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Although Schoener [1968] reported four species of Anolis lizard in the paper, only two species were included in the data set. A Poisson regression does not constrain the total number of lizards in a given habitat condition, which may be misleading because the number of lizard cannot be unlimited. A different way of analyzing the same data is using logistic regression. Assuming that there are only two competing species, we can model one species habitat preference as the probability of seeing one species over the other. The data set we used has two parts. The first 24 rows are the number of times of species A. opalinus and the second 24 rows are the number of times of species A. grahamii. Use logistic regression to predict the probability of seeing one species (e.g., A. grahamii). That is, use response of A. grahamii as success and the response of A. opalinus as failure, and develop a model to predict the probability of success. If the probability of success is high for one condition, the habitat defined by this condition is preferred by A. grahamii; if the probability is low, the habitat is preferred by A. opalinus; if the probability is close to 0.5, the habitat is shared by both. Discuss whether your interpretation of species habitat preference changes from the results using a Poisson regression model.

 
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Atlantic sturgeon (Atlantic sturgeon ( ) is a long-lived, estuarinedependent, anadromous fish. They were once a...) is a long-lived, estuarinedependent, anadromous fish. They were once a valuable and abundant resource along North America’s east coast. Habitat degradation, direct harvesting, and by-catch resulted in substantial declines in Atlantic sturgeon stock. In 2012 a segment of Atlantic sturgeon (New York Bight distinct population segment) was listed as a U.S. endangered species. Monitoring changes in sturgeon population is often done by sampling juvenile populations because juvenile sturgeons stay in their natal reproductive habitat for two to six years before migrating as mixed stocks along Atlantic coastal areas. The New York State Department of Environmental Conservation monitors juvenile sturgeon abundance in the tidal portion of the Hudson River. The data (sturgeon.csv) used in this problem are from 2006 to 2015, including counts of juvenile sturgeons caught (CATCH), effort (Effort), water chemistry (dissolved oxygen (DO), conductivity (COND), salinity (SALINITY)), tidal stage (STide), distance to salt front (DTSF), and sampling month (MON) and year (YEAR).

(a) Use Poisson regression to model the changes in sturgeon abundance over time (year, i.e., Catch ~ YEAR) using effort as the offset.

(b) Use GAM to explore the nature of the temporal trend, after the effects of other factors (e.g., temperature, salinity, distance to salt front) are accounted for.

(c) Revise the Poisson regression model based on GAM output and check if overdispersion is a problem.

 
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A storm on July 4, 1999 with wind speeds exceeding 90 miles per hour hit the Boundary Waters Canoe Area Wilderness (BWCAW) in northeastern Minnesota, causing serious damage to the forest. A study of the effects of the storm surveyed the area and counted over 3600 trees to determine whether each of them was dead or alive (data blowdown from package alr3). One of the objectives of the study is to learn the dependence of survival on species, size of the tree, and on local severity. The data set includes results from 3666 trees, including whether a tree was dead or alive (y=1 or y=0), its diameter (D in cm), local severity (S proportion of trees killed), and species (SPP: BF= balsam fir, BS= black spruce, C= cedar, JP= jackpine, PB= paper birch, RP= red pine, RM= red maple, BA = black ash, A= aspen). Fit a logistic regression model and discuss the dependence of survival on the three potential predictors (size, local severity, and species).

 
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Routine water quality data are used in the U.S. by state agencies for assessing environmental standard compliance. Frey et al. [2011] collected water quality and biological monitoring data from wadeable streams in watersheds surrounding the Great Lakes to understand the impact of nutrient enrichment on stream biological communities. Because sample sizes for different streams vary greatly, assessment uncertainty also fluctuates. Qian et al. [2015b] recommended that similar sites be partially pooled using multilevel models for improving assessment accuracy. Water quality monitoring data from Frey et al. [2011] are in file greatlakes.csv. The data file includes information on sites (e.g., location), sampling dates, and various nutrient concentrations. Of interest is the total phosphorus concentration (Tpwu). Detailed site descriptions are in file greatlakessites.csv, including level III ecoregrion, drainage area, and other calculated nutrient loading information.

When assessing a water’s compliance to a water quality standard, we compare the estimated concentration distribution to the water quality standard. The U.S. EPA recommended TP standard for this area is 0.02413 mg/L. We can use monitoring data from a site to estimate the log-mean and log-variance to approximate the TP concentration distribution (a log-normal distribution) and can calculate the probability of a site exceeding the standard.

• Use linear regression to estimate site means simultaneously (with site as the only predictor variable) and estimate the probability of each site exceeding the standard assuming a common within-site variance.

• Use the multilevel model to estimate site means and estimate the probability of each site exceeding the standard. Compare the multilevel model results to the linear regression result and discuss the difference.

 
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