Nathalie ; Allain
email: nathalie.leclanche@hotmail.fr

PharmD Cyril Leven

MD, PhD Bruno Falissard

J.-S Allain

MD Jean-Marie Batail

PharmD Elisabeth Polard

F Montastruc

MD, PhD Dominique Drapier

MD, PhD Florian Naudet

MD Jean-Sébastien Allain

Manic switches induced by antidepressants: an umbrella review comparing randomized controlled trials and observational studies

Keywords: Meta analysis, Randomized Controlled Trial, Affective disorders, Depression

published or not. The documents may come

Significant outcomes

We found a difference in reported manic switch rates under antidepressants between Randomised Controlled Trials (RCTs) and observational studies (OS), with an underestimation in RCTs (or an overestimation in OS).

Overestimated prevalence of manic switch was related to imipraminics, to Serotonin-Norepinephrine Reuptake Inhibitors and to other classes of drugs, in comparison with placebo treatment.

The prevalence of manic switch was lower among adults than children and higher in case of bipolar disorder.

Limitations

We cannot conclude formally as to whether there is an underestimation in RCTs or an overestimation in OSs.

We disrupted the randomisation with our meta-regression model, so that caution is required for comparisons between treatments, and any differences found can, in part, be explained by an indication bias.

Ethical statement

This article does not include any studies involving human participants or animals performed by any of the authors.

INTRODUCTION

Recently, there has been a debate about psychiatric drugs, which have been accused of doing more harm than good in many situations [START_REF] Gøtzsche | Does long term use of psychiatric drugs cause more harm than good?[END_REF]. While these drugs have gone through strict regulatory controls, their risk-benefit balance is nevertheless still a subject of controversy, particularly in terms of Adverse Drug Reactions (ADRs) over the long term [START_REF] Gøtzsche | Why I think antidepressants cause more harm than good[END_REF][START_REF] Nutt | Attacks on antidepressants: signs of deep-seated stigma?[END_REF][START_REF] Naudet | Antipsychiatry and the antidepressants debate[END_REF]. This state-of-the-art fundamentally questions the evaluation process of the risk-benefit balance of these drugs, which is usually assessed, in a first step, using Randomised Controlled Trials (RCTs). However, it is well known that these studies are underpowered to detect rare adverse events when they are less frequent than the occurrence of the primary outcome of the study (generally response) (5; 6). Less is known on whether RCTs might also underestimate the prevalence of certain ADRs [START_REF] Tang | Comparison of serious adverse events posted at ClinicalTrials.gov and published in corresponding journal articles[END_REF] because of 1/ an overselection of "supranormal" and healthy participants and 2/ a limited follow-up duration. Observational studies are thus thought to be complementary, in a second step, to explore the prevalence of ADRs in a real-life setting.

We chose to explore this issue by focusing on antidepressants, the effectiveness of which is still controversial, and on prevalence of manic switch as an outcome. In the management of antidepressant pharmacotherapy, preventing this ADR is an important issue for patients. Indeed, it is frequently a "serious" ADR, often resulting in hospitalisation [START_REF] Edwards | Adverse drug reactions: definitions, diagnosis, and management[END_REF]. While it is not infrequent in day-to-day clinical practice, information on its prevalence is conflicting across the literature (simple or systematic reviews and meta-analyses) with no consensus and even very heterogeneous estimations. In fact, causality assessments ("imputability" or "imputation") for this ADR are often controversial. Clinical judgment, history of mood disorder, family psychiatric history, the existence of a hyperthymic temperament, time-lapse after a prescription of antidepressant and evolution could all explain such discrepancies. All these characteristics might be very different across RCTs and observational studies. We thus designed an umbrella review on aggregated data to compare manic switch prevalence between RCTs and observational studies and to reconcile the apparently divergent conclusions observed in the literature by modeling the prevalence of manic switch according to the different study designs, in a meta-epidemiological perspective.

METHOD

The methods of this umbrella review of aggregated data and the inclusion criteria were specified in advance and documented in a protocol (Systematic Reviews Registration -PROSPERO 2013:CRD42013004383).

Eligibility criteria

We aimed to explore discrepancies in prevalence of manic switch reported in simple, systematic reviews and meta-analyses, so these studies were searched for in a first step. In a second step, references in these studies that met the following selection criteria were extracted.

Types of participant

We reviewed all studies involving children, adolescents and adults with a diagnosis of unipolar or bipolar depression (according to international classifications such a DSM-IV or according to the severity of symptoms measured with depression scales), and treated with antidepressants. Studies involving patients with other psychiatric or medical comorbidities were considered.

Types of intervention

We focused our attention on antidepressant treatment (Selective Serotonin Reuptake Inhibitor (SSRI), Serotonin-Norepinephrine Reuptake Inhibitors (SNRI), Imipraminics (IP), other classes of antidepressant, all classes).

Where appropriate, in RCTs, we also extracted data about placebo arms. Patients treated with MAOIs (monoamine oxidase inhibitors) were excluded from this review, because these drugs are currently rarely used in clinical practice [START_REF] Thomas | Combination therapy with monoamine oxidase inhibitors and other antidepressants or stimulants: strategies for the management of treatment-resistant depression[END_REF].

Types of outcome

The primary outcome was the rate of manic switch reported in each arm of each study.

Types of study

In this review, we considered RCTs and observational cohorts (longitudinal non-randomised and non-blinded studies) that provided data about rates of manic switch after antidepressant treatment. Only study reports in English, French and Spanish were included.

Search strategy

Eligible studies were identified from PubMed/Medline, the Cochrane library, and Embase, including congress abstracts. We searched all simple reviews, systematic reviews or meta-analyses published between 1990 and 2013 to identify all trials included in these studies. We used the following algorithm: (mania OR manic OR

Study selection

Study selection was performed in two steps. In a first step, meta-analyses, simple and systematic reviews were identified in a blinded standardised manner by two reviewers (CL, FN). In a second step, all relevant references were extracted (NA, FN). A comparison across all the individual studies was performed, checking authors, treatment comparisons, sample sizes and outcomes, to avoid duplicates and compilations of data from several reports on the same study. Thus no individual study was counted more than once.

Assessment of methodological quality

Each paper was assessed for methodological quality prior to inclusion in the review, using two appropriate standardized critical appraisal instruments (10), one for RCTs and one for observational studies.

Data collection process

Data synthesis

To adjust our comparison of observational studies and randomised controlled trials on identified sources of heterogeneity, and to quantify the impact of certain variables on the manic switch rate (MSR), we performed a meta-regression. The dependent variable was MSR and the following pre-specified explanatory variables were considered: study type; type of treatment; diagnosis; year of publication; study duration; age classes; proportion of women, patient recruitment. This meta-regression was performed by using a random-coefficient Poisson model with the "study" factor specified as a random effect. Exp(betas) are reported with their 95% confidence interval. Exp(betas) can be interpreted as "adjusted rate ratios". Multiple imputation of missing data was performed using a Gibbs sampler.

To assess the robustness of our results, sensitivity analyses were performed: 1) by removing each arm in turn, and 2) using quality assessment to adjust the weight of a given study.

Analyses were performed using R [START_REF] Higgins | Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF] and the libraries meta (Schwarzer G), lme4 (Maechler D), and MICE (Van Buuren S, Groothuis-Oudshoorn K). Results are presented according to PRISMA statements (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) [START_REF] Liberati | The PRISMA statement for reporting systematic reviews and meta-analyses of studies that evaluate health care interventions: explanation and elaboration[END_REF].

Minor changes to our initial protocol

We planned to consider age and baseline severity as continuous variables, but there was too much missing data.

We replaced these variables in our model by the following proxies: age class and previous antidepressant resistance.

In response to one reviewer's comment, we conducted a post hoc analysis by adding an interaction term to the main model between study design (observational or RCT) and illness (MDD or bipolar disorder or both).

RESULTS

Study selection

The search of Medline, Cochrane and Embase databases provided a total of 1195 citations, respectively 486, 256 and 453 citations. After adjusting for duplicates, 971 remained. Of these, 950 studies were discarded because, after reviewing the abstracts, it appeared that these papers did not meet the eligibility criteria. Our first search identified 21 reviews and/or meta-analyses. In these reviews, 101 references were extracted. After reviewing full texts of these references in detail, it appeared that 44 articles did not meet the inclusion criteria as described.

Our second search step thus provided 57 studies, which were included in the quantitative review. A PRISMA chart detailing the study selection process for RCTs and observational studies is given in Figure 1.

Study characteristics and risk of bias within studies

The studies selected were 35 randomised controlled trials and 22 observational studies (see Web appendix for references) involving respectively 6244 and 76544 patients in 66 and 40 arms. A summary of study methodology and quality is given in Table 1. The description of arms is detailed in Table 2.

Results from individual studies and synthesis of results

As expected, using the Q statistic, significant heterogeneity was detected for: 1/ active treatment effect in RCTs, 2/ placebo effect in RCTs, and 3/ active treatment effect in observational studies. The Forest plot presenting individual arm results is reported in Figure 2.

The meta-regression (Table 3) shows that RCTs underestimate the rate of manic switch compared to observational studies [(exp(beta) = 0.53 (0.32 to 0.

Additional analyses

Sensitivity analyses taking quality into account showed the robustness of our estimation for study type, antidepressant class, diagnosis and age class. Sensitivity analysis removing each arm in turn found 4 studies (13-16) that could have impact estimations of coefficients concerning treatment (SNRI, other classes) and age classes (both). Results from the sensitivity analysis are presented in detail in the Web appendix.

In the post-hoc analysis exploring the interaction term between study design (observational or RCT) and the illness (MDD or bipolar disorder or both), the difference between RCT and observational studies appeared still present in MDD patients (exp(betas)=0.27 [0.09-0.76]) and the interaction term, although not significant, suggested that this difference might be less pronounced when the studies involved both bipolar disorder and MDD patients (interaction: exp(betas)=1.29 [0.18-9.19]) or only bipolar patients (interaction: exp(betas)=2.48 [0.79-7.83]).

