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GPS measurement error and resource selection functions in a fragmented landscape

GPS measurement error and resource selection functions in a fragmented landscape With advances in technology, ecologists are able to apply relatively new techniques, such as remote observation of species occurrence or an individual organism's location, to important issues concerning the management of small or declining populations. Habitat loss and fragmentation have been cited as major contributors to the decline of many species ( Wilcox and Murphy 1985 , Saunders et al. 1991 , Huxel and Hastings 1999 ). Yet for many rare species, even a basic understanding of species distributions, or of which resources and habitats are required to maintain a persistent population, are unknown. In situations where preferred habitats are not contiguous and exist as small patches, there is often a need to prioritize areas for conservation ( Palmonares 2001 , Dickson and Beier 2002 ). Determining where to focus management effort is often species‐specific and reliant on species‐habitat relationships ( Telleria and Santos 1995 , Shochat and Tsurim 2004 ). Management of many species is therefore driven by what researchers know about the associations between a particular species and the landscapes they inhabit. In most cases these associations are derived from some form of species distribution modeling or habitat selection studies linking the presence of individuals to the prevailing landscape conditions ( Dickson and Beier 2002 , Gavashelishvili 2004 ). However, in some cases the habitat fragmentation experienced by local populations forces the remaining individuals into small patches containing preferred habitat or into secondary habitat. When this occurs, remote observations of individual occurrences have an increased possibility of being misclassified due to the spatial proximity to adjacent habitat types. Measurement error, the difference in the remotely observed location from truth, can be a major cause of this misclassification. Modeling habitat associations and species distributions can be done with a wide variety of methods and models ( Boyce and McDonald 1999 , Austin 2002 , Guisan et al. 2002 , Lehmann et al. 2002 , Guisan and Thuiller 2005 ). Each method, however, uses data collected on occurrences or observations of a particular organism's spatial location. These locations, if remotely observed, introduce measurement error that can potentially influence the conclusions drawn from the models ( Hunsaker et al. 2001 ). To illustrate the influence measurement error may have on the conclusions drawn from models of species distributions and individual organism's habitat selection, I use resource selection functions (RSFs) generated from hypothetical data representing data remotely obtained by GPS radiotelemetry. While RSFs are a single example of a modeling method used to describe species‐habitat associations, the general results of this study hold for other modeling techniques when measurement error of remotely observed locations results in misclassification. A resource selection function is any function used to measure selection by an individual proportional to available resources, including habitat ( Manly et al. 2002 ). They provide researchers with the ability to link spatial locations and frequency of use to resources that may be of particular importance for an organism ( Boyce and McDonald 1999 , Manly et al. 2002 ). The ability of RSFs to inform conservation efforts has been shown for various organisms and spatial scales ( Edwards et al. 1996 , Meyer et al. 1998 , Johnson et al. 2002 , 2004 ). However, in none of these cases were the implications of measurement error on inference discussed (but see Rettie and McLoughlin 1999 and Staples et al. 2004 ). In this simulation, I highlight the tradeoff between habitat fragmentation and measurement error, and determine the robustness of error distributions to a loss of inference in selection. Lastly, I determine how measurement error effects selection strength under differing initial selection strengths, spanning the range from a habitat generalist to habitat specialist. Methods Landscape and error free locations An approximately 35 km 2 landscape located in the west central foothills region of Alberta Canada was reclassified to be dichotomous, representing hypothetical preferred habitat (1) and non‐selected matrix (0) based on realistic cover types. The proportion of the landscape that was preferred accounted for 23.7% of the area. The spatial location of a hypothetical individual was simulated for 500 locations across the landscape. The location of these points within the preferred habitat and matrix was controlled by a binomial probability function based on the habitat type, allowing the selection strength for the preferred habitat to be controlled. Four levels of selection were simulated ranging from low to high selection strength (based on the proportion of total points found in the preferred habitat). Thus simulated selection strength increased as the probability of locating an observation in the preferred habitat approached 1. Additionally, approximately no selection was simulated: in this case the hypothetical individual was distributed in accordance to the proportion of the habitats in the landscape. These points represent “use” locations as would be derived from GPS telemetry, and were assumed to have no associated location error (i.e. they represented truth). An additional 500 points were randomly distributed across the landscape, with no bias, these points represented the “availability” points required to calculate a RSF ( Manly et al. 2002 ). As in all RSF studies, availability points were generated by randomly selecting exact coordinates and were assumed to have no location error. Measurement error and fragmentation To investigate the influence of measurement error on habitat selection, measurement error was introduced to the 500 use points in incrementally larger amounts. It