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Consider the model you developed in question 8 of Chapter 8 and perform a simulation to see if the model you developed adequately describes the response variable data distribution. A potential problem with this data set is the limited variability in the response variable. This could be caused by the difficulty in accurately recording the number of mates a frog had; either the duration of observation is too short, or there might be mates that were not observed. The consequence of this problem is the underreporting of the number of mates, and the resulting model is likely to underestimate the number of mates (and producing too many 0s). Arnold and Wade [1984] discussed other problems with such data.

 
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Qian et al. [2003a] proposed a “nonparametric deviance reduction” method for detecting ecological threshold. The method is based on the CART model, but uses only one predictor representing the environmental gradient. The first split point is used as the threshold. In the paper, the authors suggested that a χ 2 test can be used to test whether the resulting split point is “statistically significant.” Because the split point is the point that results in the largest difference in deviance, it is highly likely that such a test will have a highly inflated type I error probability. Design a simulation to estimate the type I error probability of such a test. In the simulation, we can assume that the response variable is a normal random variable, such that the χ 2 test is reduced to a two-sample t-test. As the method is used to detect a threshold, the null hypothesis should be that a threshold does not exist, or the response variable distribution does not change along the gradient.

 
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Many institutions around Lake Erie have long-term water quality monitoring programs. These institutions, however, use several different water sampling methods and several chemical analytical methods for measuring water quality variables. The data file Eriecombined.csv include monitoring data from six institutions collected from 2010 to 2013. Figure 3.4 shows data from NOAA. One way to examine institutional differences is to use a multilevel model with institution as a factor variable, after other factors affecting water quality (TP and chlorophyll a concentrations in particular) are accounted for. These factors include (1) distance to the source of phosphorus (the Maumee River), (2) year, and (3) season (months).

(a) Use exploratory data analysis tools (e.g., Q-Q plots) to determine the nature of the difference (e.g., multiplicative or additive differences). Based on the exploratory analysis, recommend an appropriate transformation for the two water quality variables of interest (TP and chlorophyll a concentrations).

(b) Fit multilevel models for TP and chlorophyll a concentrations (TP and CHLA, respectively), using distance to Maumee River mouth (DISTANCE) as a continuous predictor and INSTITUTION, YEAR, and SEASON as three factor variables. Describe model outputs in plain language.

(c) Present the differences among institutions graphically.

Figure 3.4

Many institutions around Lake Erie have long-term water quality monitoring programs. These...

 
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A frequently used indicator of stream ecosystem conditions is the species richness (number of species or taxa found in a sample). In the EUSE example, we often use (1) total number of macroinvertebrate taxa (richness) and (2) the number of taxa belonging to three orders known to be sensitive to pollution. These three orders are Ephemeroptera (mayflies), Plecoptera (stone flies), and Trichoptera (caddisflies). Taxa in these three orders are collectively known as EPT taxa. EPT taxa richness is often more indicative of water quality than total taxa richness. The data file posted on USGS web page (http://pubs.usgs.gov/ sir/2009/5243/data/EUSE_USGSReportData.csv) includes total richness (RICH) and EPT taxa richness (EPTRICH).

(a) Develop a multilevel model to model the response of total taxa richness (RICH) to changes in watershed urbanization represented by the national urban intensity index (NUII), as well as to regional level climate conditions.

(b) Develop a multilevel model to study the response of EPT taxa richness (EPTRICH) to changes in urban intensity, as well as to regional level variables such as mean temperature and precipitation.

(c) Another way to examine the changes in biological community is to examine the changes in the relative EPT taxa richness (i.e., EPT taxa richness as a fraction of total number of macroinvertebrate taxa) along the urban gradient. Develop a multilevel logistic regression model to study the changes of relative EPT taxa richness and compare the results to the EPT taxa richness model.

(d) Considering the connection between multinomial and Poisson models discussed in Chapter 8, discuss how the connection can be used in a multilevel setting.

 
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