DISCUSSION

Summary of evidence

Our results highlight a difference in rates of manic switch after antidepressants between RCTs and observational studies, with an underestimation in RCTs or an overestimation in OSs. This result is stable between the main analysis and the sensitivity analysis, demonstrating its robustness. It has also been suggested in a previous comprehensive review [START_REF] Tondo | Mania associated with antidepressant treatment: comprehensive meta-analytic review[END_REF]. The difference is not trivial; without adjustment it is nearly a 10 percent difference and, after adjustment, the exp(betas) are in the same order of magnitude as the differences estimated between antidepressants and placebo in our model. In RCTs, inclusion criteria are more restrictive than those in observational studies, which could provide a better reflection of real-life settings. For example, patients with comorbid substance abuse or dependence are often excluded from RCTs. And precisely, comorbid substance abuse or dependence, in particular alcohol, is associated with an increased risk for (hypo) mania during antidepressant use [START_REF] Goldberg | The association between substance abuse and antidepressant-induced mania in bipolar disorder: a preliminary study[END_REF]. In addition, we found that study duration was associated with an increased risk of manic switch. Patients exposed to antidepressants over a long period are at higher risk for manic switch. RCTs are shorter that observational studies and could underestimate the occurrence of manic switch.

Other study factors associated with significant variations in the rates of manic switch were evidenced with our meta-regression. In our meta-analysis treatment with SNRI, IP or other antidepressants (often bupropion, a psychostimulant) are associated with an increased risk of manic switch compared to placebo, in line with the results in the literature (19; 20). Manic switch was also more frequent with all antidepressant classes than with placebo, but this did not reach statistical significance. Manic switch was estimated to be more frequent in bipolar patients, compared to unipolar patients, providing some clear external validity for our model.

In a post-hoc analysis, we found no significant interaction between study design and illness. This might suggest that the difference between observational studies and RCTs is consistent across the different disorders.

Nevertheless, the result should be interpreted cautiously since our study might be underpowered to explore this issue.

Interestingly, we identified a decrease in the rate of manic switch for each year of study publication. This result is obtained after adjustment and takes into account the fact that earlier studies used IP and possibly SNRI, and later studies avoided SNRI, and examined the safer SSRIs. First, manic switch is an entity that is somewhat difficult to define from a nosographic perspective and this result could be related to changes in diagnostic criteria, and to the changing concept of manic switch. In older studies [START_REF] Keup | Inpatient treatment of depressive states with tofranil (imipramine hydrochloride)[END_REF], the psychomotor agitation associated with IP was liable to be labelled as a manic switch, while today the diagnostic criteria are more restrictive.

Criteria have changed over time. In the DSM III-R, antidepressant-induced agitation was considered as a real manic episode, while in the DSM IV, it is no longer a subtype of bipolar disorder, but a substance-induced mood disorder. In the present meta-analysis, no study was based on the DSM V criteria. Nonetheless, views have recently changed about manic switch. DSM V adopts the DSM III-R views and adds the following note to the diagnostic criteria for manic episodes: "A full manic episode that emerges during antidepressant treatment but at a fully syndromal level beyond the physiological effect of that treatment is sufficient evidence for a manic episode and, therefore, a bipolar I diagnosis". Hypomanic episodes have a similar note in DSM V. These notes seem to widen the concept of bipolar disorder, because patients developing mania or hypomania during antidepressant treatment can be diagnosed as bipolar I or II in DSM V, while they were diagnosed as suffering from substance-induced mood disorder in DSM-IV-TR. Since the so-called "physiological effect of that treatment" is rather vague, it is hard to say to what extent this new definition will broaden the concept of bipolar disorder by including "true" bipolar disorders but also "true" ADRs resulting directly from a given treatment.

Second, the inclusion criteria may have changed across time, resulting in populations with different risks for manic switch. In addition, in the sensitivity analysis, where quality was taken into account, the effect of publication year disappeared suggesting that this effect, and indeed the assessment of manic switch, can be linked to study quality.

We also found an effect of the age class. Adults have a lower risk of manic switch than children and adolescents, which is in agreement with clinical experience [START_REF] Martin | Age effects on antidepressant-induced manic conversion[END_REF]. Antidepressants should therefore be used with even more caution in this population.

Limitations

Our method of extracting studies from meta-analyses and reviews about manic switch does not enable an exhaustive review as commonly performed in meta-analysis. This could have led to a biased sample. But this biased sample is a straighforward reflection of the literature. Our aim was indeed to understand the literature on rates of manic switch from a meta-epidemiological perspective.

Our sensitivity analyses identified four studies with an impact on a few coefficient estimations. After careful examination, three of these studies were in fact large (RCT or observational). Their impact on our model estimates are thus to a degree to be expected. Nonetheless, the following limitations were identified: 1/ in the first observational study [START_REF] Martin | Age effects on antidepressant-induced manic conversion[END_REF], even after mailing study authors, it was not confirmed that all participants were depressed, and the proportion of subjects with depression was not clearly defined; 2/ the two other studies (14; 22) were derived from the same program (the STEP-BD program), and there may have been some overlap of subjects in the two studies (i.e. the same subject could have been studied twice). The author was contacted by email and indicated that there was probably not a great deal of overlap between the two studies. The last influential study [START_REF] Shirazi | An open trial of citalopram in children and adolescents with depression[END_REF] we identified was a small observational study in children and adolescents, and could only have affected the estimation of coefficients related to age class. Its influence on the other coefficients was not significant.

It should be kept in mind that, to a certain extent, we disrupted randomisation with our meta-regression model, so caution is required for comparisons between treatments, and any differences found can, in part, be explained by an indication bias. Nonetheless, our results are in accordance with the results of the literature (17; 20) which is a sign of external validity.

Nevertheless, the main limitation is probably not statistical but rather conceptual, and concerns our interpretation of differences between RCTs and observational studies. While our interpretation is in favor of an underestimation of MSRs in RCTs, one can argue, conversely, that there is an overestimation in observational studies, or that both phenomena are at play here. In observational studies, it may be more difficult to differentiate mania induced by antidepressants and mania arising from the natural course of bipolar disorder or anxious comorbidity. Observational studies are thought to be more representative of "real life", but bias can also contribute to an overestimation of manic switch. On the other hand, the widely discussed GSK Study 329 [START_REF] Keller | Efficacy of paroxetine in the treatment of adolescent major depression: a randomized, controlled trial[END_REF][START_REF]Canadian Medical Association or its licensors: Drug company experts advised staff to withhold data about SSRI use in children[END_REF][START_REF] Noury | Restoring Study 329: efficacy and harms of paroxetine and imipramine in treatment of major depression in adolescence[END_REF] is good example of underestimation by misreporting of ADRs in RCTs. Additionally, rare ADRs cannot be evidenced in RCTs and only pharmacoepidemiological studies are able to explore them.

Perspective

Safety is a cornerstone of drug evaluation. The development of evaluation techniques in pharmacology has been linked to safety issues [START_REF] Avorn | Two Centuries of Assessing Drug Risks[END_REF] and for instance the focus on efficacy in the 1962 amendments to the Food and Drugs Act after thalidomide aimed to improve safety. The wide adoption of RCTs was intended to promote an appropriate evaluation of the risk-benefit balance of treatments. In practice, RCTs have not proved ideal, since they do not function well for safety assessment purposes. Randomisation can introduce confounding factors wherever a treatment and an illness are liable to produce similar or fairly similar adverse events (27) as for example in the case of manic switch in bipolar participants. Secondly RCTs are underpowered to detect adverse events when these events are less frequent than the occurrence of the primary outcome. On the basis of a limited example (manic switch with antidepressants in major depressive disorder), our results suggest a difference in between RCTs and observational studies, but we can't conclude formally whether there is an underestimation in RCTs or an overestimation in OSs. On the other hand, previous work (28; 29) suggested that RCTs could overestimate the beneficial effects of antidepressants in major depressive disorder. Overall this suggests that the evidence used for drug approval could lead to a biased estimation of the risk-benefit balance of treatments such as antidepressants. While the extent of any such distortion is still unclear and very hard to explore, it is a crucial issue for meta-research. It could explain distinct points of view and disagreements between trialists and pharmacoepidemiologists, against a backdrop of ideological and/or financial conflicts of interest. Beyond these scientific aspects, this distortion can be of really practical interest for clinicians, who have to cope, in their dayto-day practice, with this area of uncertainty: when choosing a treatment for a given patient, if the clinician chooses to treat him "scientifically" he/she must accept that this is an experiment, and must be aware of what happens for the individual patient. From the patient perspective, we have indeed no better evidence than the patient's experience (30).

CONCLUSION Implications for research

In our umbrella review, we only studied one ADR, and it might be tempting to extrapolate our findings to other ADRs under antidepressants, some of which are still being debated in the literature. For example, in the context of antidepressant trials, a new meta-regression would be interesting to determine whether the rate of suicide under antidepressants is underestimated in randomised-controlled studies in major depressive disorder.

Implications for practice

It is important to take into account the results derived from different study designs and the specific characteristics of patients when assessing the risk-benefit balance for drugs. Observational studies should be used as a necessary complement to RCTs in the evaluation of the risk-benefit ratio for a given drug. Once prescribed, strict monitoring of effectiveness and ADRs for a given drug is essential. The reporting of ADRs to pharmacovigilance system should be encouraged among practitioners.

  mood) AND switch AND ((Clinical Trial[ptyp] OR Meta-Analysis[ptyp] OR Review[ptyp] OR systematic[sb] OR Clinical Trial, Phase I[ptyp] OR Clinical Trial, Phase II[ptyp] OR Clinical Trial, Phase III[ptyp] OR Comparative Study[ptyp] OR Controlled Clinical Trial[ptyp] OR Clinical Trial, Phase IV[ptyp])) AND ("1990/01/01"[PDAT] : "2013/12/31"[PDAT])). We considered every additional clinical study on manic switch identified with the previous search.

