TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters



## Appendix A-6: Multiple Regre ssion Models and CriteriaTSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-1 

### Appendix A-6. Multiple Regre ssion Models and Criteria
Multiple regression models can provide a means of more accurately modeling the relationships between nutrient concentr ations and biological responses, an d potentially can provide a more robust approach for using these relationships to set criteria. In Florida lakes, most research indicates that chl. a are influenced by both total phosphorus (TP) and total nitrogen (TN). The effects of these nutrients on chl. a concentration can be modeled simultaneously in clear lakes (color ≤ 40 platinum-cobalt units (PCU)).

Data exploration suggests that the relationship between chl. a , TP, and TN is not linear, nor are the residuals normally distribute d, so linear regression assumptions are likely not supported. An alternative approach is to model th e probability of an exceedance of a chl. a threshold, which does not require normally distributed residuals. Furthermore, non-parametric smooth models can be used to more accurately model non- linearities in the re sponse of chl. a . Hence, a more robust model can be written as follows:

]1[ ),( 1 ln TNTPs p p

        



Where p is the probability that chl. a > 5 μ g/L, s(.) is a nonparametric smooth function, and the function on the left side of the e quation is the log it transform of p

. The results of this model are plotted in Figure A6-1, where sampled values of TN and TP are shown as open circles and the plotted contours show probabilities that chl. a > 5 μ g/L. To set a criterion, we would select a probability of exceedance that conforms with management expectations. In this case, p = 0.25 has been selected for illustrative purposes. This contour line then defines combinations of TN and TP values that correspond with a 25% chance of the sample exceeding the chl. a threshold.

To set a criterion, single values of TN and TP n eed to be specified. One way to do this is to identify the central tendency of the relationshi p between TN and TP (show n in the plot as the solid, straight line). This line indicates the aver age ratio between ln(TN) and ln(TP) values, and in terms of management, setting a criterion on this line assumes th at management actions likely influence both TN and TP in proportions similar to the ratios of TN and TP observed in the data. The intersection between this “a verage ratio” line and the 0.2 5 contour then provides single criterion values for TN and TP (shown as dashed horizontal and vertical li nes). These criteria are as follows: ln(TN) = -0.53 and ln(TP) = -4.31.

The criteria for TN and TP define two distinct regions in the plo t: The lower left quadrant is the region in which both TN and TP cr iteria are satisfied, and the rest of the plot corresponds with one or both of these criteria being exceeded. By counting samples in which the chl. a threshold is exceeded in each of these regions, the effectiveness of the selected criteria can be measured. That is, in the lower left quadrant, 2 out of 127 samples exceed the chl. a criterion, whereas in the remaining three quadrants, 306 out of 399 samples exceed the chl. a criterion.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-2 

### Within-lake Models of TN-TP-Chl. a Relationships One of the most robust approaches for modeling the relationship between nutrient concentrations and chl. a is to model these relationships using data co llected over time from a single lake. When the data set is limited to a single lake, the pot ential for confounding by va riations in conditions across different lakes is eliminated. Hence, the ob served relationship is more likely to represent a cause-effect relationship.

The Florida lake data set is extensive, and a num ber of lakes have enough data to support their own model. For example, in one lake (REACHCODE = 3080102004935), 206 samples of TN, TP, color, and chl. a were collected. The effects of TN and TP on chl. a in this lake can be modeled using a multiple regression model of the following form:

]2[ )ln()ln( )ln( 2 10 TNbTPbbachl  

Both TP and TN are signif icantly related to chl. a , and the model accounts for 59% of the observed variability in chl. a in this lake (Figure A6-2).

Color in this lake does not exceed 40 PCU, so a target chl. a concentration might be 5 μg/L (ln(5) = 1.6). The regression model described above could be used to se t nutrient criteria by extrapolating the modeled relationship to predic t concentrations of TN and TP necessary to maintain the target chl. a concentration.

In Figure A6-3, the regression model described in Equation 2 is used to calculate concentrations of TN and TP that would give chl. a = 5μg/L (shown as a red lin e on the contour plot). As expected with this linear model, the red line is pa rallel to the other contou r lines. The dashed line represents how TP and TN co-vary across the entir e data set, and as outlined in the previous section, the intersection of the red contour li ne and the dashed line provides one way of identifying separate candidate nutrient criteria fo r TN and TP. Note, however, that these criteria are located beyond the observed range of TN and TP. That is, to establish these criteria, the model must be extrapolated beyond the limits of th e observed data. Uncertainty is inherent in any extrapolation, and in this case th e magnitude of uncertainty is la rge because of the extent to which the estimated criteria depart from the observed data. Adding data from other lakes to expand the range of observed TN, TP, and chl. a might appear to alleviate some concern regarding extrapolation, but data from other lakes introduces the possibility that unmodeled factors differing across lakes might influence the results. This additional across-lake variability substantially increases uncertainty in model pred ictions. Furthermore, in setting criteria, one could argue that one inherently must always extrapolate (or predict) chl. a concentrations at lower nutrient levels. That is, prediction uncertainty is inherent in using regression models to set criteria, regardless of how these models are specified.

