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CHAPTER 12 Interpreting Single-Variable Models Models are metaphorical (albeit sometimes accurate) descriptions of nature, and there can never be a “correct” model. —Ray Hilborn and Marc Mangel (1997:xii) [W]e see 2 major errors of omission [in model selection studies]. The first is a general failure to fully interpret “best” or “plausible” models. —Fred S. Guthery, Leonard A. Brennan, Markus J. Peterson, and Jeffrey J. Lusk (2005:461) I’ve had 10⫹ college credits in mathematics and I still couldn’t apply most of what I learned to the wildlife profession. I simply didn’t learn it beyond what was required to get an A or B in the classroom. At this point I lack the aptitude to sit down with a mathematics book and make the leap from the mechanics of solving a problem to the abstract thought required to apply it to wildlife science. —Grant Mecozzi (M.S. Candidate in Wildlife Ecology and Management, personal communication) The last chapter concluded with the observation that “significance” or “Akaike bestness” does not preclude a model from being weak, useless , or even humorous. That circumstance provides a sort of negative motivation for model interpretation. We do not want worthless models to enter the permanent record, nor do we wish to appear doltish by publishing a worthless model. To prevent these undesirable outcomes, we must interpret models. We have positive motivations for interpreting models, too. Models extracted from biological data contain biological information . That information is available by interpreting intercepts, coef- ficients, and, in some cases, the nature of a relationship between 2 variables. Sometimes models contain unexpected, hidden information that may be derived from manipulation of the model itself. Before stating the objectives of this chapter, I offer an aside. Many natural resource scientists, younger and older, feel the same frustration over mathematics expressed by Grant Mecozzi (see epigraph ). Many others look upon equations as a form of quantitative voodoo. I have been there. The first time I saw ␭ in a line transect density estimator, I felt as if I had lost before starting the game. So I have tried in chapters 12 and 13 to assist individuals with less developed quantitative skills by defining terms and operators and providing example calculations. This chapter introduces the interpretation of a class of models used extensively in natural resource science. These models are selected 126 Practice equations where the value of a dependent variable is predicted based on the value of a single independent variable; such widely used models include linear (with or without an intercept), semi-log, and log-log. I include polynomials here because these equations simply use transformations (square, cube, . . .) of a single independent variable. The next chapter introduces interpretation of multivariable models. Linear Models Simple linear models (regression analysis) receive widespread use in natural resource science. They are useful when a dependent variable (y) is proportionally related to an independent variable (x). A no-intercept linear model passes through the origin (y ⫽ 0 when x ⫽ 0). These models are of the form y ⫽ bx. Statistical packages provide the option of estimating the coefficient (b) for no-intercept models. The coefficient is the change in y for a 1-unit increase in x. No-intercept models are appropriate whenever y must equal 0 when x equals 0. For example: (1) In predicting absolute abundance based on an index of abundance, one assumes the index would be 0 if true abundance was 0. We will see an example of this in the next chapter. (2) In estimating the relation between age estimated from dental annuli and true age, one assumes the number of annuli would be 0 if the true age was 0. An interesting application of the no-intercept model appeared as an early mark-recapture estimator of population size (Hayne 1949). The dependent variable was the proportion marked in the population based on a sample, and the independent variable was the number previously marked and released into the population. Recall that the regression coefficient (b) gives the increase in y for a unit increase in x. So in the mark-recapture application, b gives the increase in the proportion marked by the mark and release of 1 animal. In other words, it estimates the proportion that 1 animal represents in the population (b ⫽ 1/N), so by algebra N ⫽ 1/b. Simple linear models with intercepts are of the form y ⫽ a ⫹ bx, or y ⫽ a ⫺ bx, [3.147.104.248] Project...

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