A

  data extraction sheet based on the Cochrane Handbook for Systematic Reviews of Interventions guidelines (11) was used. Three reviewers (NA, CL, JSA) performed data extractions. Disagreements were resolved by consensus or through discussion with a fourth reviewer (FN). For each arm of each study included, we extracted information on: 1/ characteristics of the study (type [RCT, observational], year, blinding, duration), 2/ characteristics of trial participants (number of patients included in the analysis, diagnosis [unipolar or bipolar or both], age classes [children/ adolescents or adults or both], proportion of women, initial severity, previous antidepressant resistance (resistance to at least two distinct classes), recruitment [inpatients or outpatients or both]), 3/ type of intervention (treatment [SSRI, SNRI, IP, other classes, all classes, placebo], duration of treatment), 4/outcome measure (rate of manic switch, type of evaluation of manic switch).

  87)]. Overestimated prevalence was attributable to IP [exp(beta) = 1.85 (1.22 to 2.79)]; to SNRI [exp(beta) = 1.74 (1.06 to 2.86)]; and to other classes [exp(beta) = 1.58 (1.08 to 2.31)], compared to placebo treatment. Regarding the patients, the prevalence of manic switch was lower in adults than in children and adolescents [exp(beta) = 0.2 (0.07 to 0.59)]; and higher [exp(beta) = 20.58 (8.41 to 50.31)] in bipolar disorder. There was a decrease in manic switch prevalence [exp(beta) = 0.97 (0.96 to 0.98)] for each year of study publication and an increase [exp(beta) = 1.01 (1 to 1.03)] for each additional week of duration.

Figure 1 Figure 2 .

 12 Figure 1. PRISMA Chart

Figure 1

 1 Figure 1. Prisma Flow-chart

Table 1 . Study characteristics

 1 Quality score is computed out of 100 points from the two Joanna Brigs Institute instruments

	Identification
	Included Screening and	eligibility

Table 3 . Meta-regression analysis. Main analysis

 3

	SSRI= Selective Serotonin Reuptake Inhibitor
	SNRI=Serotonin-Norepinephrine Reuptake Inhibitor
	IP= Imipraminics

ACKNOWLEDGEMENTS

This work was supported by a local grant from Rennes CHU (CORECT: COmité de la Recherche Clinique et Translationelle). We would like to thank Angela Swaine Verdier for revising the English.

CONTRIBUTORS

Study

Random effects model

Events


Nathalie Klement
email: nathalie.klement@ensam.eu

Nathalie Grangeon

Michel Gourgand
email: gourgand@isima.fr

Medical Imaging: Exams Planning and Resource Assignment Hybridization of a Metaheuristic and a List Algorithm

Keywords: Hybridization, Metaheuristic, List Algorithm, Medical Imaging

The presented work is about optimization of the hospital system. An existing solution is the pooling of resources within the same territory. This may involve different forms of cooperation between several hospitals. Problems of sizing, planning and scheduling may be considered. We define the problem of activities planning with resource assignment. To solve this problem, we propose a hybridization between a metaheuristic and a list algorithm. Single based metaheuristics are used. This proposition requires a new encoding inspired by permutation problems. This method is easy to apply: it combines already known methods. With the proposed hybridization, the constraints to be considered only need to be integrated into the list algorithm. For big instances, the solver used as a reference returns only lower and upper bounds. The results of our method are very promising. It is possible to adapt our method on more complex issues through integration into the list algorithm of the constraints. It would be particularly interesting to test these methods on real hospital authorities to assess their significance.

INTRODUCTION

Given the current economic situation, everything is done to move towards a better use of goods and services production systems. The hospital system also follows this trend as much or less resources are allocated to it but it should work more efficiently to meet a demand that is increasing. To do so, in 2015, the french government defined the HGT: Hospital Group of Territory, an evolution of the HCT previously presented (Gourgand et al., 2014a). It is a cooperation between public institutions, which are at different places, that implement a common strategy and jointly manage some functions and activities through delegations or skills transfer between them. Some decision support tools are needed to manage this new kind of organization.

The aim of our work is the development of a decision support tool to help to manage HGTs or any hospital cooperations. This tool should be used at different levels: strategic, tactical or operational, to deal with problems of sizing, planning, resources assignment or scheduling. It should be used to anticipate the creation of a new cooperation, to manage this organization day to day, or to react in case of hazard or crisis situation. In this paper, we take the problematic of medical imaging over a HGT as an example. Some material resources, such as X-ray, scanner, MRI, are located at different places belonging to the HGT. Human resources work there and have specific competences on these material resources. Some patients need to pass an exam on such a material resource. The planning horizon can be some days or weeks, divided in periods of half-days. Some incompatibilities and time constraints are defined. The tactical level will be discussed in this paper: the objective is to assign the exams to human and material resources during a period. The other levels can easily be solved by our method which uses some operational research purposes. doen et al., 2010) referenced more than one hundred and twenty articles about planning and scheduling of operating theater. [START_REF] Van Den Bergh | Personnel scheduling: A literature review[END_REF] made a literature review about scheduling human resources and referenced around three hundred articles.

But articles dealing with multi-place system in hospital network are scarce, or their case study are quite limited. Planning surgical vacations by specialty is dealt by [START_REF] Santibáñez | Surgical block scheduling in a system of hospitals: an application to resource and wait list management in a british columbia health authority[END_REF]. Availability of operating theater, beds capacity, surgeons preferences and waiting list are considered. The proposed model allocates specialties to operating theater, and is applied to a Canadian study case, composed by eight hospitals, over four weeks. [START_REF] Everett | A decision support simulation model for the management of an elective surgery waiting system[END_REF] developed a support aid tool to manage the waiting list of patients who need a surgical act. The system is made up by several hospitals, working in cooperation. The waiting list is common for all the hospitals. Each day, each patient is assigned to one place according to the availability of the hospitals. If no hospital is available on one day, patients are assigned the next day. Assignment of resources is not considered. [START_REF] Vanberkel | A comprehensive simulation for wait time reduction and capacity planning applied in general surgery[END_REF] developed a tool to reduce the waiting time and to plan beds capacity in a surgery service, over several places. It is a problem of allocation of the fixed resources, in an other Canadian example. This tool aims at studying a redistribution of postoperative beds between the places, using simulation tools. Problem of capacity in intensive care unit can result in the cancellation of programmed acts, an overload of medical team or a reject of urgent patients. Thus, urgent patients could be transferred to further places. A cooperative solution is studied by [START_REF] Litvak | Managing the overflow of intensive care patients[END_REF], taking into example a case in the Netherlands. Some hospitals belonging to the same territory share some beds to urgent patients.

Researchs are done about the pooling of resources. [START_REF] Trilling | Nurse scheduling using integer linear programming and constraint programming[END_REF] dealt with the problem of scheduling human resources over different services in one hospital. The objective is to share resources within larger surgical suite, in order to reduce the costs. At another level, it can be seen as a sharing of resources from different places within larger organization such as HGT. In this paper, concerned resources are stretchers and nurses, who are common resources used for any hospital services and locations.

A lot of researches about hospital system are dedicated. Articles consider three problems: sizing, planning and scheduling. Most of the papers focuses on one particular problem. Their models and resolution methods are not easily reusable. Our proposed model and tool are generic, so they could be reused as often as possible.

ANALYSIS

To analyze our system, we split the system into three subsystems: the physical subsystem (physical entities used to perform all the activities, their geographical distribution and their interconnections), the logical subsystem (flows that the system should treat, all activities concerning the treatment of these flows and all entities in the system relating to them) and the decision subsystem (center of decision which contains all the decision rules).

Physical Subsystem

The HGT is composed by several places. There is a known distance between each place.

On each place, there are one or several material resources. A material resource belongs to a type (for instance X-ray, scanner or MRI). Each material resource has an opening schedule which defines the times when the material resource is available over each period. For example, a given material resource may only be available five hours on Monday because it needs a maintenance operation or because an external doctor reserved it. Overtime may be allowed but is limited in time.

Human resources compose a medical team. The composition of this team depends on the considered exam. This team should have a specific number of stretchers, specialist doctors, nurses, etc. Human resources belong to a given place but can work on other places belonging to the same HGT, allowing a pooling of human resources over the HGT. Moves are not allowed within the same period, but between two periods. A time is given to human resources to go from a place to another. A human resource can use one or several types of material resource according to its skills. A skillful human resource, who can work on several types of material resources, is potentially less efficient than a human resource who can only work on one type of material resource, or one particular material resource. This efficiency should be translated in the processing time of the concerned exams. Each human resource has a planning which defines its regular work time, taking into account break times and holidays. Time to move from one place to another is included in the work time of human resources. Overtime may be allowed but is limited in time.

Logical Subsystem

The logical subsystem defines the flow: the set of exams to plan and assign, and the relationship between these exams and the resources previously defined.

An exam should be done before a period at the latest, called a due date. Each exam has a known processing time which depends on the assigned human and material resources. Each exam starts at one period and ends at the same one. Each exam has a reference place, where it should be done, if possible.

An exam needs a given number of human resources and one material resource. All required resources must be compatible with each other. By definition, an exam is compatible with some material resources, so the assigned material resource must be compatible with the exam. This material resource belongs to a type, so the assigned human resources must have the needed skill to use this type of material resource. The place where the exam is done is deduced from the one where is located the assigned material resource.

Decision Subsystem

The objective is to develop a model which, from a set of exams, builds a planning associating the triplet {exam, human resources, material resource} to a period. The study is made in a predictive approach, all the exams can be treated since the beginning of the planning horizon. The objective of this model is not to schedule exactly the exams but to assign one period to each exam. This planning must optimize some criteria and respect some constraints.

Criteria

Three categories of criteria can be defined: economical, about the comfort of the patient, and about the proper functioning of the HGT.

