Niche Modeling: Predictions From Statistical Distributions - Chapter 7

Chapter 7 Bias A bias is generally understood as a prejudice in the sense for having a predilec-tion to one particular point of view or ideological perspective. Statistical bias is a quantifiable tendency of a statistical estimator to over or underestimate the quantity that is being estimated. In formal terms, if θ is an estimator of a parameter θ, then θ is biased if the expected value of θ, E(θ), is not equal to θ. Bias is a concern when any process is supposed to be objective and fair. Similarly statistical bias of an estimator should be of concern, as that bias can influence results and undermine the objectivity of conclusions. For example, systematic errors increase the likelihood of assigning statisti-cal significance to chance events (Type I error). Given an hypothesis H0 : θ = θ and the alternative Ha : θ = θ then bias would tend to increase the chance of rejecting H0 when H0 is true, or committing a Type I error. Bias is more insidious than uncertainty introduced by small samples. For example, errors in estimation of a mean value are generally symmetric, that is, equally as likely to be above or below the actual value. Thus while a small sample introduces uncertainty, it does not necessarily bias the result. The detection of bias in modeling should be a primary concern. While in-creasing the sample size can reduce uncertainty in symmetric errors, averaging is not a solution to asymmetric errors. The variation in symmetric sampling errors reduces our power to detect statistical patterns in data (Type II error). Type II errors only lead us to reserve judgment, but Type I errors cause us to draw incorrect conclusions. In most cases, drawing incorrect conclusions should be a greater concern than failing to find significance. 115 © 2007 by Taylor and Francis Group, LLC 116 Niche Modeling 7.1 Range shift We look at potential bias in the range-shift methodology. This methodology examines effects of potential shifts in the ranges of species that may be brought on by changes in the environment. In order to better understand the response of species to environmental change, and potentially predict responses, an ecological niche model is devel-oped based on present day environmental variables and then reapplied using the different variables, either from the past or future. This often leads to a ‘shift’ in the predicted distribution of the species. Shift-modeling is the name for comparing predicted ranges of species in the past and future with the present. 7.1.1 Example: climate change The main applications of shift-modeling to date have been the reconstruc-tion of past distributions of species under paleo-climatic conditions, and the possible effects climate change on biological communities. Climate is always changing. Since the last major extinction at the start of the Pleistocene ice-age at 700kyr BP, the earth’s climate system has os-cillated between glacial and interglacial states approximately every 200,000 years [PJR+99]. Temperatures in Antarctica varied by about 10◦C from peak to trough, and transitions of at least 2◦C occur on average every 10,000 years [AMS+97, AMN+03, MGLM+04]. Local climate changes can be even more abrupt. The Younger Dryas event at about 12kyr BP changed local temperatures 7◦C in 1000 years, and events at 8.2kyr BP cooled Europe even more abruptly. In response to these changes, the fossil record shows changes to faunal composition and richness with a few recent extinctions attributable to recent climate changes [JW99]. There is some observational evidence that ranges of some fauna and flora have changed in response to the increase of 0.6◦C in the last century and will continue to change in response to climate changes [PY03, RPH+03, TWC+04]. © 2007 by Taylor and Francis Group, LLC Bias 117 7.2 Range-shift Model Here we develop a simple geometric model to estimate the asymmetric error in simple range-shift niche models. The particular statistic of concern is the change in the potential range area available to species. Note the range-shift methodology does not explicitly represent other factors such as: • dispersal ability, • rate of change, or • threshold effects. Variation in dispersal ability of individual species in response to change is modeled by two geometric combinations of ranges: new ranges represent-ing free dispersal and the intersection of new and old areas representing no dispersal. These alternatives are intended to ‘bracket’ the possible range of behaviors. Simple shift models incorporate rate of change in temperature only by way of dispersal and no-dispersal scenarios. They contain no history effects that could incorporate a threshold effect, and potential novel combinations of cli-mate. As bias is the tendency for a statistic to over or underestimate the true value, bias is quantified by showing a particular statistic does not estimate the correct value, by comparing the true value with the estimated value. Deviation in the expected value of the new or intersection area due to errors quantifies the bias. A theoretical model including errors in the parameters allows calculation of both the true area, and the change in area resulting from errors in those parameters. Range area predictions are regarded as biased if the observed value of range area Ae does not equal the expected value Ae. Thus area can be treated as a statistical parameter like accuracy [SP02a]. Consider a shape such as a circle with width, or radius r. Shifting the shape laterally by s produces areas: • old AO, • new AN, • an intersection area AI, and • a union AU. © 2007 by Taylor and Francis Group, LLC 118 Niche Modeling I O N FIGURE 7.1: Theoretical model of shift in species distribution from change in climate. Dashed circle marked O is old range, solid circle marked N is new range and I is intersection area. These areas can be calculated for squares and circles using the usual for-mulas. In this case the area of the square is AO = (2r)2. The formula for intersection areas of a circle for shift s when radius r remains a constant value of 1 is: AI = 2r2cos−1(s2/2sr) The equation for intersection of a square is somewhat simpler: AI = 2r(2r −s) Using the formulation of the square leads to simpler calculations. Fig-ure 7.2 shows the intersection areas for squares and circles at different shifts for squares (solid line) and for circles (dashed line). The intersection area of a square is only slightly greater than the intersection area of the circle. We will consider only the square in order to simplify calculations. We now incorporate three error terms into the equations: • error in the shift term se, © 2007 by Taylor and Francis Group, LLC Bias 119 r=1 r=0.9 r=0.8 0.0 0.5 1.0 1.5 2.0 Shift s FIGURE 7.2: The change in the areas of intersection of a square and circle for different shifts (s) and widths (r). © 2007 by Taylor and Francis Group, LLC ... - tailieumienphi.vn