becomes @xmath36 . to illustrate the sampling framework we are going to adopt here ,
let us consider the following situation : we can label population units of type @xmath37 only when they are isolated from units of type @xmath38 .
this can be formalized by introducing a binary stratum variable @xmath39 such that @xmath40 indicates when an observation is drawn from the entire population @xmath5 while @xmath41 denotes the sampling from the sub - population @xmath9 .
remark that @xmath41 implies @xmath1 whilst @xmath40 implies that @xmath0 is an unknown value @xmath42 .
moreover , by construction @xmath39 is independent from the covariates @xmath4 , given the response @xmath0 .
the introduction of the variable @xmath39 allows us to define the structure of the data at the population level and at the sample level in terms of presences / absences ( @xmath0 ) and known / unknown data ( @xmath39 ) , as reported in table [ tab1 ] and table [ tab2 ] .
di  lorenzo , b. , farcomeni , a. , and golini , n. ( 2011 ) . a bayesian model for presence - only semicontinuous data with application to prediction of abundance of taxus baccata in two italian regions .
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novel methods improve prediction of species distribution from occurence data . , * 29 * , 129151 .
presence - only data are referred to situations in which , given a censoring mechanism , a binary response can be observed only with respect to on outcome , usually called _ 
 presence_. in this work we present a bayesian approach to the problem of presence - only data based on a two levels scheme . a probability law and a case - control design are combined to handle the double source of uncertainty : one due to the censoring and one due to the sampling . 
 we propose a new formalization for the logistic model with presence - only data that allows further insight into inferential issues related to the model . 
 we concentrate on the case of the linear logistic regression and , in order to make inference on the parameters of interest , we present a markov chain monte carlo algorithm with data augmentation that does not require the a priori knowledge of the population prevalence . a simulation study concerning 24,000 simulated datasets related to different scenarios 
 is presented comparing our proposal to optimal benchmarks .    
 [ [ keywords ] ] keywords : + + + + + + + + +    bayesian modeling , case - control design , data augmentation , logistic regression , markov chain monte carlo , population prevalence , presence - only data , simulation .