Simple Sum and PCA

Abstract

The first step in ReVA analysis is a spatial overview of environmental quality. There may be dozens or even hundreds of individual values describing specific aspects of a watershed, from the number of native fish to the concentration of ozone in the air. ReVA integrates the information into a single index for each watershed and produces an overview map. Two complimentary methods are used and greater confidence can be placed in results where the methods agree.

Method Details

A straightforward approach is to sum the coded values. Each variable is coded to range from 0 (good) to 1 (bad) and therefore the values can be summed for each watershed. In this Simple Sum method, a smaller sum represents better overall environmental condition. The range of the sums is divided into seven equal intervals and each watershed is assigned to a septile. The watersheds can then be mapped with a color assigned to each septile from good (green) to poor (red) condition.

The second method first performs a Principle Components Analysis (PCA) on the data. The first five principle components are then used to weight the variables before they are summed. The result of this PCA/Sum method is that the covariances between variables are taken into account.

The two methods are complimentary in the sense that they are sensitive to different peculiarities in the data set. The Simple Sum is not affected by skewed distributions (e.g. many watersheds in good condition and only a few in poor) in the variables but is sensitive to covariances. The Simple Sum will tend to overly penalize a watershed where several stressors co-occur. The PCA/Sum Method accounts for covariance but is sensitive to skewed distributions because the PCA step assumes normal distributions.

Limitations

Using any single indicator always has the associated problem of occlusion (Suter 1983). This occurs when one or two variables that are clearly unsatisfactory and require immediate remedial action cannot be detected because they occur on the same watershed along with one or two variables indicating good conditions. The sum produces an intermediate value that occludes the danger. The problem is minimized in ReVA where the user has immediate access to all of the variables in the radar plot. Nevertheless, any method as simple as summing incurs this problem. Therefore, interpretations should not be drawn with these simple methods without careful searching of the data for occluding.

Basically, the Simple Sum method is an overly simplistic approach to integration. Because it ignores the complex relationships among the variables, any assessment based on this method should be done with extreme caution and only accepted if backed up with confirming independent evidence. Because the Simple Sum and PCA/Sum methods are sensitive to different problems with the data they should be used together.

One disadvantage of the simple methods is that they only give a relative ranking of the watersheds. There is no objective standard to which all the watersheds can be compared. Thus, the best watershed may not be objectively satisfactory and the worst watershed may not be objectively problematic. Therefore, the simple methods should not interpreted in isolation. They should be used in conjunction with other methods that provide more objective standards.

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