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Multivariate Analysis of Ecological Data
Version 2

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The Case for Nonparametric Multiplicative Habitat Models

Ecologists readily accept the concept of complex species response functions to multiple interacting factors. In representing those species responses, however, ecologists usually fall back on simplistic statistical models that cannot hope to capture the complexity of a species in relationship to its habitat. The models usually lack interaction terms and the default response shapes are typically linear (as in multiple linear regression) or sigmoid (as in logistic regression). Yet the viewpoint most widely accepted among ecologists is that species have hump-shaped response functions to environmental gradients. Furthermore, we expect the shape of this response to depend on other factors. In other words, factor interactions should be expected. Linear models may be appropriate in some cases, such as species responses to short environmental gradients. Likewise, logistic response functions are sometimes appropriate, for example, with a sigmoid relationship between a species probability of occurrence and a successional gradient. Many other possibilities exist, however (see illustration below).

The standard ecological concept for the relationship of species to environmental gradients is a unimodal, hump-shaped curve, such as those popularized by Whittaker. Though widely accepted as a theoretical model, where are the statistical models of single-species response functions that incorporate a hump-shaped response? These are surprisingly rare in the ecological literature. Even more rare are models where the shape and size of the unimodal response depends on another variable, yet this should be the norm.

The challenge for habitat modeling is exactly the same as that expressed for data analysis in general by Scott (1992, p. 5): "The modern challenge in data analysis is to be able to cope with whatever complexities may be intrinsic to the data. The data may, for example, be strongly non-Normal, fall onto a nonlinear subspace, exhibit multiple modes, or be asymmetric [all of these are commonly true of species responses]. Dealing with these features becomes exponentially more difficult as the dimensionality of the data increases, a phenomenon known as the curse of dimensionality."

This curse applies to all data sets on species performance in relation to multiple habitat factors. As the number of factors increases, the number of potential interaction terms increases exponentially. The number of transformations or combinations of transformations similarly inflates. The number of possible combinations of factors to include or exclude balloons.

Huston (2002) described well the some of the pointless arguments and faulty conclusions that have emerged from using simple, inappropriate statistical models to represent complex systems of interacting factors. He encouraged us to recognize that "the interactive effects of multiple limiting factors require new statistical approaches for quantifying ecological processes..."

HyperNiche offers a new approach to solving this problem: multiplicative models. Nonparametric Multiplicative Regression (NPMR; McCune 2006) effectively represents species responses to multiple habitat factors. Factor interactions are accommodated automatically and the overall form of the response surface need not be specified in advance. With a built-in cross-validation procedure to reduce problems of overfitting, NPMR promises models that fit better and are more parsimonious than traditional methods.

With multiplicative models, the effect of each variable can depend on the value of other variables. This is simple conceptually but mathematically difficult for traditional modeling methods. A practical solution is provided by adapting nonparametric curve fitting techniques, the components being combined multiplicatively rather than additively - this is NPMR.

The main purpose of HyperNiche is to offer this practical solution. HyperNiche incorporates NPMR into an easy-to-use yet powerful package, complete with 3D and 2D graphics, GIS interface, easy connections to spreadsheets and community analysis, data transformation, and predictions for new cases.


Huston, M. A. 2002. Critical issues for improving predictions. Pages 7-21 in J. M. Scott, P. J. Heglund, M. L. Morrison, J. B. Haufler, M. G. Raphael, W. A. Wall, & F. B. Samson, eds., Predicting Species Occurrences: Issues of Accuracy and Scale. Island Press, Washington.
McCune, B. 2006. Non-parametric habitat models with automatic interactions. Journal of Vegetation Science 17: 819-830
Scott, D. W. 1992. Multivariate Density Estimation: Theory, Practice, and Visualization. John Wiley, New York. 317 pp.

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