Concerning the economical aspects, criteria are about the costs. Occupation rates of each place, each material resource and each human resource help to ensure the proper use of these entities. To be the most economic, these occupation rates have to be maximized. However, it can be preferable to define a security margin, so the HGT can be reactive in case of hazard. All exams are planned during the considered planning horizon. The makespan is the period assigned to the last exam. It ensures that all exams are assigned as soon as possible. The smaller the makespan is, more time remains free at the end of the planning horizon to potentially treat the next exams. The number of moves of human resources in the HGT should be considered, as well as overtime of human and material resources. Overtime and moves have a cost for the HGT. It is better to minimize them. But it can be interesting to allow some overtime or some moves to increase the number of exams during the considered period. About the comfort of the patient, the criterion is the number of exams done at their reference place. If some exams cannot be done at their reference place, the distances between the reference places and the effective ones may be minimized.

About the medical criterion, the number of exams done before their due dates has to be maximized. Thus, if a patient needs other exams, the next ones could be done on time. If an exam is planned after its due date, the tardiness may be minimized.

Constraints

Constraints that Must Be Respected • Each exam must be assigned to human resources, one material resource and one period. The considered human resources must be assigned during the period at the place where the considered material resource is located. • Compatibility between the material resource and the exam: the assignment must satisfy the given list of incompatibilities between exams and material resources. • Compatibility between skill of the human resource and the type of the material resource: for each exam, the human resource must be able to work on the considered material resource. • If a human resource can move during the planning horizon, its moves are constrained: a human resource can only work at one place during one period.

Constraints that May Be Respected

• Exams should be done before their due dates and at their reference place. • Material resources and human resources may be used or work during their opening schedule. Otherwise, additional time is considered as overtime.

Overtime is limited in time.

CONSIDERED PROBLEMS

The considered problem is defined as follow: exams planning and assignment of needed material and human resources. The previous analysis was about the complete problem. Some hypothesis are made to divide this complete problem into three problems.

Hypothesis

The following hypothesis are made:

• Only one human resource and one material resource are needed to perform exams. Human resources are compatible with one or several types of material resources. • Processing time of the exams are given and fixed.

• Opening schedules of material resources are equal in every periods. • To each exam, the release date is equal to the date of appointment decision. These dates are equal to zero: all exams are known at the beginning of the planning.

• Distances between places are taken into account in the time allocated to the human resources to move from one place to another. This time is assumed to be constant, all places are equidistant to the others. • Overtime is not allowed.

Definition of the Considered Problems

The complete problem is divided into three problems of increasing difficulty:

• Problem 1 is the more basic: human resources are not considered. Only the material resources are considered. • Human resources are considered in Problem 2.

They can work on one or several types of material resources. They cannot move, they work all the time at the same place. The assignment of the human resources at the places is given. • Human resources are mobile in Problem 3. They can work on several places, they can move from one place to another. The assignment of the human resources at the places has to be built by the model.

Two criteria are used in the following study:

• Sum of assigned periods to all exams, which ensure that all exams are planned as soon as possible.

• Number of exams assigned before their due date.

Analogy with the Bin Packing Problem

Our problem can be seen as a bin packing problem (Gourgand et al., 2014a). The bin packing problem considers N items, with a given size, and some bins with the same capacity. The aim is to pack all the items in a minimum number of bins. The size of the packed items has to respect the capacity of the bins.

Each item has to be assigned once and only once.

Without Human Resources

Considering Problem 1, the aim is to assign exams to a material resource during a period. The planning horizon is made by couples (resource, period). The objective is to assign exams to couples (resource, period). Exams have to be done as soon as possible: the aim is to minimize the number of couples, (= the number of bins). An example is given by Figure 1, where the assignment of exams to material resources MR 1 , MR 2 and MR 3 is considered.

With Human Resources

Exams have to be assigned to material and human resources during one period. Analogy is made between this problem and interdependent bin packing problem. Let's take p 1 and p 2 two bin packing problems, with a given number of bins. Groups of bins are defined in both problems. A group should be made by one or several bins. Number of bins can be different for both problems, but number of groups is the same. Each item is assigned to one and only one bin in each problem. Interdependence between both problems is defined as follow: if an item is assigned to a bin from group g in problem p 1 , it must be assigned to a bin from the same group g in problem p 2 . The aim is to assign items in bins of both problem, by minimizing the number of used bins, satisfying capacity constraints and interdependence between both problems.

In our case, both problems p 1 and p 2 can be defined like this:

• p 1 : assignment of each exam to a material resource during a period, respecting the opening schedule of the material resource during the period and the compatibility between the exam and the material resource.

• p 2 : assignment of each exam to a human resource during a period, respecting the work time of the human resource during the period and the compatibility between the exam and the human resource.

Compatibility between exam and human resource is not directly defined but can be deduced: an exam is compatible with a human resource if and only if this exam is treated by a material resource from one type and this human resource can work on this type of material resource.

Thus, group g is the couple (period, type). In both problems p 1 and p 2 , exam has to be assigned during the same period to the same type of material resource. Figure 2 illustrates the interdependent bin packing problem in the HGT case. The lower portion of the figure is the assignment of exams to material resources and periods, in the same way as Figure 1. The upper portion is the assignment of exams to human resources and periods. In both portions, each exam has to be assigned to the same period and the same type (according to the type of the material resource and the competencies of the human resource to use this type).

RESOLUTION METHOD

The bin packing problem is NP-Hard [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NPcompleteness[END_REF]. Our problems are an extension of the bin packing problem, so our problems of exams planning with resources assignment are also NP-Hard. In the following, approximate methods are used to solve them.

Our proposed method is a hybridization of a metaheuristic and a list algorithm. Our tool is convenient because one part is generic: it can be used for any of the considered problems. Only the list algorithm needs to be specific to the considered problem.

Genericity

The proposed tool, illustrated by Figure 3, uses a hybridization of a metaheuristic and a heuristic, more precisely a list algorithm. A single solution based metaheuristic or a population based metaheuristic can be used. The encoding used by the metaheuristic is a list Y of exams. The list algorithm L considers the exams according to their order in list Y to plan and assign them to the required resources, considering the problem constraints. This builds a solution X. The objective function H evaluates the solution X. According to this evaluation, the solution is chosen or not by the metaheuristic. At the end of the running, the given solution by the hybridization is the best list Y * of exams: the one which optimizes the objective function by applying the list algorithm. This hybridization can be used to solve many problems: the specificity of a given problem is only considered in the list algorithm.

A List Y of Exams

The general scheme of the encoding is given by Equation (1), with Ω the set of all the lists Y and S the set of all the admissible solutions X built by the list algorithm L.

Y ∈ Ω -→ Heuristic L L(Y ) = X ∈ S -→ Criterion H H(X) (1)
The set Ω is the set of all permutations of exams. Cardinal of Ω is N! with N the number of exams. One solution Y ∈ Ω is a list of exams. More details about the encoding are given in [START_REF] Gourgand | An analogy between bin packing problem and permutation problem: A new encoding scheme[END_REF].

Metaheuristic

The metaheuristic performs in the set of solutions Ω. An initial solution is randomly computed: a list of exams randomly sorted between one and the number of exams. Several metaheuristics have been used: some single solution based metaheuristics such as iterated local search or simulated annealing. A neighborhood system is used to visit the set of solutions, it allows to go from one solution to another one. Neighborhood system V is a permutation of two exams in the list Y : the exam at position i permutes with the one which is at position j. V satisfies the accessibility and reversibility properties.

List Algorithm

A list algorithm is used to build the solution X from the list Y : it assigns the exams to resources and to periods.

List scheduling algorithms are one-pass heuristics that are widely used to make schedules. A standard list scheduling algorithm constructs a schedule by assigning each job in listed order to the first machine that becomes idle [START_REF] Zhu | Scheduling and lot sizing with sequence-dependent setup: A literature review[END_REF]. It is important to work with a list algorithm, because the metaheuristic browses the set of solutions. So the used algorithm needs to consider the order of the list to assign exams to resources and periods.

Our problem has be analyzed as a bin packing problem and some list algorithms have been proposed since the definition of this problem [START_REF] Johnson | Near-optimal bin packing algorithms[END_REF]. So our developed list algorithm is inspired by them. For instance, considering Problem 1, in which human resources are not considered, Algorithm 1 is an extension of the First Fit algorithm for the bin packing problem. Other list algorithms can be adapted from Algorithm 1 to solve the cases of Problems 2 and 3, considering human resources.

Objective Function

Solutions are compared according to an objective function which characterizes the quality of the solution. In our case, the objective function represents the timing aspect of our problem. Exams have to be done as soon as possible, thus the makespan, the period assigned to the last exam, should be considered. Because many solutions may have the same makespan, we choose instead the sum of assigned periods to all exams, so the solutions can be dissociated. This criterion is written H S . Another criterion is considered to ensure that most of the exams are assigned before their due date. This criterion, written H D , is computed as the number of exams assigned after their due date. The weighed criteria method is used [START_REF] Coello | An updated survey of ga-based multiobjective optimization techniques[END_REF]. The objective function is a weighed sum between both criteria, defined by Equation (2). ω D is chosen equal to 5 because H S is always smaller than 10 5 so both criteria are easily readable. This function has to be minimized.

H(X) = 10 ω D × H D (X) + H S (X)
(2)

The Best List Y *

Algorithm 2 describes the whole method with the example of stochastic descent as the used metaheuristic. Stochastic descent may be used in an iterated local search. The set Ω of the lists of exams is browsed thanks to the metaheuristic using neighborhood system V . Lists are compared thanks to the list algorithm L and the objective function H. According to an acceptance criterion, some lists are selected. At the end, the metaheuristic gives the best found list Y * .

Algorithm 2: Hybridization between stochastic descent and a list algorithm.

Data: Initial solution Y 1 X := L(Y ) 2 while necessary do 3

Choose uniformly and randomly

Y ′ ∈ V (Y ) 4 X ′ = L(Y ′ ) 5 if H(X ′ ) ≤ H(X) then 6 X := X ′ 7 Y := Y ′ Result: Y * = Y

EXPERIMENTS

The data are randomly generated but the characteristics and the size of the data represent real instances. The HGT is composed by 3 places. The planning horizon is made by 8 to 40 periods. As a remind, one period represents one half-day, thus the planning horizon is between 4 and 20 days. 4 to 8 resources are available. 50 to 500 exams need to be planned and assigned. Incompatibilities between exams and resources are randomly generated. Each processing time is between 5 and 100. Each material resource has an opening schedule equal to 300 minutes.

