## Introduction

Visitation rate is a measure of the frequency with which animals visit specific locations or objects of interest. This information is important for many management and conservation issues and also for many ecological questions, such as human–wildlife conflicts, measuring risks of predation or disease transmission. For example, estimating visitation rates and their dependence on pond/landscape factors can be critically important to assessing the conflict between otters (*Lutra lutra*) and pond owners in commercial fish farms (Schwerdtner & Gruber 2007). Similarly, the attack rate of bears on sheep depends on feeding possibilities, vegetation features (Wilson *et al*. 2006) and individual type of bears. Further examples are visitation rates of predators to nesting sites [e.g. lizards on turtle nests (Doody *et al*. 2006) or foxes and crows to ground breeding birds (Summers *et al*. 2004)].

Examples where the frequency of interaction and its distribution is more important than absolute numbers of individuals include visitation frequency of pollinators in different habitats (Tylianakis, Tscharntke & Klein 2006), the frequency of bites of malaria-transmitting mosquitoes (Ye *et al*. 2007), transmission of tuberculosis by badgers (*Meles meles*) (Tuyttens *et al*. 2001) and the risk of foxes (*Vulpes vulpes*) spreading rabies (Webbon, Baker & Harris 2004). In general, visitation rates convey important information whenever the exposed objects or areas have a scale at which the dispersion of individuals or their spatial use matters more than their absolute number.

Ideally, visitation rates are measured by direct observation, video surveillance (Binner, Henle & Hagenguth 1996), scent stations (Sargeant, Johnson & Berg 2003) or radio-telemetry of all individuals that might visit a location of interest. Unfortunately, these approaches are time-consuming, expensive and impossible for many systems, especially for cryptic animals. Due to these difficulties, cryptic animals are studied frequently with the use of indirect methods, such as counting tracks, faeces, hairs, dens or nests (hereafter ‘signs’). Data based on signs of animals are used to monitor the distribution and abundance (for a review see Wilson & Delahay 2001), presence/absence (Binner, Henle & Hagenguth 1996; Macdonald & Mason 1987) and the visitation probability to particular locations and their relationships to landscape factors.

Current methods to estimate visitation rates for cryptic animals are based on repeated sampling of the site(s) of interest. Usually, the number with positive records (presences) of a sign divided by the total number of sampling occasions at that location is taken as an estimate of the probability of visitation (Rowe-Rowe 1992; Madsen & Prang 2001; Klenke 2002; Rostain *et al*. 2004). Note that a similar approach is used to estimate abundance of animals, but that in this case the detection of signs is analysed across multiple sites (Caughley 1980; Wilson & Delahay 2001) instead of separately for each location, as in the case of estimating site-specific visitation rates. This visitation rate estimator has been used extensively in the literature (Tuyttens *et al*. 2001; Sadlier *et al*. 2004; Prokesova, Barancekova & Homolka 2006).

The method in current use does not discriminate between aged and new signs in the estimation of visitation rates/probabilities. This means that in surveys in which it is possible to make such a distinction, information that could potentially improve estimates is neglected. Here, we propose a new maximum-likelihood approach, which uses additional information on the age of signs to improve the precision and efficiency of estimates. We show that the current estimation technique is a special case of our more general framework and thus our work unifies the new method with existing techniques under the same mathematical umbrella. We derive the new estimator in the context of a particular field survey design, quantify the performance of the method using simulated data sets and, finally, apply the method to an empirical example to highlight its utility. To identify study designs that allow the efficient estimation of visitation rates/probabilities, we quantified the performance of the new estimator across a wide range of different sampling schemes using simulated data.