Primer on Natural Resource Science

Chapter 13. Interpreting Multivariable Models

CHAPTER 13 Interpreting Multivariable Models Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. —Edwin A. Abbot (1992:viii) It is often helpful to think of the four coordinates of an event as specifying its position in a four-dimensional space called space-time. It is impossible to imagine a four-dimensional space. I personally find it hard enough to visualize three-dimensional space! —Stephen W. Hawking (1988:24) Edwin A. Abbott (1838–1926) published Flatland: A Romance in Many Dimensions in 1884. The book was about a culture that lived in 2 dimensions . This sexist culture was highly class conscious. Women, the lowest class, were straight lines. The social scale for men ranged from lower-class isosceles triangles to middle-class squares to upper-class circles. It was heresy in Flatland to consider a third or, heaven forbid, a fourth or higher dimension. Welivedin2dimensions(Flatland)inchapter12.Nowwemove into Spaceland (3 dimensions) and Hyperspaceland (⬎3 dimensions). Vexing concepts, such as infinity, will attend our immigration, and we will come to see that Hyperspaceland may be viewed only in an arbitrary and incomplete fashion. This chapter begins with a discussion of dimensions and de- fines the concept of hyperspace. Then I provide examples of multivariable models from multiple and logistic regression and show how these models may be interpreted analytically and graphically. Dimensions of Relations With respect to models, an equation with a single dependent and independent variable has 2 dimensions. A model with a single dependent variable and 2 independent variables has 3 dimensions. If the number of independent variables exceeds 2, the dimension of the model exceeds 3. We can graphically visualize relationships in 2 (chapter 12) or 3 dimensions on 2-dimensional paper (Fig. 13-1). Commercial software is available for plotting 3-dimensional graphs. There are 2 other ways to visualize 3 dimensions on 2-dimensional paper. We could view a curvilinear response surface 138 Practice (prediction surface for an equation; Fig 13-1 is an example) as a homologue to landscape topography and then plot contours as is done on topographic maps. The technique first involves setting a contour value (arbitrary values of the dependent variable). Then we select an arbitrary set of values for 1 independent variable and solve for the corresponding value of the other independent variable. The pairs of values for the 2 independent variables along a contour are then plotted . The contour technique would show peaks, valleys, and flatlands in the response surface just as a topographic map shows these landscape features. I have not seen this technique used in natural resource science, but it is available. The third alternative is simpler. It involves plotting the dependent variable as a function of 1 independent variable at different fixed values of the second independent variable. We will see application of this technique in the logistic regression example that follows. For ⬎3 dimensions, we can graphically visualize only slices of hyperspace. By slices of hyperspace, I mean portions of the response surface that are exceedingly thin because additional independent variables are held constant. For example, given a dependent variable and 3 independent variables, we could hold one of the independent Fig. 13-1 Three-dimensional graphic in 2 dimensions and an example of a response surface. 139 Interpreting Multivariable Models variables constant, which yields a specific number. This number is added to the intercept, and the resulting model (1 dependent, 2 independent variables) can be plotted as in Fig. 13-1. There are infinitely many slices in hyperspace, and it would take eternity to view them all. Nonetheless, we can make rational decisions on what the values of constants are; we would suppose modes or means would capture the general nature of the response to variables not held constant . However, it must be realized that our ability to reliably interpret multivariable models wanes rapidly as the number of independent variables increases, and eventually, the models are beyond human comprehension . Many models appearing in the ecological literature today are incomprehensible. Multiple Linear Regression Multiple linear regression involves ⱖ2 independent (x) variables in a model of the form y ⫽ a ⫹ b1 x1 ⫹ b2 x2 ⫹ . . . ⫹ bk xk , where y ⫽ the predicted value of the dependent variable, a ⫽ the y-intercept or the value...