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In order to perform a logistic regression, it was necessary that the locations of some thistles as well as an approximately equal number of thistle-free locations be known. The known thistle locations were derived from a horseback GPS survey conducted in 1998 by BADL within the Burns Basin Wilderness area. The survey tallied a total of 120 point locations representing thistle infestations 0-5 acres in size, and mapped approximately 40 polygons of intensive thistle growth with a total area of 1.4 x 106 m2. To produce point locations free of thistles, we generated approximately 160 randomly distributed locations within the study area, excluding those that occurred within the mapped thistle polygons, or within 90 meters of a mapped thistle point. (A five-acre circle corresponds to a circle 80 m in diameter.) However, one must be aware that there is a chance that a thistle-free location actually contains thistles that were not mapped by the BADL survey. The random non-thistle and known thistle locations were then combined to produce a single data set for the logistic regression.
Polygon coverages of each of the variables were provided by BADL. Each coverage was clipped to the study area boundary. Area fractions qi of each occurring class (e.g. geologic unit) were computed for the study area. The number of thistle points Ni falling within each class was determined, with the total observed points N = SNi. The predicted number of thistle points for each class Ei was also determined as the mean of the binomial distribution function Ei = Nqi, and a 2s confidence interval about this mean was calculated using the formula for the standard deviation of the binomial distribution function s = [Nqi(1-qi)]0.5. Classes for which the observed thistle locations fell outside the 2s confidence interval of the predicted locations were considered to have a significant correlation (positive or negative) with thistle growth.
For each cover type a Z-score was calculated using Z = (Ni - Ei) / [Nqi(1-qi)]0.5 to show the number of standard deviations that the observed value falls from the expected value. A positive Z-score greater than 2 indicates a less than 5% probability that this number of points occurred in that particular cover type by chance. A negative Z-score less than •2 indicates that far fewer thistle points were observed in that unit than would be expected by chance. Thus a negative Z-score less than 2 suggests that thistles show an aversion to the cover type, and positive Z-score greater than 2 suggests that thistles show an affinity for that cover type.