Analysis of Environmental Data Problem Set Conceptual Foundations: De te rm in istic fu n c tio n s Answers

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1 Analysis of Environmental Data Problem Set Conceptual Foundations: De te rm in istic fu n c tio n s Answers 1. The following real data set contains data on marbled salamander abundance (abund=mean number of breeding females over 10 years), fecundity (fecund=mean number of metamorphs produced per breeding female over 10 years), and an index of coarse woody debris (CWD) on the forest floor in the neighborhood of each of 14 ponds. pool abund fecund CWD Based on this dataset, answer the following questions: a. Consider the relationship between salamander abundance (abund; independent variable) and fecundity (fecund; dependent variable) as shown in the scatterplot. Identify at least three plausible deterministic functions (using the bestiary of functions presented in lecture and found in Bolker) that could be used to model this relationship as part of a phenomenological model. In other words, find a plausible relationship between abundance and fecundity; i.e., how is fecundity expected to change as abundance increases? Note, here you are seeking to describe the deterministic component of the statistical model; we are going to ignore the error or stochastic component of the model for the time being (but we will come back to this in the next problem set). How will your choice of deterministic function affect the expected value of fecundity for abundances >>100? Since we do not have a theoretical basis for choosing a particular model at least none was given, there are lots of mathematical functions to choose from to model the apparent relationship. Some logical choices are the monomolecular, Beverton-Holt, Ricker, and Holling type III, since these are either saturating response functions that rise and approach an asymptote or rise to a peak and then decline (Ricker). There are

2 Deterministic Functions: Problem Set Answers 2 of course other possibilities depending on what you see or want to see in the data. For example, you might consider the right-most point an outlier and ignore it, in which case a simple linear function would probably fit well. Or you might consider a piecewise polynomial model with a linear fit on the left-side and a constant on the right side, although an abrupt transition is probably hard to justify given the smooth saturating functions that essentially do the same thing but without the abrupt threshold. Lastly, the choice of deterministic function will clearly have an affect on the expected value of fecundity for abundances >>100 because the shape of the function determines whether the expected value will stay roughly the same (e.g., monomolecular), continue to increase (e.g., Beverton-Holt), or decrease (e.g., Ricker). Thus, while the choice of deterministic function may not matter too much within the range of observed data, it can make a huge difference if the model is going to be used to extrapolate beyond the range of the observed data. b. Consider the relationship between coarse woody debris (CWD; independent variable) and salamander abundance (abund; dependent variable) as shown here. If we posited a power function as the deterministic component of our statistical model and estimated (in a later step) the parameters of this model to be: a=0.1 and b=2.8, what would the expected value be for a CWD level of 10? Note, the use of the power function here to explain the deterministic relationship between CWD and abundance is purely phenomenological. Can you explain why this is a phenomenological model and not a mechanistic model? The expected value of abundance given a CWD=10 (independent variable) and parameters a=0.1 and b=2.8 (values of the deterministic model parameters) is easily computed by plugging in these numbers to the power function given: b 2.8 abundance=a*x =0.1*10 =63.09.

3 Deterministic Functions: Problem Set Answers 3 The expected value can also be obtained graphically by drawing vertical line up from CWD=10 to where it intersects the fitted line and then drawing a horizontal line over to where it intersect the y-axis, which is the expected abundance. Lastly, the power function is a phenomenological deterministic model in this example because there is no theoretical basis give for why we should expect a power law relationship between CWD and abundance, making the interpretation of the model parameters a and b somewhat abstract. 2. The following real data set (depicted here using a box-and-whisker plot) derives from a manipulative arboriculture experiment on the resistance of various climbing ropes to injury from various pruning blade types, in order to determine if some blades are safer than others (see Brian Kane for details). Briefly, the dataset contains data on pruning blade type (blade; categorical variable with 4 levels: F1, F2, F3 and SI) and percent of the rope cut by the blade (p.cut; continuous variable, 0-1) for 30 replicates (n=30 reps x 4 blades=120 observations). In the figure below, a box-andwhisker plot is depicted for each blade type and shows the distribution of p.cut values for each blade type. Based on this dataset, answer the following questions. a. What is the deterministic function for the model describing the relationship between blade type (independent variable) and percent of rope cut (dependent variable)? Note, here you are seeking to describe the deterministic component of the statistical model; we are going to ignore the error or stochastic component of the model for the time being (but we will come back to this in the next problem set). The deterministic function for this model is simply an indexed vector of means; i.e., a vector of means containing a mean for each blade type. Here, the hypothesis although not stated explicitly is that the mean p.cut (i.e., expected value) differs among blade types. Thus, the expected value is simply a different mean for each blade type. b. Based on the deterministic model above and the data depicted in the figure, what is the approximate expected value of p.cut for an F2 blade? The expected value of p.cut for an F2 blade is simply the mean value of p.cut for the F2 observations, depicted here as box-and-whisker plot. Based on a visual approximation from the box-and-whisker plot, the mean p.cut for an F2 blade is approximately Note, the distribution of p.cut for an F2 blade appears to be

4 Deterministic Functions: Problem Set Answers 4 roughly normally distributed based on the fact that the median is roughly centered in the box and the whiskers are approximately of the same length. Thus, the mean and median are approximately the same, so we can estimate the mean value of p.cut using the median value depicted by the solid line in the box. 3. The following real dataset (depicted here as a scatterplot) contains data on black oak (Quercus velutina) abundance (Quercus.velutina=percent cover on 25 m radius plot) and a GIS-derived index of ecological integrity (iei) for each of 98 upland forest plots in the Deerfield watershed in western Massachusetts. The data were collected as part of the CAPS (Conservation Assessment and Prioritization System) project as a means of verifying the CAPS index of ecological integrity. Based on this dataset, answer the following questions. a. Identify at least three plausible deterministic functions (using the bestiary of functions presented in lecture and found in Bolker) that could be used to model this relationship as part of a phenomenological model. In other words, find a plausible relationship between iei and percent cover of this tree species; i.e., how is cover of this species expected to change as iei increases? Note, here you are seeking to describe the deterministic component of the statistical model; we are going to ignore the error or stochastic component of the model for the time being (but we will come back to this in the next problem set). Since we do not have a theoretical basis for choosing a particular model at least none was given, there are lots of mathematical functions to choose from to model the apparent relationship. Some logical choices would be functions that decline monotonically (e.g., negative exponential), have a hump-shaped or unimodal distribution (e.g., Ricker), or exhibit threshold-like behavior (e.g., 3 parameter logistic in this case, hockey stick). Three of these possibilities are shown in the scatterplot. Note the strong influence of the abundant zeros in pulling the curve down towards zero, which we will discuss below. There are of course many other possibilities depending on what kind of shapes you think might be ecologically realistic in this context. b. Given the rarity of this species (i.e., it is only present in 7 of 98 plots), how sensitive is your choice of deterministic model to the value of these 7 non-zero values? In other words, if we changed one or more of these values would your best model change? Can you imagine a deterministic model that might deal with the large number of zeros?

5 Deterministic Functions: Problem Set Answers 5 The 7 non-zero points clearly have high leverage on the choice of the deterministic function and on its eventual optimal fit to the data; in other words, removing any of these points or moving their position could change the choice of best model and its optimal fit. It turns out that datasets like the one shown here with lots and lots of zeros are quite common in environmental studies, especially ecological studies involving the species occurrence because most species don t occur in most places. Fortunately, methods exist for handling these so-called zero-inflated datasets. Briefly, the excessive number of zeros are modeled separately from the rest of the data. This type of model is known as a multi-level model (or hierarchical model), which we will discuss in more detail at the end of the semester.