The results are detailed in Table 2. The host machine is powered by an Intel Xeon X5687 quad-core CPU running at 3.6 GHz. The computation has been stopped after thirty minutes. Each reported result is the value of the objective function for the best solution found in less than thirty minutes, but most of the time, the best solution is found in a few minutes. The results are presented as a couple of values (H D ; H S ) with H D the number of exams assigned after their due dates, and H S the sum of assigned periods to all the exams. The results compare three methods:

• The resolution of the mathematical model with an exact method by using the solver CPLEX. If no optimal solution has been found in less than thirty minutes by the solver, no result is written.

• Our results from the method previously published in (Gourgand et al., 2014a), using two single solution based metaheuristics (iterated local search and simulated annealing) in a classical way: the best value found by all these methods is reported.

• Our results from our proposed method detailed in the current paper. The used metaheuristics are distinguished: iterated local search and simulated annealing, written ILS* and SA*.

The results are promising. Firstly, this problem has been solved by CPLEX thanks to our mathematical model previously proposed. The solver finds an optimal solution only for small size of problems (less than two hundred exams over four days). The solver does not find any solutions when the size of the problem increases. Then, it has been solved with two approximate methods: in a classical way, and with a hybridization. Both methods find an optimal solution for the small instances. For biggest instances, the hybridization between a metaheuristic and a list algorithm outperforms our previous method. Simulated annealing seems to work better than iterated local search.

CONCLUSION AND PERSPECTIVES

The current hospital context needs to find solutions to improve efficiency of hospital systems. Hospital cooperation has emerged, as Hospital Group of Territory. A pooling of resources may cause a better use of the different places in a same territory. But this cooperation needs some decision support tools to improve or optimize their running. In this paper, we defined the general problem of activities planning with resources assignment in a multi-place hospital context.

Because this problem is NP-Hard, we propose an approximate method to solve it: a hybridization between a metaheuristic and a list algorithm. The results are promising: our method finds good results in a few minutes. An improvement of the results is in process, using a population based metaheuristic: Particle Swarm Optimization. Using PSO, the results are very good for small instances: an optimal solution is found in a few seconds, but the method still needs some tuning for the biggest instances.

Thanks to the hybridization, our method can be easily reusable. Indeed, to solve other problems, only the list algorithm needs to be modified. The metaheuristic part will still be the same. Any kinds of planning, assignment or scheduling problem can be solved thanks to this tool by changing the list algorithm: for instance, it has been used to solve an industrial problem [START_REF] Silva | A generic decision support tool for lot-sizing and scheduling problems with setup and due dates[END_REF]. Problems with human resources can easily be solved by developing some new list algorithms dedicated to them. Then, a direct application to a hospital system could be envisaged. Other applications in the hospital field could be done, in other hospital services, with other resources, etc. We could extend our current work about medical imaging to medical surgeries. More constraints about medical team should be considered. The next problematic is to consider patients with several exams or surgeries, by taking into account precedence constraints between them.

Figure 2 :

 2 Figure 2: Interdependent bin packing problem.

Figure 3 :

 3 Figure 3: Hybridization metaheuristic -List algorithm.

Algorithm 1 :

 1 List Algorithm First Fit HGT. Data: List of exams (Y i ) i∈{1,N} ; opening schedule of all resources during all periods; processing time of all exam 1 Occupied time := 0 for all resources and all periods 2 forall the i do 3 First resource, first period, assigned := f alse 4 while (assigned = f alse) AND current period ≤ max of periods do 5 while (assigned = f alse) AND current resource ≤ max of resources do 6 if exam Y i is compatible with current resource then 7 if exam Y i fits in couple (resource, period) then 8 Assign exam Y i to couple (resource, period)

Table 1 :

 1 Table 1 summarizes analogies between bin packing problem and Problem 1: exams planning with resources assignment. Analogies between both problems.

	Time					Opening
						schedule
						of resources
		6
			2		11	Caption:
		1	3	7	5	X	Exam number X
			4
	Resource MR 1 MR 2 MR 3	MR 1 MR 2 MR 3	Bin
	Period	Monday AM	Monday PM
	Figure 1: Representation of Problem 1.
			Bin packing	Problem of exams planning
			problem		with resources assignment
			Item			Exam
			Bin		Couple (resource, period)
			Size of an item	Processing time of an exam
	Data		Capacity of a bin	Opening schedule of resources
			-		Due date
			-		Reference place
	Problem		Assign items to one bin	Assign exams to one couple (resource, period)
	Constraints	Capacity of bins -	Opening schedule of resources Constraint of compatibility
	Criteria		Minimize the number of used bins -	-Minimize the sum of assigned periods

Table 2 :

 2 Results: (number of exams assigned after their due dates; sum of assigned periods to all the exams).

	Number of exams CPLEX (Gourgand et al., 2014a)	ILS*	SA*
	50	(0; 51)	(0; 51)	(0; 51)	(0; 51)
	50	(1; 150)	(10; 147)	(1; 151)	(1; 150)
	100	(0; 131)	(0; 131)	(0; 131)	(0; 131)
	100	(0; 517)	(2; 535)	(1; 516)	(0; 518)
	200	(0; 266)	(0; 266)	(0; 266)	(0; 266)
	200	-	(3; 1197)	(0; 1154)	(0; 1135)
	300	-	(0; 548)	(0; 537)	(0; 534)
	400	(0; 830)	(0; 890)	(0; 841)	(0; 835)
	500	-	(0; 1350)	(0; 1241)	(0; 1234)
	500	-	(194; 8218)	(19; 6382) (18; 6659)

STATE OF THE ARTSince the last ten years, many methods of operational research have been used to solve hospital system problems.[START_REF] Rais | Operations research in healthcare: a survey[END_REF] referenced around two hundred and fifty articles about operational research for hospital system in general. (Car-


Anaïs Charbonnel
email: anais.charbonnel@espaces-naturels.fr

Aurélien Frank D'amico

Frédéric Besnard

Laetitia Blanc

Mélanie Buisson

Pascal Némoz

Laffaille

Ana€ Is Charbonnel

Frank D'amico

Aur Elien Besnard

Fr Ed Eric Blanc

La€ Etitia Buisson

Pascal Laffaille

Spatial replicates as an

Keywords: autocorrelation, detection probability, Galemys pyrenaicus, Markovian model, multiscale model, sign survey

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Presence-absence data are widely used to determine the habitat requirements of species or predict their distribution in unexplored areas [START_REF] Guisan | Predictive habitat distribution models in ecology[END_REF]. One widely acknowledged major issue is that false absences are unavoidable during fieldwork [START_REF] Gu | Absent or undetected? Effects of non-detection of species occurrence on wildlife-habitat models[END_REF][START_REF] Gibson | Dealing with uncertain absences in habitat modelling: a case study of a rare ground-dwelling parrot[END_REF][START_REF] Lobo | The uncertain nature of absences and their importance in species distribution modelling[END_REF]. In species distribution modelling, failure to take imperfect detection into account can introduce a considerable degree of error into spatial and temporal distribution patterns [START_REF] Yoccoz | Monitoring of biological diversity in space and time[END_REF]. For example, false conclusions can be reached about the suitability of areas for conservation actions [START_REF] Rondinini | Tradeoffs of different types of species occurrence data for use in systematic conservation planning[END_REF]. Correctly accounting for false absences is thus important, especially for the conservation of cryptic and secretive species (e.g. [START_REF] Gibson | Dealing with uncertain absences in habitat modelling: a case study of a rare ground-dwelling parrot[END_REF][START_REF] Durso | Needles in haystacks: estimating detection probability and occupancy of rare and cryptic snakes[END_REF], for which surveys often rely on recording indirect signs.

Over recent decades, many methods have been developed to deal with detection issues and to correct the biases they induce, for example distance sampling [START_REF] Buckland | Introduction to Distance Sampling -Estimating Abundance of Biological Populations[END_REF] or capture-recapture [START_REF] Pollock | Large Scale Wildlife Monitoring Studies: Statistical Methods for Design and Analysis[END_REF]. More recently, [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF][START_REF] Mackenzie | Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence[END_REF] developed site occupancy modelling (hereafter referred to as 'standard site occupancy modelling'), a statistical method that models the probability that a species occupies some sites even though it has not been detected with any certainty. This method is increasingly being successfully used to model the distribution of various species (e.g. K ery, Gardner & Monnerat 2010; Olea & Mateo-Tom as 2011; K ery, Guillera-Arroita & Lahoz-Monfort 2013). Site occupancy modelling requires replication of detection-non-detection data at sampled sites. Replications at site level are usually temporal (i.e. the sites are monitored over several successive visits). They may, however, also be spatial (i.e. sampling several spatial subunits at each site during a single visit).

Standard site occupancy modelling [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF][START_REF] Mackenzie | Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence[END_REF] relies on the assumption that there is no change in site occupancy during the survey [START_REF] Rota | Occupancy estimation and the closure assumption[END_REF]. Violation of this assumption of closure leads to an underestimation of the detection probability and an overestimation of the occupancy. When temporal replicates are used, successive visits must therefore be conducted over a relatively short period. When spatial replicates are substituted for temporal ones, all the subunits must have a non-negligible occupancy probability. Thus, the sampling design must take into account the movement patterns of a species to adhere to the model assumptions [START_REF] Kendall | A cautionary note on substituting spatial subunits for repeated temporal sampling in studies of site occupancy[END_REF][START_REF] Guillera-Arroita | Impact of sampling with replacement in occupancy studies with spatial replication[END_REF]. [START_REF] Nichols | Multi-scale occupancy estimation and modelling using multiple detection methods[END_REF] also developed a multiscale occupancy model (hereafter referred as 'standard multiscale occupancy model') which can be used when both spatial and temporal replicates are available for the same sites. This model relaxes the closure assumption by dealing with the situation in which the species is present at a site, but may be absent from some survey subunits.