is assumed that there is no map error in order to limit measurement error to that due to a GPS collar. Measurement error was imposed using three different approximate distributions truncated at 5 standard deviations; uniform, bivariate normal, and exponential (or Laplace) assuming no directional bias in error (but see Frair et al. 2004 ). These distributions were selected as they are realistic (bivariate normal and exponential) in terms of the distribution of GPS collar location error or are often used in RSF studies to quantify the immediate conditions around a location or account for error (uniform, as a buffer around a point location; Zimmerman and Powell 1995 , Kenow et al. 2001 , Dickson and Beier 2002 , McLoughlin et al. 2002 , Jerde and Visscher 2005 ). At each level of measurement error, for each distribution, 100 realizations (two times that recommended by Samuel and Kenow 1992 ) were located within the corresponding probability density function (PDF; Fig. 1 ). 1 Probability density functions (A) of the distributions used to add error to truth locations in the simulation. The exponential (or Laplace) distribution is given by the short dashed black line, the bivariate normal by the solid light gray line, and the uniform distribution by the long dashed dark gray line. The dot indicates the true location. Theoretical schematic representing the possibility of misclassifying truth given heterogeneity in the landscape (B). The circle indicates the area integrated to ±5 standard deviation of the error distribution within which locations (realizations) could be misclassified (x's) from truth (dot). Misclassification and switching from the preferred habitat (white) to matrix (grey) results in reduced selection. Error was added as increasing multiples of measurement error. Unlike many studies which record error in the natural units of GPS locations, meters, I employed error standard deviations. The error standard deviations units are independent of map units (meters) and are relative to the error associated with the GPS collar, a metric dependent on the specific error associated with a spatial observation or GPS collar (see Jerde and Visscher 2005 for further explanation). For example, if a collar has 50 m of error at one standard deviation then 2 error standard deviations equates to 100 m. Therefore, this metric allows tailoring to the error associated with each researcher's GPS collars and situation. In the example described below, measurement error was added in multiples corresponding with the pixel size of the habitat map. Measurement error was introduced to all use points by allowing the habitat classification to vary from truth within the error associated with each distribution's PDF ( Fig. 1 ). The probability of being correctly classified between the preferred and matrix habitat was calculated to be proportional to the amount of matrix within the circular area weighted by the uniform, normal or exponential distribution of error. A Bernoulli trial was then conducted with a random number generator between 0 and 1. Regardless of the true location, if the value of the random number was less than the proportion of matrix within the error distribution, the point was assumed to be in the matrix. If greater, it was assumed to be in the preferred habitat type. This was repeated 100 times for each of the 500 use points thus creating data with introduced error for comparing selectivity to truth using RSFs. The proportion of preferred habitat was quantified within each error PDF at all error increments surrounding the points generated to be moderately selective. The coefficient of variability (CV) was calculated as: 1 where σ is the standard deviation of the proportion of the error circle that is preferred habitat, and is the mean proportion of preferred habitat within the error circle for the 500 use points. As is common, the CV was multiplied by 100 to express (and graph) it as a percentage. The CV was calculated for each radius of measurement error. Areas that contained a single homogeneous habitat type of either preferred or matrix were given a value of 1 since the introduction of error would have no effect on these points. Thus as the landscape becomes more homogeneous the approaches 1 as σ goes to 0 causing the CV to approach 0 (0 being a landscape of only a single habitat type). This index to fragmentation adequately describes the variation in habitat types found within the error distributions by indexing the possibility for a point to be misclassified into the wrong habitat. This is the primary interest of this study ( Fig. 1 ). Resource selection Selection for the preferred habitat was calculated comparing use locations to random available locations (type III design, Manly et al. 2002 ) using binary logit regression calculated as: 2 where selection, ω(x) , for some covariate, x , is measured by the coefficient, β . For this simulation this equation was simplified to only look at the selection for a single covariate, x = {0 matrix, 1 preferred}, habitat type as a dichotomous variable. This model was fit to the use points with no measurement error as a measure of truth for comparison purposes. The model was then fit to the realizations of the use locations with introduced measurement error from each of the three distributions. In all cases the available points were the same and were considered to contain no location error. For each increment of measurement error, in each of the distributions, 100 RSFs were calculated and the average β value (and 95% confidence interval of the distribution) calculated. For comparison purposes the results were displayed as: 3 where β * is the proportion of true selection (i.e. error free) remaining, observed selection calculated as the mean β of the 100 RSF calculated using the realizations within the error distributions for the preferred habitat, and β t expected selection calculated from truth (no error). For illustrative purposes the comparison of the three distributions is restricted to the points generated to be moderately selective, while the exponential distribution was used to compare the impact of the strength of selection and measurement error on the loss of inference from truth. All spatial analysis was done in ArcMap ver. 9.0 (ESRI 2004) while habitat modeling was conducted in Stata ver. 8.0 (Stata Corporation 2003). Results The variability (CV) in habitat composition, plotted for the moderately selective individual, increased to a maximum with increasing measurement error ( Fig. 2 ). The trend was nearly identical for each of the three distributions and levels of selectivity. This intuitive result reflects the inclusion of more habitats into an area and the increased possibility of misclassifying a location ( Fig. 1 ). 