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

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

A6-1 

Appendix A-6. Multiple Regression Models and Criteria 

Multiple regression models can provide a means of more accurately modeling the relationships 

between nutrient concentrations and biological responses, and 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 distributed, so linear regression assumptions are likely not supported. An 

alternative approach is to model the 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 response of chl. a. Hence, a more robust 

model can be written as follows: 

),( ]1[ 1ln TNTPs

p

p 

















Where p is the probability that chl. a > 5 μg/L, s(.) is a nonparametric smooth function, and the 

function on the left side of the equation is the logit 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 μg/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 need to be specified. One way to do this is to 

identify the central tendency of the relationship between TN and TP (shown in the plot as the 

solid, straight line). This line indicates the average ratio between ln(TN) and ln(TP) values, and 

in terms of management, setting a criterion on this line assumes that 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 “average ratio” line and the 0.25 contour then provides single 

criterion values for TN and TP (shown as dashed horizontal and vertical lines). 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 plot: The lower left quadrant is the 

region in which both TN and TP criteria 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 collected over time from a single lake. When 

the data set is limited to a single lake, the potential for confounding by variations in conditions 

across different lakes is eliminated. Hence, the observed relationship is more likely to represent a 

cause-effect relationship. 

The Florida lake data set is extensive, and a number 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: 

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

Both TP and TN are significantly 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 set nutrient criteria by 

extrapolating the modeled relationship to predict 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 line on the contour plot). As 

expected with this linear model, the red line is parallel to the other contour lines. The dashed line 

represents how TP and TN co-vary across the entire data set, and as outlined in the previous 

section, the intersection of the red contour line and the dashed line provides one way of 

identifying separate candidate nutrient criteria for 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 the observed data. Uncertainty is inherent in any 

extrapolation, and in this case the magnitude of uncertainty is large 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 predictions. 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-lake and across-lake models, but the within-lake 

model offers distinct advantages over a model that is based on many different lakes. As 

discussed above, the potential that other, unmodeled factors will confound the relationship is 

greatly reduced. Furthermore, the remaining variability in the model can be treated as sampling 

variability and the mean prediction 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 within-lake modeling approach is that only a small number of 

lakes in the data set have enough samples to support 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 clear 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 “loosely” related. For example, one could 

assume that all the regression coefficients for TN (b2 in Equation 2) for all lakes are drawn from 

the same normal distribution. Thus, values of b2 across different lakes would be similar, but not 

identical, and a model for a lake with a relatively small sample size (e.g., N = 10) can “borrow 

strength” from models fit for other 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 are 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 above 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 criteria by selecting a percentile of each of these 

distributions. For example, one could argue that the 50th 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: 

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

Forty-nine colored lakes had more than 10 complete 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 significantly and negatively associated with chl. a in many 

lakes, and accurately estimating the relative contributions of color and nutrient 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 relationships are likely due to errors in model 

specification or other unmodeled factors that change within a given lake. For example, increased 

nutrients and changes in color likely interact to cause the observed concentrations of chl. a, but 

here they have been modeled as independent predictors. 

Nutrient criteria estimated for colored lakes exhibit a wider range of values 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 regression. The points indicate sampled values of TN 

and TP. The contour lines indicate modeled probability that chl. a > 5 μg/L. The points colored 

red indicate samples in which observed chl. a > 5 μg/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 modeled values of ln(chl. a). 

Figure A6-3. Extrapolation of regression model shown 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 for clear lakes. The symbols indicate values of 

TN and chl. a. The solid lines indicate model fit for a single lake. The red horizontal line shows a 

possible chl. a threshold of 5 μg/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 for a single lake. The red horizontal line 

shows a possible chl. a threshold of 20 μg/L. 

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