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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.