2 The results of the simulations taking into account the influence on measurement error (error standard deviations), relative to minimum pixel size, on the degree of fragmentation, as indexed by the coefficient of variability (CV) in the proportion of error distribution containing preferred habitat. The CV calculated under three approximate distributions; uniform (circles and solid lines), bivariate normal (triangles and dotted lines), and exponential (squares and dashed lines). Truth is considered to be 0 error standard deviations and values were calculated for the moderately selective case. The accuracy of inferences on resource selection relative to truth is a tradeoff between GPS measurement error and the inherent fragmentation and complexity of the landscape. Large amounts of measurement error results in the observed location being misclassified from the habitat it was truly found in. This resulted in a reduced (from truth) degree of selection for the preferred habitat for the moderately selective case ( Fig. 3 ). While the observed selection from the uniform distribution simulations was much reduced from truth with the introduction of error, both the exponential and bivariate normal distribution resulted in similar rates of decrease in selection. In the case of the largest measurement error simulated for the uniform distribution, selection was reduced by 38% suggesting that buffering points (uniform distribution) is not a good option for dealing with location error ( Fig. 3 ). Both the exponential and bivariate normal distribution are more robust to this decrease in selection but a sizeable decrease in selection of 23% remains. Under the exponential distribution, the effect of increasing measurement error is proportionally stronger when the degree of selectivity is higher, as indexed by the initial odds ratio ( Fig. 4 ). Therefore measurement error is a particularly important consideration for researchers investigating the selection patterns of habitat specialists inhabiting small habitat patches. The odds ratio is the exponentiated β coefficient and indexes how many times more the preferred habitat is selected in relation to the matrix (e.g. an odds ratio of 4 indicates the preferred habitat is selected 4 times more than the matrix). While for graphical reasons the 95% confidence intervals were not given for the hypothetical case where selection was approximately random (odd ratio=1.02) the proportion of selection remaining did fluctuate (0.88–1.11) corresponding to a range in odds ratios of 0.89 (suggesting avoidance) to 1.13 (suggesting selection) for the largest amount of measurement error simulated. Even at the smallest error simulated for this case the potential for switching selection was present (95% CI range from 0.94 to 1.03). 3 The results of the simulations taking into account the influence on measurement error (error standard deviations) on selection inference, as indexed by the proportion of the original “true” selection remaining (β* and 95% distribution interval) for the preferred habitat due to the possibility of misidentifying the habitat in which the location may have occurred. Error was estimated using three approximate distributions; uniform (circles and solid lines), bivariate normal (triangles and dotted lines), and exponential (squares and dashed lines). Truth is considered to be 0 error standard deviations and values were calculated for the moderately selective case. 4 The results of the simulations showing the influence of measurement error (error standard deviations) and selection strength (initial condition given as a odds ratio on the right side) on the proportion of the original “true” selection remaining (β * and 95% distribution interval, dotted line) using an exponential distribution. Truth is considered to be 0 error standard deviations. Figure symbols indicate the error free selection strength of a hypothetical organism. Diamonds indicate approximately no selection (habitat generalist; odds ratio=1.02), circles represent low selection (odds ratio=1.52), triangles indicate moderate selection (odds ratio=2.32), and squares a highly selective organism (habitat specialist; odds ratio=10.07). Discussion When models are used to interpret the ecological context for the location of particular species or individuals from remotely observed data the possibility for incorrect interpretations can occur. This can be due to the misclassification of species‐habitat interactions from location error (as in this study), map error, and classification error of the underlying landscape (or vegetation type; Hunsaker et al. 2001 ). In all cases, the failure to incorporate, investigate, or acknowledge these forms of measurement error can result in the misidentification of important habitats or resources. In the example given, the difference between true and observed selection results in a reduced estimation of true habitat selection strength. The introduction of measurement error into spatial models causes the observed selection strength to change by misclassifying the true underlying habitat type. This misclassification, however, is accelerated through landscape fragmentation. Increasing fragmentation puts different habitat types in close proximity, allowing for increased misclassification of the spatial context of the true location