Prediction uncertainty exists in both within-lak e and across-lake models, but the within-lake model offers distinct advantages over a mode l that is based on many different lakes. As discussed above, the potential that other, unmode led factors will confound the relationship is greatly reduced. Furthermore, the remaining variability in the m odel can be treated as sampling variability and the mean pred iction of the model can be used for setting criteria.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-3 The main disadvantage of the w ithin-lake modeling approach is that only a small number of lakes in the data set have enough samples to sup port the model. At a minimum, 10 observations are required per degree of freedom of a regression model, and so at least 30 samples are required to model responses in any single lake. Only 18 cl ear lakes meet this requirement. One way to expand the number of lakes that can be modeled is to assume that relationships between nutrients and chl. a , though not identical across lakes, are “l oosely” related. For example, one could assume that all the regression coefficients for TN ( b 2 in Equation 2) for all lakes are drawn from the same normal distribution. Thus, values of b 2 across different lakes would be similar, but not identical, and a model for a lake with a relativ ely small sample size (e.g., N = 10) can “borrow strength” from models fit for ot her lakes. Using this approach, models estimating the response of chl. a to nutrients within 93 different lakes with at least 10 samples can be fit.

Hierarchical Bayesian models provide a means of fitting the system of regression models described above, and the results for all 93 lakes ar e shown in Figure A6-4. In this figure, TN is plotted against chl. a . Modeled responses to TN are shown as solid black lines. The length of each black line extends over the range of TN and chl. a values observed for a single lake. To calculate these predictions, the effect of TP on chl. a was controlled by fixing TP at its mean value for a given lake.

Then, the same extrapolation approach outlined a bove can be applied to estimate nutrient criteria for each lake (Figure A6-5). These candidate criteria are specific to each lake in the data set. However, if one assumes that the set of lakes in the data set are representative of the lakes in the region, one could set overall regional nutrient crit eria by selecting a perc entile of each of these distributions. For example, one could argue that the 50 th percentiles of these distributions (ln(TN) = -0.85 and ln(TP) = -4.8) would protect 50% of the lakes at the specified chl. a level.

To account for the effects of color on chl. a in colored lakes (color > 40 PCU), a color term can be added to the regression equation:

]3[ )ln()ln()ln( )ln( 3 2 10 ColorbTNbTPbbachl   

Forty-nine colored lakes had more than 10 co mplete samples for modeling. The Hierarchical Bayesian model for these lakes yielded a larger range of responses to increased nutrients with some lakes (Figure A6-6). Color is a signifi cantly and negatively associated with chl. a in many lakes, and accurately estimating the relative contri butions of color and nutr ient concentrations to chl. a is difficult. The average effects of TN and TP across all colored lakes was similar to that observed for clear lakes; however, in certain lakes, increased TN was associated with decreases in chl. a concentrations. These unexpected relationshi ps are likely due to errors in model specification or other unmodeled factors that change within a give n lake. For example, increased nutrients and changes in color likely interact to cause the observed concentrations of chl. a , but here they have been modele d as independent predictors.

Nutrient criteria estimated for colored lakes exhi bit a wider range of valu es than that observed with clear lakes (Figure A6-7). Median criterion values were ln(TN) = 0.50 and ln(TP) = -2.7.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-4 Figure A6-1. Results from multiple logistic regre ssion. The points indicate sampled values of TN and TP. The contour lines indicate modeled probability that chl. a > 5 μ g/L. The points colored red indicate samples in which observed chl. a > 5 μ g/L.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-5 Figure A6-2. Relationship between chl. a , TN, and TP in a single lake. The symbols indicate observed values of TN and TP, and the contour lines indicate modele d values of ln(chl. a ).

Figure A6-3. Extrapolation of regression model show n in Figure A6-2 to set nutrient criteria. The red line is the predicted combinations of TN and TP that result in chl. a = 5 μg/L. The dashed line indicates how TN and TP co -vary across the entire data set. The solid circle indicates possible TN and TP criteria.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-6 Figure A6-4. Hierarchical Bayesian model results fo r clear lakes. The symbol s indicate values of TN and chl. a . The solid lines indicate model fit for a si ngle lake. The red horizontal line shows a possible chl. a threshold of 5 μ g/L.

Figure A6-5. Candidate nutrient criteria for clear lakes.TSD for U.S. EPA’s Proposed Numeric Nutrient Criteria for FL Inland Surface Fresh Waters

A6-7 Figure A6-6. Hierarchical Bayesian model results for colored lakes. The symbols indicate values of TN and chl. a . The solid lines indicate model fit fo r a single lake. The red horizontal line shows a possible chl. a threshold of 20 μ g/L.

Figure A6-7. Candidate nutrient criteria for colored lakes.