Besides the closure assumption, for the standard site occupancy model [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF][START_REF] Mackenzie | Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence[END_REF], the detection events of the species must be independent. In the context of temporal replicates, this assumption may be violated, for instance, if the same observer visits the same site several times and remembers where to look for signs of the species' presence (observer bias hereafter). In the case of spatial replicates, dependence could arise if the subunits are not randomly selected (e.g. if they are spatially close together or sequentially sampled; [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF][START_REF] Anderson | Occupancy modeling and estimation of the holiday darter species complex within the etowah river system[END_REF] and the observer becomes more attentive after the first detection event. In such situations, the spatial or temporal correlation between detection events can be explicitly modelled using a Markovian detection model [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]. When data are collected using a spatial cluster of sampling subunits, spatial autocorrelation can also arise between adjacent replicates with regard to species' presence. To deal with such situations, [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF] developed a Markovian occupancy model.

To our knowledge, only one study has directly compared the parameter estimates of occupancy and detection of temporal vs. spatial replicates [START_REF] Parry | A systematic re-sampling approach to assess the probability of detecting otters Lutra lutra using spraint surveys on small lowland rivers[END_REF]. Our work was prompted by a field survey of a small semi-aquatic mammal, the Pyrenean desman Galemys pyrenaicus E. Geoffroy Saint-Hilaire, 1811. Repeated visits to stream reaches in the Pyrenees to get temporal replicates are timeconsuming and expensive, because of being frequently difficult to access, and they have to be sampled meticulously by wading. Spatial replicates therefore offer an appealing, cost-effective alternative. Since sites are river transects, the least expensive approach is to use successive portions of rivers as adjacent spatial replicates. The Markovian occupancy model, originally developed for large and highly mobile mammals using trails such as tigers Panthera tigris, may also be the most appropriate method to model data collected along river transects [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF][START_REF] Karanth | Monitoring carnivore populations at the landscape scale: occupancy modelling of tigers from sign surveys[END_REF]. The goals of our study were to (i) directly compare occupancy and detection estimates obtained from temporal vs. spatial design, and (ii) evaluate whether the Markovian occupancy model (with autocorrelation) is supported when spatial adjacent replicates are used, for a species with a small linear home range.

Materials and methods

S T U D Y A R E A

The French Pyrenees (W1°40 0 -E3°10 0 , N43°08 0 -N42°23 0 ) are a range of mountains approximately 400 km long, covering an area of 18 176 km² and reaching a maximum elevation of 3298 m (Fig. 1). Climate and topography are the main factors influencing the dynamics and flow regime of Pyrenean rivers.

B I O L O G I C A L M O D E L
The Pyrenean desman is a small, semi-aquatic mammal with a range restricted to the Pyrenees (Andorra, France and Spain) and some areas in northern and central Spain, and northern Portugal. It lives in mountain brooks, cold and well-oxygenated water courses, and its altitudinal range extends from 15 m (western French Pyrenees) or 450 m (eastern French Pyrenees) to 2700 m [START_REF] Bertrand | Plan National d'Actions en faveur du desman des Pyr en ees (Galemys pyrenaicus), 2009-2014. Soci et e Franc ßaise pour l'Etude et la Protection des Mammif eres/Minist ere de l'Ecologie, de l'Energie, du D eveloppement Durable et de l'Am enagement du Terri[END_REF]. The Pyrenean desman has declined substantially in its distribution, triggering several conservation regulations [START_REF] Fernandes | Galemys pyrenaicus[END_REF]. Like many secretive species, searching for faeces is the most efficient and least cost-and timeconsuming method for surveying this across a wide geographic extent (e.g. [START_REF] Kindberg | Monitoring rare or elusive large mammals using effort-corrected voluntary observers[END_REF]. Being elusive and cryptic, its ecology is still largely unknown. It is not yet understood whether this species is territorial or why it leaves its faeces on emergent items.

F I E L D S U R V E Y P R O T O C O L
Thirty-four sites (i.e. river transects) were surveyed for this study (Fig. 1). Previous sampling had reported presence of the Pyrenean desman in all these sites, as it is more efficient to select sites where the animal is known to be present than a random selection when modelling detection probabilities [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF]. Skilled observers waded river transects to search for faeces. They meticulously inspected each emergent rock, tree root or branch in the stream. Pyrenean desman faeces were identified from their colour, size, position and smell. Surveys were conducted during the summer, when faeces seem to persist longest [START_REF] Bertrand | R epartition g eographique et ecologie alimentaire du desman des Pyr en ees Galemys pyrenaicus (Geoffroy, 1811) dans les Pyr en ees franc ßaises[END_REF], to maxi-mize detectability. The same observer visited five times each site, between June 7th and October 5th 2012, to obtain temporal replication data over one season. For each site, visits were at least 1 week apart to limit observer bias. To maximize the abundance of available signs, we avoided surveys during or after a period of fluctuating water levels or heavy rainfall [START_REF] Bertrand | R epartition g eographique et ecologie alimentaire du desman des Pyr en ees Galemys pyrenaicus (Geoffroy, 1811) dans les Pyr en ees franc ßaises[END_REF].

Each site was a riverbed transect, 500 m long, which approximately matches the mean home range of the species determined by radiotracking (523 m; SE 50Á85; [START_REF] Melero | New insights into social and space use behaviour of the endangered Pyrenean desman (Galemys pyrenaicus)[END_REF]. Each sampling site had five subunits (i.e. segments) of equal length (100 m) that constituted the adjacent spatial replicates. As ecological knowledge for the Pyrenean desman is limited (N emoz et al. 2011), the optimum segment length was estimated. We chose a length of 100 m as the best compromise between the need to detect signs when animals are in fact present and that of having more than three spatial replicates, as recommended by [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF].

S I T E O C C U P A N C Y A N A L Y S I S

The data configurations used in the analyses are shown in Fig. 2: the first had both spatial and temporal replicates at the 34 sites (Fig. 2a), the second had spatial replicates during a single visit (Fig. 2b), the third had temporal replicates for a single segment (Fig. 2c), and the fourth had temporal replicates at the site level (i.e. segments pooled; Fig. 2d). The multiscale models [START_REF] Pavlacky | Hierarchical multi-scale occupancy estimation for monitoring wildlife populations[END_REF] were fitted to both spatial and temporal replicates, that is the first data configuration. We fitted the standard site occupancy model [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF], the Markovian detection model and the Markovian occupancy model [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF] to the last three data configurations.

Standard occupancy model -w(.), p(.)

This modelling approach estimates two key parameters: p, the probability of detecting the species if the species is in fact present on the site (i.e. the probability of detection), and w, the probability that a site is occupied or used by a species (i.e. the probability of site occupancy).

Standard multiscale occupancy model -w(.), h(.), p(.)

This model has been used for presence-absence data collected at two spatial scales [START_REF] Pavlacky | Hierarchical multi-scale occupancy estimation for monitoring wildlife populations[END_REF]. N sample sites are subsampled by R spatial segments (spatial replicates). All sampled sites (and consequently all segments) are also repeatedly visited (temporal replicates). The parameters of the model are: p, the probability of detection on a segment at one visit if the sample site and the segment are in fact occupied; h, the probability of occupancy for one survey segment if the sampling site is in fact occupied; and w, the probability of occupancy of a sampling site. The small scale occupancy, h, is an availability parameter to account for situations in which the species is present in the sampling site and may be present in some survey segments but not in others.

Multiscale occupancy model with Markovian process for detection -w(.), h(.), p 0 (.), p 1 (.)

This multiscale model takes into account the autocorrelation in detection events between spatial or temporal occasions. Two parameters are linked to detection: p 0 , the probability of detection on one occasion if the site is in fact occupied and the species was not detected on the previous occasion, and p 1 , the probability of detection on one occasion if the site is in fact occupied and the species was detected on the previous occasion.

Markovian detection model -w(.), p 0 (.), p 1 (.) This is derived from the standard occupancy model [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF] to account for autocorrelation in detection between occasions. Detection is modelled as an observable Markov process, which allows the detection probability for each occasion to depend on whether or not the species was detected on the previous occasion. The two parameters linked to detection, p 0 and p 1 , are similar to those presented for the multiscale occupancy model with Markovian process for detection.

Markovian occupancy model -w(.), h 0 (.), h 1 (.), p(.)

This model was developed for adjacent spatially autocorrelated selected replicates (i.e. segments; [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]. This model assumes spatial closure (i.e. occupancy of one spatial replicate assumes that the other spatial replicates are similarly occupied; [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]. That is why this model design was specifically developed for highly mobile species, and the spatial scale of the sampling sites must be the same as the home range of the species (or as close to it as possible). In addition to the occupancy probability measured at the site level (w) and the detection probability measured at the segment level (p), two probabilities are estimated: h 0 , the probability that a species is present on a segment if the site is actually occupied and the species was absent on the previous segment, and h 1 , the probability that a species is present on the segment if the site is occupied and the species was present on the previous segment. This model could also be useful in occupancy studies that use temporal replication to test for the temporal pattern of local occupancy [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]).

M O D E L C O M P A R I S O N A N D E V A L U A T I O N

We used Akaike information criterion (AIC) to assess the relative fit of the models within each candidate data configuration.