of the point. The tradeoff between the degree of fragmentation and spatial accuracy, and the resultant conclusions drawn from spatial models, can allow researchers to tailor sampling protocols to the species in question. For instance, a model created for a landscape dominated by a single habitat type (approaching homogeneous) will be relatively unaffected by even a large amount of measurement error. Conversely, in a highly fragmented landscape or a landscape composed of a large number of habitat types, there is a need for high spatial accuracy (low measurement error) in order for the conclusions to represent truth. While the results may seem intuitive, the impact of measurement error has not, to date, been taken into account and studies continue without consideration of the consequences of measurement error. Measurement error could have been introduced using other distributions. However, the bivariate normal and exponential distributions are realistic in terms of the distribution of GPS collar location error ( Zimmerman and Powell 1995 , Kenow et al. 2001 ) and the uniform distribution, as a buffer around a point location, is often used in RSF studies to quantify the immediate conditions around a location ( Dickson and Beier 2002 , McLoughlin et al. 2002 ). Both the bivariate normal and exponential have very similar results in terms of the reduction in observed selection strength. This suggests the introduction of error has less to do with the type of distribution than the “peakedness” or level of variance of the distribution (controlled by ? in the exponential and σ in the bivariate normal). A buffer around a location (a uniform distribution of error) is simple to characterize, and is often used to accommodate error ( Dickson and Beier 2002 , McLoughlin et al. 2002 ). This study, however, has shown that a uniform error distribution may lead to a decrease in selection approximately double that of a normal or exponential distribution and is likely unsuitable for approximating the distribution of measurement error in a GPS collar. This study highlights a number of issues researchers should consider when drawing conclusions from models when the landscape is highly fragmented and/or measurement error is large. In the example given above, a reduction in the selection coefficient may be only a reduction in statistical power. However, the chance that a coefficient may change sign (from selection to avoidance) is a real possibility given the range of reduction in selection observed in this study. The probability of making an incorrect conclusion is increased when the landscape is fragmented or selection is weak. Not surprisingly, this change in selection from selected to avoided was only witnessed in the situation when the initial selection was extremely weak (odds ratio=1.02). The possibility of observing reduced selection strength (from truth), however, is more likely when the true selection is strong. This is because a strongly selective individual (a habitat specialist) living in small or isolated patches of preferred habitat will necessarily be located adjacent to the matrix. This problem will increase as the total proportion of the landscape that is preferred habitat declines or become increasingly fragmented into patch sizes under the minimum resolution of the measurement error. The resultant considerations must be: what reduction from truth are we willing to accept and to what level do we need to reduce measurement error Consider the case presented here ( Fig. 4 ): assume we are willing to accept an observed reduction in selection of ca 5% (not unlike a standard statistical acceptance (α) level), which is indicated by the solid line bisecting the curved confidence intervals in Fig. 4 . In this case we would require a level of accuracy which equates to where the solid line bisects the curve of the initial selection strength our individual shows for the preferred habitat. For example, if our individual is very selective (odds ratio=10.07) we require accuracy of under 1 error standard deviations. However, if our individual is a habitat generalist and only slightly selects the preferred habitat (odds ratio=1.52) then we require an accuracy of approximately 3.5 error standard deviations. This simple example highlights the “rules of thumb” that can be derived from these simple relationships. However, the tradeoff between measurement error, habitat fragmentation and inferences on selection is a general result which must be considered in species‐specific conservation planning. Researchers employing remotely observed data must remember that these data are prone to error. While for simplicity's sake remote observations are considered truth, the interpretations and conclusions from models may change when location error or fragmentation are taken into account. Researchers should understand that habitat association and species distribution models calculated from remotely observed data are a single realization from a distribution of measurement error. The consequences of both measurement error and fragmentation on resource or habitat selection inference is of particular importance for the conservation of rare or threatened species which exist in remnant patches throughout a fragmented landscape. Conservation actions arising from models of species distributions and habitat associations are questionable unless both fragmentation and measurement error are accounted for. Acknowledgements Special thanks to C. Jerde for thoughtful discussions on measurement error and comments on previous drafts and the Merrill lab for commenting on an early presentation on this topic. P. DeWitt, M. Araújo, and L. Kowolik also provided helpful comments on an earlier draft. Funding was provided by the Univ. of Alberta's Dep of Biological Science. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Ecography Wiley

GPS measurement error and resource selection functions in a fragmented landscape

R. Visscher, Darcy
Ecography , Volume 29 (3) – Jun 1, 2006
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Wiley
