## Introduction

A central question in ecology is how observed patterns in the spatial distribution of individuals within populations are determined by the interactions between individuals and their environment (Lima & Zollner 1996; Turchin 1998; Wiegand *et al*. 1999; Matthiopoulos 2003b). A useful approach to address this question is to understand the dynamics of animal movements in relation to state-dependent social and ecological factors (Hebblewhite *et al*. 2005; Whitehead & Rendell 2004). Most animals use the same areas repeatedly over time (Darwin 1861), hence animal movements are often defined using the home range concept (Tufto, Andersen & Linnell 1996; Crook 2004; Jetz *et al*. 2004; Anderson *et al*. 2005), where the home range is the area used by an animal over a given time interval (Burt 1943; White & Garrot 1990). Home range is characterized typically with descriptors of its size, shape and structure (Kenward 2001). In this paper we concentrate on measures of home range size, which is the most commonly used animal space use estimator in ecological research. The utility of the home range concept has been questioned, with various methodological issues concerning its estimation having recently been raised (White & Garrot 1990; Powell 2000; Kenward 2001; Kernohan, Gitzen & Millspaugh 2001).

Given these issues, it is surprising that few non-simulation-based studies have explored how home range estimation varies with method, sample size and sampling regime, and how this affects statistical inference from data. In addition, there is currently no method to quantify the contribution of each component of a sampling regime including the number of marked individuals, the number of sampling bouts and the number of locations to the variance in home range estimates. However, any method that enables such a variance decomposition could enable the efficient allocation of (usually) limited field resources to each component of the sampling variance. We present a quantitative approach that will facilitate this allocation of resources and which should help provide unbiased estimates of animal space use. The approach is applied to measures of home range size, but it can be used to address the effects of the sampling regime on other measures of animal home ranges, such as shape and structure.

We outline briefly the main methodological issues associated with the estimation of home range size: first, there is considerable discussion in the literature over the utility of the home range concept and the way in which home ranges should be measured. The whole field has been preoccupied until recently with the issues of autocorrelation of animal movements and the concept of time-to-independence (e.g. Swihart & Slade 1985; and see Powell 2000; Kenward 2001; Kernohan *et al*. 2001). However, this focus has recently been shown to be misguided: De Solla *et al*. (1999) and Blundell *et al*. (2001) demonstrated that removing autocorrelation removes the biological signal of interest, and Otis & White (1999) showed that the conclusions of the analyses of Swihart & Slade (1985) were based on a methodological error. In general, current work has illustrated that ‘the concept of time-to-independence or distance-to-independence is mistaken’ (Fortin & Dale 2005). In fact, recent studies have started to show how much biologically relevant information is contained in the autocorrelation structure of animal movements (Cushman, Chase & Griffin 2005).

As well as concerns about autocorrelation, other concerns have been raised about the utility of the home range approach. White & Garrot (1990) concluded their authoritative review by saying that ‘home range estimates are a poor substitute for good experimental protocol’, and Gautestad & Mysterud (1995) even questioned the existence of a measurable home range (see also Gautestad & Mysterud 2005). However, recent consensus suggests that this conclusion, based on the observation that home range area increases linearly with the number of sampled animal locations, is an artefact of the minimum convex polygon (MCP) approach they used (Powell 2000; Kenward 2001; see also Powell, Zimmerman & Seaman 1997). Despite inherent problems with the MCP method (Anderson 1982; White *et al*. 1990) it is still widely used, especially in comparative studies, even though kernel methods are now increasingly favoured (see also Gitzen & Millspaugh 2003; Matthiopoulos 2003a; Barg, Jones & Robertson 2005; Hines *et al*. 2005; Katajisto & Moilanen 2006).

A second problem is that research investigating the performance of home range estimation methods generally bases sampling recommendations solely on the minimum necessary number of fixes to calculate home range size (but see Girard *et al*. 2002), even though the sampling interval between fixes may influence home range estimates (Hansteen, Andreassen & Ims 1997). From a practical point of view, it would be useful to determine easily the most efficient combination of data collection protocol and home range estimation method applied to the data collected, as there is an inevitable trade-off between the number of locations sampled per individual animal and the number of animals monitored (Otis & White 1999). Most researchers addressing these issues used simulated data sets of animal locations (e.g. Worton 1995b; Burgman & Fox 2003; Getz & Wilmers 2004). These conclusions need to be verified with real animal location data (Seaman *et al*. 1999), as the problem of using simulated data is that the estimators may simply reflect the parametric distribution functions generating the simulated data (Kenward 2001; Horne & Garton 2006) and that computer-simulated data can be an inadequate representation of real animal space use behaviour (Blundell *et al*. 2001; Hemson *et al*. 2005).

Lastly, and perhaps most importantly, there is a general tendency in the literature on home range estimation methods to identify the ‘true’ home range size *per se*, or to search for increasing precision of estimates. However, often the primary question of interest is to understand the factors determining the variance within and between individuals in home range size. Generally, the natural experimental unit for longitudinal data (the category for most home range studies) is the sequence of measurements on an individual animal, and inferences are based on the partitioning of both the within- and among-individual variances (Diggle *et al*. 2002). Therefore, the fact that estimated home range size is not the ‘true’ value is of less importance than any artefact in the variance structure of the data, i.e. the ‘true’ variance differences within and between individuals (see also Carroll 2003).

In conclusion, besides technical improvements, the unsolved questions in animal home range studies are which methods perform best given the data and the research questions (Hemson *et al*. 2005), and how differences in the sampling regime affect statistical inferences. Horne & Garton (2006) recently presented a statistical approach, based on a new application of information–theoretic model selection procedures, that overcomes several of the limitations of traditional methods for evaluating which home range estimation methods perform best given the data. However, this approach does not consider how the relationships between sampling regime and home range estimation may affect subsequent statistical inferences on the home range estimates itself, as well as on the model selection procedure proposed by the authors, nor does it allow the identification of an efficient allocation of field resources to data collection. Our approach potentially allows improvements towards answering these questions. A variance components analysis using mixed effects models (Pinheiro & Bates 2000) allows the variance in home range size to be partitioned into contributions from each component of the sampling regime, allowing the best combination between estimation methods and sampling regime to be identified from data.

We applied this approach to a detailed data set of 19 736 fixes collected on 32 (16 males and 16 females) European roe deer *Capreolus capreolus* (Linnaeus 1758). We evaluated four home range estimation methods and four sampling regimes over four time-scales and with different utilization distribution (Worton 1989) density isopleths. We also investigated whether the results obtained from the roe deer were consistent within a second species with different life-history by using data from the kestrel *Falco tinnunculus* (Linnaeus 1758) as a second case study. We used 5201 fixes on 21 individuals (18 males, three females) from two different study areas, selected from a data set obtained from the long-term individual based time-series (LITS) project website (NERC Centre for Population Biology 2006).

We address three specific questions:

- 1What are the contributions of different components of the sampling regime to the total variance in home range size?
- 2Is it sufficient to standardize the number of fixes collected to obtain unbiased statistical inference?
- 3Which home range estimation method is the most efficient and unbiased, and which should be used for comparative studies?