0-1 0-1 0-1 0-1 0-1 One segment Temporal replicates 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1

Spatial replicates

Temporal replicates

One segment Table 1. Occupancy and detectability estimates obtained with the five models (SE: standard error). The two multiscale models [w(.), h(.), p 0 (.), p 1 (.)] and [w(.), h(.), p(.)] were applied on both spatial and temporal replicates (Fig. 2a). The Markovian occupancy model [w(.), h 0 (.), h 1 (.), p(.)], the Markovian detection model [w(.), p 0 (.), p 1 (.)] and the standard occupancy model [w(.), p(.)] were applied on spatial replicates (Fig. 2b), temporal replicates of a single segment (Fig. 2c) and temporal replicates at the site level (i.e. segments pooled; Fig. 2d). The relative differences in AIC val- Models were ranked, with the lowest AIC value indicating the best fit to the data. We did not evaluate the value of the AIC per se, but the difference between the value for the model of interest and that for the best-supported model (Di; [START_REF] Burnham | Model Selection and Multi-model Inference: A Practical Information-Theoretic Approach[END_REF]. A fitted model with a Di value of less than two has more substantial support for the data than the other models. The relative differences in AIC values for the fitted models can be used to calculate the weight (Wi) of evidence of each model in the model set. We fitted all the models using the freeware PRESENCE v. 5.5 [START_REF] Hines | PRESENCE 2: Software to Estimate Patch Occupancy and Related Parameters[END_REF].

0-1 0-1 0-1 0-1 0-1 Spatial replicates One segment 0-1 0-1 0-1 0-1 0-1 Temporal replicates One site (b) (a) (d) (c)
ues
(.) 44Á92 0Á00 - - 0Á41 (0Á04) 0Á34-0Á48 0Á75 (0Á02) 0Á70-0Á79 0Á77 (0Á04) 0Á68-0Á84 1Á00 - - - - - Standard w(.), h(.), p(.) 103Á84 0Á00 0Á60 (0Á02) 0Á56-0Á64 - - - - 0Á72 (0Á03) 0Á65-0Á79 1Á00 - - - - - Spatial w(.), h 0 (.), h 1 (.), p(.) 0Á00 0Á60 0Á93 (0Á11) 0Á33-0Á99 - - - - - - 0Á85 (0Á18) 0Á26-0Á99 0Á24 (0Á19) 0Á04-0Á70 0Á85 (0Á09) 0Á59-0Á95 w(.), p 0 (.), p 1 (.) 0Á80 0Á40 - - 0Á45 (0Á08) 0Á31-0Á61 0Á79 (0Á05) 0Á68-0Á88 - - 0Á77 (0Á08) 0Á57-0Á90 - - - - w(.), p(.) 12Á56 0Á00 0Á64 (0Á04) 0Á55-0Á72 - - - - - - 0Á74 (0Á08) 0Á57-0Á86 - - - - Temporal (segment) w(.), p 0 (.), p 1 (.) 0Á00 0Á84 - - 0Á38 (0Á08) 0Á24-0Á55 0Á73 (0Á06) 0Á60-0Á83 - - 0Á81 (0Á10) 0Á55-0Á93 - - - - w(.), h 0 (.), h 1 (.), p(.) 3Á43 0Á15 1Á00 - - - - - - - 0Á83 (0Á11) 0Á51-0Á96 0Á34 (0Á10) 0Á17-0Á55 0Á73 (0Á06) 0Á60-0Á83 w(.), p(.) 10Á45 0Á00 0Á58 (0Á05) 0Á48-0Á66 - - - - - - 0Á75 (0Á08) 0Á57-0Á87 - - - - Temporal (site) w(.), p 0 (.), p 1 (.) 0Á00 0Á88 - - 0Á55 (0Á06) 0Á43-0Á66 0Á82 (0Á04) 0Á74-0Á89 - - 1Á00

Results

For spatial replicates and temporal replicates, we applied analyses successively to the five visits and segments, respectively. Given that the trends were similar, we only give here the results for the third visit (spatial replicates) and the first segment (temporal replicates) (see Tables S1-S3, Supporting Information for the other visits and segments).

M O D E L S E L E C T I O N

In all cases, the standard occupancy model was the leastsupported model according to the AIC, with a DAIC greater than 10 relative to the best model and with a null AIC weight (Table 1). The multiscale model, which took spatial dependence of detection into account, was the most-supported model (i.e. spatial and temporal replicates; W = 1; Table 1). For temporal replicates at both spatial levels (site and segment), comparison of AIC values indicated that the best-supported model was the Markovian detection model (W = 0Á88 and W = 0Á84 for site and segment, respectively), followed by the Markovian occupancy model (W = 0Á12 and W = 0Á15 for site and segment, respectively; Table 1).

This result suggests a possible observer bias between detection events, with p 0 (SE, standard error) = 0Á55 (SE 0Á06) and p 1 = 0Á82 (SE 0Á04) and p 0 = 0Á38 (SE 0Á08) and p 1 = 0Á73 (SE 0Á06) at the site and segment level, respectively (Table 1). For spatial replicates, the Markovian occupancy model was more highly supported than other models (W = 0Á60; Table 1), suggesting that spatial autocorrelation existed in the presence of desman faeces between segments and that this should be modelled, with h 0 = 0Á24 (SE 0Á19) and h 1 = 0Á85 (SE 0Á09; Table 1).

P R O B A B I L I T Y E S T I M A T E S

Site occupancy

The na€ ıve site occupancy (i.e. the proportion of sampling sites with at least one detection event during the survey) was a priori known to be equal to one; hence, the site occupancy probability (w) estimated by all models applied to temporal replicates and to temporal and spatial replicates was, not surprisingly, equal to one. For the spatial replicates only, na€ ıve occupancy was equal to 0Á74, because some sites had no detection event during this sampling visit (the third one). Hence, estimated values of w were lower for all models applied to spatial replicates (Table 1). w was highest when fitted using the Markovian occupancy model (w = 0Á85; SE 0Á18). However, large standard error estimates made it impossible to make strong inferences.

Segment occupancy

On the local scale, estimates of segment occupancy (h or w) were roughly similar between all analyses and models, with w or h ranging from 0Á72 (SE 0Á03) to 0Á83 (SE 0Á11; Table 1). For spatial and temporal replicates, the mostsupported multiscale model which accounts for spatial correlation in detection events estimated a h value of 0Á76 (SE 0Á04). For temporal replicates, w was higher when correlation for detection [w(.), p 0 (.), p 1 (.)] or presence [w(.), h 0 (.), h 1 (.), p(.)] was accounted for, with w = 0Á81 (SE 0Á10) and 0Á83 (SE 0Á11), for each model, respectively (Table 1).

Detectability at the site level

For temporal replicates, the Markovian occupancy model gave an estimate of detection probability equal to one (Table 1). The standard model, which was the leastsupported model according to the AIC, estimated a detection probability that was 29% lower ( p = 0Á71; SE 0Á03).

Detectability at the segment level

The Markovian occupancy model applied to different data configurations (spatial vs. temporal replicates) gave roughly similar estimates. With this model, p ranged from 0Á93 (SE 0Á11) to 1Á00, whereas the standard occupancy model gave estimates ranging from 0Á58 (SE 0Á05) to 0Á64 (SE 0Á04; Table 1) depending on data configurations. These estimates were comparable to those at the site level. Models that took into account the autocorrelation between detection events suggested that the detection probability was lower when there was no detection on the previous occasion than when the species had been detected on the previous occasion (Table 1). For example, for the Markovian detection model applied to spatial replicates, p 0 = 0Á45 (SE 0Á08) and p 1 = 0Á79 (SE 0Á05).

Discussion

O C C U P A N C Y A N D D E T E C T A B I L I T Y E S T I M A T E S

When the same occupancy model was fitted to several different data configurations, we found that the estimated detection probabilities were relatively similar, whatever the replication design (i.e. spatial or temporal) and the scale (i.e. site or segment) used. This indicates that the detectability of the Pyrenean desman in this data set appears to be fairly constant when a single segment of 100 m was surveyed on five successive occasions, when a site of 500 m was surveyed on five successive occasions or when five adjacent 100-m segments were surveyed on one occasion. The most parsimonious model was the Markovian detection model with temporal replication, while the Markovian occupancy model was the most supported with only spatial replicates. Detection probabilities were high using the Markovian occupancy model (0Á91-1), which is likely to be more accurate than the standard occupancy model (0Á58-0Á71). These results are consistent with those of [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF] showing that species detectability is underestimated in the presence of dependence between sampling occasions with the standard occupancy model.

The probabilities of detection estimated for the Pyrenean desman were relatively high (P ≥ 0Á58), but their precision and inferences are limited by the small size of the samples collected during this study (34 sites). The Pyrenean desman seems to have moderate to high detectability when its faeces are present on emergent rocks, roots or branches in the stream. In comparison, the probability of detecting the Eurasian otter Lutra lutra on the basis of spatial replicates (600-m stream transects for sites and 50-m segments for subunits) has been estimated to be 0Á26 on the Gower peninsula (UK; [START_REF] Parry | A systematic re-sampling approach to assess the probability of detecting otters Lutra lutra using spraint surveys on small lowland rivers[END_REF]. Given that the lowest estimated probability of detection for the Pyrenean desman equals 0Á58, only five sampling occasions will be sufficient to get an almost perfect probability of detection (P = 1À(1À0Á58) 5 = 0Á99) when Pyrenean desman signs are in fact present on the sampled site. However, we acknowledge that the sites sampled in this study may contain a high abundance of signs of the Pyrenean desman since they were already known to be occupied prior to sampling. If sign abundance positively influences sign detectability, the latter may be overestimated in this study [START_REF] Mccarthy | The influence of abundance on detectability[END_REF].

Our results show that temporal replication at site level is more accurate than spatial replication for estimating occupancy which is not surprising as all sites (but not all segments) have at least one detection event among the five visits. This is not necessarily the case when using spatial replication because some sites may have no detection event during the selected sampling visit (the third one here but see Tables S2-S3, Supporting Information). When using temporal replication at site level, all occupancy models (i.e. Markovian occupancy model, Markovian detection model and standard occupancy model) do not appear to underestimate occupancy (equalling 1), while temporal sampling restricted to a 100-m segment underestimates occupancy (0Á81 for the Markovian detection model which is the most-supported model). Estimates of occupancy with spatial adjacent replicates are also underestimated in a magnitude comparable to those estimated when successively sampling a 100-m segment (0Á77 for the Markovian detection model). However, when using adjacent spatial replicates, the Markovian occupancy model is the most appropriate and minimizes the underestimation of occupancy (0Á85) which becomes even higher than temporal replicates at the segment level. Thus, there appears to be a trade-off between spending more time sampling with temporal replication to achieve a more accurate estimate of occupancy, or sampling with adjacent spatial replicates which is more cost-effective but with a slight underestimation of site occupancy.