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Copyright © 2006 Wiley Subscription Services, Inc., A Wiley Company
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0906-7590
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1600-0587
DOI
10.1111/j.0906-7590.2006.04648.x
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Abstract

With advances in technology, ecologists are able to apply relatively new techniques, such as remote observation of species occurrence or an individual organism's location, to important issues concerning the management of small or declining populations. Habitat loss and fragmentation have been cited as major contributors to the decline of many species ( Wilcox and Murphy 1985 , Saunders et al. 1991 , Huxel and Hastings 1999 ). Yet for many rare species, even a basic understanding of species distributions, or of which resources and habitats are required to maintain a persistent population, are unknown. In situations where preferred habitats are not contiguous and exist as small patches, there is often a need to prioritize areas for conservation ( Palmonares 2001 , Dickson and Beier 2002 ). Determining where to focus management effort is often species‐specific and reliant on species‐habitat relationships ( Telleria and Santos 1995 , Shochat and Tsurim 2004 ). Management of many species is therefore driven by what researchers know about the associations between a particular species and the landscapes they inhabit. In most cases these associations are derived from some form of species distribution modeling or habitat selection studies linking the presence of individuals to the prevailing landscape conditions ( Dickson and Beier 2002 , Gavashelishvili 2004 ). However, in some cases the habitat fragmentation experienced by local populations forces the remaining individuals into small patches containing preferred habitat or into secondary habitat. When this occurs, remote observations of individual occurrences have an increased possibility of being misclassified due to the spatial proximity to adjacent habitat types. Measurement error, the difference in the remotely observed location from truth, can be a major cause of this misclassification. Modeling habitat associations and species distributions can be done with a wide variety of methods and models ( Boyce and McDonald 1999 , Austin 2002 , Guisan et al. 2002 , Lehmann et al. 2002 , Guisan and Thuiller 2005 ). Each method, however, uses data collected on occurrences or observations of a particular organism's spatial location. These locations, if remotely observed, introduce measurement error that can potentially influence the conclusions drawn from the models ( Hunsaker et al. 2001 ). To illustrate the influence measurement error may have on the conclusions drawn from models of species distributions and individual organism's habitat selection, I use resource selection functions (RSFs) generated from hypothetical data representing data remotely obtained by GPS radiotelemetry. While RSFs are a single example of a modeling method used to describe species‐habitat associations, the general results of this study hold for other modeling techniques when measurement error of remotely observed locations results in misclassification. A resource selection function is any function used to measure selection by an individual proportional to available resources, including habitat ( Manly et al. 2002 ). They provide researchers with the ability to link spatial locations and frequency of use to resources that may be of particular importance for an organism ( Boyce and McDonald 1999 , Manly et al. 2002 ). The ability of RSFs to inform conservation efforts has been shown for various organisms and spatial scales ( Edwards et al. 1996 , Meyer et al. 1998 , Johnson et al. 2002 , 2004 ). However, in none of these cases were the implications of measurement error on inference discussed (but see Rettie and McLoughlin 1999 and Staples et al. 2004 ). In this simulation, I highlight the tradeoff between habitat fragmentation and measurement error, and determine the robustness of error distributions to a loss of inference in selection. Lastly, I determine how measurement error effects selection strength under differing initial selection strengths, spanning the range from a habitat generalist to habitat specialist. Methods Landscape and error free locations An approximately 35 km 2 landscape located in the west central foothills region of Alberta Canada was reclassified to be dichotomous, representing hypothetical preferred habitat (1) and non‐selected matrix (0) based on realistic cover types. The proportion of the landscape that was preferred accounted for 23.7% of the area. The spatial location of a hypothetical individual was simulated for 500 locations across the landscape. The location of these points within the preferred habitat and matrix was controlled by a binomial probability function based on the habitat type, allowing the selection strength for the preferred habitat to be controlled. Four levels of selection were simulated ranging from low to high selection strength (based on the proportion of total points found in the preferred habitat). Thus simulated selection strength increased as the probability of locating an observation in the preferred habitat approached 1. Additionally, approximately no selection was simulated: in this case the hypothetical individual was distributed in accordance to the proportion of the habitats in the landscape. These points represent “use” locations as would be derived from GPS telemetry, and were assumed to have no associated location error (i.e. they represented truth). An additional 500 points were randomly distributed across the landscape, with no bias, these points represented the “availability” points required to calculate a RSF ( Manly et al. 2002 ). As in all RSF studies, availability points were generated by randomly selecting exact coordinates and were assumed to have no location error. Measurement error and fragmentation To investigate the influence of measurement error on habitat selection, measurement error was introduced to the 500 use points in incrementally larger amounts. It is assumed that there is no map error in order to limit measurement error to that due to a GPS collar. Measurement error was imposed using three different approximate distributions truncated at 5 standard deviations; uniform, bivariate normal, and exponential (or Laplace) assuming no directional bias in error (but see Frair et al. 2004 ). These distributions were selected as they are realistic (bivariate normal and exponential) in terms of the distribution of GPS collar location error or are often used in RSF studies to quantify the immediate conditions around a location or account for error (uniform, as a buffer around a point location; Zimmerman and Powell 1995 , Kenow et al. 2001 , Dickson and Beier 2002 , McLoughlin et al. 2002 , Jerde and Visscher 2005 ). At each level of measurement error, for each distribution, 100 realizations (two times that recommended by Samuel and Kenow 1992 ) were located within the corresponding probability density function (PDF; Fig. 1 ). 1 Probability density functions (A) of the distributions used to add error to truth locations in the simulation. The exponential (or Laplace) distribution is given by the short dashed black line, the bivariate normal by the solid light gray line, and the uniform distribution by the long dashed dark gray line. The dot indicates the true location. Theoretical schematic representing the possibility of misclassifying truth given heterogeneity in the landscape (B). The circle indicates the area integrated to ±5 standard deviation of the error distribution within which locations (realizations) could be misclassified (x's) from truth (dot). Misclassification and switching from the preferred habitat (white) to matrix (grey) results in reduced selection. Error was added as increasing multiples of measurement error. Unlike many studies which record error in the natural units of GPS locations, meters, I employed error standard deviations. The error standard deviations units are independent of map units (meters) and are relative to the error associated with the GPS collar, a metric dependent on the specific error associated with a spatial observation or GPS collar (see Jerde and Visscher 2005 for further explanation). For example, if a collar has 50 m of error at one standard deviation then 2 error standard deviations equates to 100 m. Therefore, this metric allows tailoring to the error associated with each researcher's GPS collars and situation. In the example described below, measurement error was added in multiples corresponding with the pixel size of the habitat map. Measurement error was introduced to all use points by allowing the habitat classification to vary from truth within the error associated with each distribution's PDF ( Fig. 1 ). The probability of being correctly classified between the preferred and matrix habitat was calculated to be proportional to the amount of matrix within the circular area weighted by the uniform, normal or exponential distribution of error. A Bernoulli trial was then conducted with a random number generator between 0 and 1. Regardless of the true location, if the value of the random number was less than the proportion of matrix within the error distribution, the point was assumed to be in the matrix. If greater, it was assumed to be in the preferred habitat type. This was repeated 100 times for each of the 500 use points thus creating data with introduced error for comparing selectivity to truth using RSFs. The proportion of preferred habitat was quantified within each error PDF at all error increments surrounding the points generated to be moderately selective. The coefficient of variability (CV) was calculated as: 1 where σ is the standard deviation of the proportion of the error circle that is preferred habitat, and is the mean proportion of preferred habitat within the error circle for the 500 use points. As is common, the CV was multiplied by 100 to express (and graph) it as a percentage. The CV was calculated for each radius of measurement error. Areas that contained a single homogeneous habitat type of either preferred or matrix were given a value of 1 since the introduction of error would have no effect on these points. Thus as the landscape becomes more homogeneous the approaches 1 as σ goes to 0 causing the CV to approach 0 (0 being a landscape of only a single habitat type). This index to fragmentation adequately describes the variation in habitat types found within the error distributions by indexing the possibility for a point to be misclassified into the wrong habitat. This is the primary interest of this study ( Fig. 1 ). Resource selection Selection for the preferred habitat was calculated comparing use locations to random available locations (type III design, Manly et al. 2002 ) using binary logit regression calculated as: 2 where selection, ω(x) , for some covariate, x , is measured by the coefficient, β . For this simulation this equation was simplified to only look at the selection for a single covariate, x = {0 matrix, 1 preferred}, habitat type as a dichotomous variable. This model was fit to the use points with no measurement error as a measure of truth for comparison purposes. The model was then fit to the realizations of the use locations with introduced measurement error from each of the three distributions. In all cases the available points were the same and were considered to contain no location error. For each increment of measurement error, in each of the distributions, 100 RSFs were calculated and the average β value (and 95% confidence interval of the distribution) calculated. For comparison purposes the results were displayed as: 3 where β * is the proportion of true selection (i.e. error free) remaining, observed selection calculated as the mean β of the 100 RSF calculated using the realizations within the error distributions for the preferred habitat, and β t expected selection calculated from truth (no error). For illustrative purposes the comparison of the three distributions is restricted to the points generated to be moderately selective, while the exponential distribution was used to compare the impact of the strength of selection and measurement error on the loss of inference from truth. All spatial analysis was done in ArcMap ver. 9.0 (ESRI 2004) while habitat modeling was conducted in Stata ver. 8.0 (Stata Corporation 2003). Results The variability (CV) in habitat composition, plotted for the moderately selective individual, increased to a maximum with increasing measurement error ( Fig. 2 ). The trend was nearly identical for each of the three distributions and levels of selectivity. This intuitive result reflects the inclusion of more habitats into an area and the increased possibility of misclassifying a location ( Fig. 1 ). 