For local occupancy (i.e. segment level), when signs of the Pyrenean desman are present at a site, they occur, on average, in 75% of the segments. The h 0 and h 1 estimates suggest that spatial autocorrelation between segments is very high. Indeed, when no species signs have been detected on a given segment, the next segment has only a probability of 0Á24 signs being present. In contrast, the probability of occupancy of a given segment is 0Á85 when signs of presence have been detected on the previous segment. It is still not understood why the Pyrenean desman leaves its faeces on emergent items and how it uses its home range. This finding suggests that signs of the Pyrenean desman are clustered within sites, with some adjacent segments more marked with faeces than others in a single 500-m transect. The choice of a segment length of 100 m seems to be appropriate for the Pyrenean desman when using the Markovian occupancy model, as it is long enough to detect signs when the animals are present and short enough to exhibit dependence in occupancy between adjacent segments.

Nevertheless, some caution is called for with regard to the closure assumption. In the temporal replicates design, we do not know the extent to which the Pyrenean desman shifts its home range. We kept the duration of the survey as short as possible (i.e. 4 months) to best meet this assumption [START_REF] Rota | Occupancy estimation and the closure assumption[END_REF]). In the spatial replicates design, the sampling site was 500 m in length, which corresponds to the reported mean home range size for the Pyrenean desman [START_REF] Melero | New insights into social and space use behaviour of the endangered Pyrenean desman (Galemys pyrenaicus)[END_REF]. We cannot fully address how likely it is that we met the closure assumption. Preliminary findings have suggested that the Pyrenean desman is highly mobile within its home range (Y. Melero, pers. comm.).

The detectability of a species is rarely constant across space and time and often depends on factors such as environment or species biology [START_REF] Mackenzie | Estimating site occupancy rates when detection probabilities are less than one[END_REF]. Although the spatial variation of detectability is increasingly acknowledged (e.g. [START_REF] Royle | Estimating abundance from repeated presence-absence data or point counts[END_REF][START_REF] Royle | Site Occupancy Models with Heterogeneous Detection Probabilities[END_REF], little is known about the influence of unmodelled variation between sites on parameter estimates [START_REF] Mackenzie | Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence[END_REF]. The difficultly in accessing and surveying our sampling sites mean that cost and time considerations have constrained the selection of these sites and limited their number. Adding covariates in occupancy models to a small data set might cause them to fail to converge, especially the Markovian occupancy model, which estimates five parameters even without any covariate [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]. However, since the aim of this study was not to identify the environmental factors that influence the detectability of the Pyrenean desman, the model parameters were kept constant. With a larger number of sites sampled at the scale of the Pyrenees range, for example, adding environmental covariates to the Markovian occupancy model [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF] would help to identify important factors that influence both the detectability and occupancy of the Pyrenean desman.

I M P L I C A T I O N S F O R T H E C O N S E R V A T I O N O F T H E P Y R E N E A N D E S M A N
The cryptic behaviour of the Pyrenean desman and its habitat heterogeneity clearly result in its detectability being variable in space and time, as well as being imperfect. Although it is much easier to detect the Pyrenean desman by looking for its faeces, many factors, such as the skills of the observer (e.g. [START_REF] Barber-Meyer | Influence of prey depletion and human disturbance on tiger occupancy in Nepal[END_REF], the configuration of the river, the fluctuations in water levels (natural and human-induced), the period when sampling is conducted or the meteorological conditions (e.g. [START_REF] Aing | A Bayesian hierarchical occupancy model for track surveys conducted in a series of linear, spatially correlated, sites[END_REF], may have a considerable impact on the chance of finding faeces. For example, it may be less easy to detect faeces of the Pyrenean desman in areas with higher annual rainfall or in rivers influenced by hydroelectric activities, as these factors may reduce the persistence of faeces or accelerate the process of deterioration. In this study, all the sampled stream segments had emergent items which must be present if Pyrenean desman signs are to be detected. However, the density of emergent items along the river varied between segments, and we assume that the presence and/or detectability of signs may be influenced by the proportion of the surface area covered with emergent items. A possible covariate describing, for instance, the density of emergent items along streams deserves to be further explored. It is indeed critical to understand how these different factors affect the probability of detecting the species for the design of efficient monitoring strategies in the future [START_REF] Field | Optimizing allocation of monitoring effort under economic and observational constraints[END_REF]. This study provides further insights into possible survey design and ways of monitoring this endangered and elusive species. The results of a survey conducted at a larger scale (i.e. the whole Pyrenees) will enable to increase our knowledge about the geographical range of this species in order to identify any discontinuities within this range. This in turn should allow biodiversity managers and government organizations to make informed decisions regarding the conservation of the Pyrenean desman and its habitats in Andorra, Spain and France.

I M P L I C A T I O N S F O R S U R V E Y S O F O T H E R S P E C I E S

Survey areas sometimes present difficult field conditions (e.g. topography, climate) that hinder access and signbased searches. On a large spatial scale and with a large number of sampling sites, temporal replicates could be very time-and money-consuming and consequently prohibitive. Moreover, for monitoring to be useful, it needs to be sustainable in the long term. Spatial replication seems more effective and straightforward to implement than temporal replication [START_REF] Hines | Tigers on trails: occupancy modeling for cluster sampling[END_REF]. Once observers have reached a site, the cost of performing additional survey replicates is low [START_REF] Mackenzie | Designing occupancy studies: general advice and allocating survey effort[END_REF]. This is supported by our results, suggesting that sampling with spatial replicates only slightly underestimates site occupancy compared to temporal replicates. Hence, adjacent spatial replication appears to be a good trade-off between costs of sampling (money and time) and accuracy of occupancy estimates. However, the choice between spatial vs. temporal design for the survey of other species will depend on the attention turned to each of these two aspects. Some general precautions should also be kept in mind when using the Markovian occupancy models to avoid violating the model assumptions and biasing the estimates of detection and occupancy. For instance, species behaviour, mainly in relation to the size of its home range and use (e.g. the distribution of signs within home range), must be considered when deciding the appropriate length of sampling sites and subunits. In this study, we showed that surveys based on adjacent spatial replicates along streams and the Markovian occupancy model parameter settings were adequate for a species with a small and linear home range. This approach may also be useful for a wider range of species which can be more easily detected by sign (e.g. burrows, tree scratches, dung, galleries, nests and exuviae) or use trails to move such as many large mammals [START_REF] Karanth | Monitoring carnivore populations at the landscape scale: occupancy modelling of tigers from sign surveys[END_REF][START_REF] Thorn | Brown hyaenas on roads: estimating carnivore occupancy and abundance using spatially auto-correlated sign survey replicates[END_REF][START_REF] Barber-Meyer | Influence of prey depletion and human disturbance on tiger occupancy in Nepal[END_REF] or flightless birds (e.g. cassowaries). It may also be appropriate to restrict surveys to roads or trails, for the sake of improving sign detection, particularly for species living in remote habitats, such as mountains (e.g. bears) or tropical forests (e.g. orang-utangs). Additionally, it would be interesting to test the Markovian occupancy model on other species using linear home ranges, for which traditional resource selection analyses are often inappropriate (e.g. [START_REF] Slaght | Home range and resource selection by animals constrained by linear habitat features: an example of Blakiston's fish owl[END_REF], such as species living along rivers (e.g. beavers, dippers, dragonflies), restricted to ecotones (e.g. edges, reeds, coastlines, continental slope) or living along hedges, cliffs or canyons.

Fig. 1 .

 1 Fig. 1. Study area and location of the sampling sites (black dots, N = 34). Grey lines indicate the stream network.

Fig. 2 .

 2 Fig. 2. Four different data configurations used in the analysis for one sampling site: (a) spatial and temporal replicates; (b) spatial replicates (i.e. only one sampling visit is studied, here the third one); (c) temporal replicates for segments (i.e. only one segment is studied, here the first one); and (d) temporal replicates for the site (i.e. data of the five segments are pooled). Data collected are detection (1) or non-detection (0) of faeces of the Pyrenean desman.

  compared with the top ranked model (DAIC) and AIC model weights (W) are given. For each data configuration, models are ranked according to their DAIC, the best model being the top ranked. No confidence intervals (CI) were derived when the estimate equalled ), h(.), p 0 (.), p 1

  probability without detection event on the previous occasion; p 1 : detection probability with detection event on the previous occasion; h: local occupancy probability (i.e. availability parameter); Ψ: occupancy probability; h 0 : presence probability with absence on the previous occasion; h 1 : presence probability with presence on the previous occasion.

Acknowledgements

We would like to thank V. Lacaze, E. Vas, A. Bodo, M.O. Durand, C. Fonty, V. Leenknegt, B. Le Roux & S. Perr e for the field work. We also thank J.E. Hines for his helpful comments concerning the Markovian occupancy model, as well as M. Ghosh and S. Danflous who corrected the English text. The study is funded by ANRT (Cifre no. 2011/1018), EDF (Electricit e de France) and European Union (FEDER) and is part of the French conservation Action Plan for the Pyrenean desman (2010-2015) supervised by DREAL MP (Direction R egionale pour l'Environnement, l'Am enagement et le Logement de Midi-Pyr en ees) and coordinated by the CEN MP (Conservatoire d'Espaces Naturels Midi-Pyr en ees).

Supporting Information

Additional Supporting Information may be found in the online version of this article.

Table S1. Model selection results for all temporal replicates and spatial replicates.

Table S2. Minimal occupancy and detectability estimates obtained for all temporal replicates and spatial replicates.

Table S3. Maximal occupancy and detectability estimates obtained for all temporal replicates and spatial replicates.