2 The results of the simulations taking into account the influence on measurement error (error standard deviations), relative to minimum pixel size, on the degree of fragmentation, as indexed by the coefficient of variability (CV) in the proportion of error distribution containing preferred habitat. The CV calculated under three approximate distributions; uniform (circles and solid lines), bivariate normal (triangles and dotted lines), and exponential (squares and dashed lines). Truth is considered to be 0 error standard deviations and values were calculated for the moderately selective case. The accuracy of inferences on resource selection relative to truth is a tradeoff between GPS measurement error and the inherent fragmentation and complexity of the landscape. Large amounts of measurement error results in the observed location being misclassified from the habitat it was truly found in. This resulted in a reduced (from truth) degree of selection for the preferred habitat for the moderately selective case ( Fig. 3 ). While the observed selection from the uniform distribution simulations was much reduced from truth with the introduction of error, both the exponential and bivariate normal distribution resulted in similar rates of decrease in selection. In the case of the largest measurement error simulated for the uniform distribution, selection was reduced by 38% suggesting that buffering points (uniform distribution) is not a good option for dealing with location error ( Fig. 3 ). Both the exponential and bivariate normal distribution are more robust to this decrease in selection but a sizeable decrease in selection of 23% remains. Under the exponential distribution, the effect of increasing measurement error is proportionally stronger when the degree of selectivity is higher, as indexed by the initial odds ratio ( Fig. 4 ). Therefore measurement error is a particularly important consideration for researchers investigating the selection patterns of habitat specialists inhabiting small habitat patches. The odds ratio is the exponentiated β coefficient and indexes how many times more the preferred habitat is selected in relation to the matrix (e.g. an odds ratio of 4 indicates the preferred habitat is selected 4 times more than the matrix). While for graphical reasons the 95% confidence intervals were not given for the hypothetical case where selection was approximately random (odd ratio=1.02) the proportion of selection remaining did fluctuate (0.88–1.11) corresponding to a range in odds ratios of 0.89 (suggesting avoidance) to 1.13 (suggesting selection) for the largest amount of measurement error simulated. Even at the smallest error simulated for this case the potential for switching selection was present (95% CI range from 0.94 to 1.03). 3 The results of the simulations taking into account the influence on measurement error (error standard deviations) on selection inference, as indexed by the proportion of the original “true” selection remaining (β* and 95% distribution interval) for the preferred habitat due to the possibility of misidentifying the habitat in which the location may have occurred. Error was estimated using three approximate distributions; uniform (circles and solid lines), bivariate normal (triangles and dotted lines), and exponential (squares and dashed lines). Truth is considered to be 0 error standard deviations and values were calculated for the moderately selective case. 4 The results of the simulations showing the influence of measurement error (error standard deviations) and selection strength (initial condition given as a odds ratio on the right side) on the proportion of the original “true” selection remaining (β * and 95% distribution interval, dotted line) using an exponential distribution. Truth is considered to be 0 error standard deviations. Figure symbols indicate the error free selection strength of a hypothetical organism. Diamonds indicate approximately no selection (habitat generalist; odds ratio=1.02), circles represent low selection (odds ratio=1.52), triangles indicate moderate selection (odds ratio=2.32), and squares a highly selective organism (habitat specialist; odds ratio=10.07). Discussion When models are used to interpret the ecological context for the location of particular species or individuals from remotely observed data the possibility for incorrect interpretations can occur. This can be due to the misclassification of species‐habitat interactions from location error (as in this study), map error, and classification error of the underlying landscape (or vegetation type; Hunsaker et al. 2001 ). In all cases, the failure to incorporate, investigate, or acknowledge these forms of measurement error can result in the misidentification of important habitats or resources. In the example given, the difference between true and observed selection results in a reduced estimation of true habitat selection strength. The introduction of measurement error into spatial models causes the observed selection strength to change by misclassifying the true underlying habitat type. This misclassification, however, is accelerated through landscape fragmentation. Increasing fragmentation puts different habitat types in close proximity, allowing for increased misclassification of the spatial context of the true location of the point. The tradeoff between the degree of fragmentation and spatial accuracy, and the resultant conclusions drawn from spatial models, can allow researchers to tailor sampling protocols to the species in question. For instance, a model created for a landscape dominated by a single habitat type (approaching homogeneous) will be relatively unaffected by even a large amount of measurement error. Conversely, in a highly fragmented landscape or a landscape composed of a large number of habitat types, there is a need for high spatial accuracy (low measurement error) in order for the conclusions to represent truth. While the results may seem intuitive, the impact of measurement error has not, to date, been taken into account and studies continue without consideration of the consequences of measurement error. Measurement error could have been introduced using other distributions. However, the bivariate normal and exponential distributions are realistic in terms of the distribution of GPS collar location error ( Zimmerman and Powell 1995 , Kenow et al. 2001 ) and the uniform distribution, as a buffer around a point location, is often used in RSF studies to quantify the immediate conditions around a location ( Dickson and Beier 2002 , McLoughlin et al. 2002 ). Both the bivariate normal and exponential have very similar results in terms of the reduction in observed selection strength. This suggests the introduction of error has less to do with the type of distribution than the “peakedness” or level of variance of the distribution (controlled by ? in the exponential and σ in the bivariate normal). A buffer around a location (a uniform distribution of error) is simple to characterize, and is often used to accommodate error ( Dickson and Beier 2002 , McLoughlin et al. 2002 ). This study, however, has shown that a uniform error distribution may lead to a decrease in selection approximately double that of a normal or exponential distribution and is likely unsuitable for approximating the distribution of measurement error in a GPS collar. This study highlights a number of issues researchers should consider when drawing conclusions from models when the landscape is highly fragmented and/or measurement error is large. In the example given above, a reduction in the selection coefficient may be only a reduction in statistical power. However, the chance that a coefficient may change sign (from selection to avoidance) is a real possibility given the range of reduction in selection observed in this study. The probability of making an incorrect conclusion is increased when the landscape is fragmented or selection is weak. Not surprisingly, this change in selection from selected to avoided was only witnessed in the situation when the initial selection was extremely weak (odds ratio=1.02). The possibility of observing reduced selection strength (from truth), however, is more likely when the true selection is strong. This is because a strongly selective individual (a habitat specialist) living in small or isolated patches of preferred habitat will necessarily be located adjacent to the matrix. This problem will increase as the total proportion of the landscape that is preferred habitat declines or become increasingly fragmented into patch sizes under the minimum resolution of the measurement error. The resultant considerations must be: what reduction from truth are we willing to accept and to what level do we need to reduce measurement error Consider the case presented here ( Fig. 4 ): assume we are willing to accept an observed reduction in selection of ca 5% (not unlike a standard statistical acceptance (α) level), which is indicated by the solid line bisecting the curved confidence intervals in Fig. 4 . In this case we would require a level of accuracy which equates to where the solid line bisects the curve of the initial selection strength our individual shows for the preferred habitat. For example, if our individual is very selective (odds ratio=10.07) we require accuracy of under 1 error standard deviations. However, if our individual is a habitat generalist and only slightly selects the preferred habitat (odds ratio=1.52) then we require an accuracy of approximately 3.5 error standard deviations. This simple example highlights the “rules of thumb” that can be derived from these simple relationships. However, the tradeoff between measurement error, habitat fragmentation and inferences on selection is a general result which must be considered in species‐specific conservation planning. Researchers employing remotely observed data must remember that these data are prone to error. While for simplicity's sake remote observations are considered truth, the interpretations and conclusions from models may change when location error or fragmentation are taken into account. Researchers should understand that habitat association and species distribution models calculated from remotely observed data are a single realization from a distribution of measurement error. The consequences of both measurement error and fragmentation on resource or habitat selection inference is of particular importance for the conservation of rare or threatened species which exist in remnant patches throughout a fragmented landscape. Conservation actions arising from models of species distributions and habitat associations are questionable unless both fragmentation and measurement error are accounted for. Acknowledgements Special thanks to C. Jerde for thoughtful discussions on measurement error and comments on previous drafts and the Merrill lab for commenting on an early presentation on this topic. P. DeWitt, M. Araújo, and L. Kowolik also provided helpful comments on an earlier draft. Funding was provided by the Univ. of Alberta's Dep of Biological Science.

Journal

Ecography – Wiley

Published: Jun 1, 2006

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