1. The Statistical Tool Kit determination of valid nutrient boundary values
Geoff Phillips

2. Objective methods Regression models Categorical methods
Numerical model (relationship) between nutrient concentration and EQR
Categorical methods
Range of nutrients observed in water bodies of different class
Nutrient class boundary values that minimise the difference in classification between biology and nutrient quality elements. Minimisation of miss-match

3. Tool Kit Excel spreadsheet Set of scripts for use in R
Simple methods to detect and remove outliers & select linear range of data
Comparison of type I and II regression models
v4 modified to now contain same range of univariate linear regression models (includes reduced major axis regression)
Categorical methods
Set of scripts for use in R
Examples of methods to detect outliers
GAM & segmented model fits to assist selection of linear range of data
Inclusion of multivariate models with N and P
Excel simple and quick
R more flexible
Comments received from AT, FI, FR, IT, LV, UK.
Generally supportive, key issues raised.
Is type II regression needed ?
Should EQR or Nutrient concentration be dependent variable ?

4. Simple ordinary least squares (OLS) linear regression
Scatter plot of EQR v Log10 TP concentration
Dependent variable is EQR
Predictor variable is log10 [TP]
We assume that TP causes EQR
Fit a line by minimising the sum of squared distance of observed values of y for given values of x
Assume this is an asymmetric relationship. We know TP which gives rise to observed EQR.
EQR = β0 + βLogTP + E
OLS is appropriate for asymmetric relationship, it assumes that the error variance of Y >> error variance of X, likely to be true because of model error ε
Error = Measurement error δTP + Model error ε

5. But ! We are predicting TP for a given EQR
The dependent variable should be log[TP]
The independent variable should be EQR
We wish to use our data to determine the likely range of TP at different EQR values
This is more likely to be a symmetric relationship as EQR is unlikely to cause TP concentration
LogTP = β0 + βEQR + E
The error variance for y (now [TP]) δTP is less likely to be >> than for X δEQR
Error = measurement error δTP (no equation error)

6. Invert the axis for comparison red line OLS of TP v EQR compared with black line OLS of EQR v TP
We are minimising the variation on the X axis (i.e. LogTP)
The outcome is a much steeper slope
Lines intersect at mean LogTP & mean EQR
Likely to produce lower TP boundary values at the GM boundary
The choice of dependent and independent variable is critical to the prediction of the slope and intercept of our model and thus boundaries.
Caused by choice of variable used to minimise variation, i.e. the response variable contains all the variability.
Difference in slope is a function of r2, Slope OLS Y on X = r2 * Slope OLS X on Y
For R2 < 0.7 differences start to become important to boundary values

7. Invert the axis for comparison red line OLS of TP v EQR compared with black line OLS of EQR v TP
True slope lies between these lines
Where is most of the error?
EQR measurement
TP measurement
Structural or equation error (only in EQR?)
Sample data suggests CV of TP >> CV EQR, but
metric is annual/growing season mean TP.
Need to consider replicate annual mean values, not the CV of sample results.
Estimate the variation without seasonality
(importance of using mean TP not spot TP concentrations)
Months
TP conc

8. Assume variation in both TP and EQR Type II regression
Minimise variation of both X and Y
Several methods
Orthogonal, major axis (MA) regression error variances of Y and X are equal, ratio λ= 1
Geometric mean, standardised major axis(SMA) or reduced major axis regression (don’t confuse with ranged major axis regression)
ratio error variance similarly proportional to variance ratio of observations x and y i.e. λ= s2y / s2x
Calculated as the geometric average of the slopes of the OLS regressions of Y on X and X on Y
Issues with averaging approach
If no relationship (random values of x and y), OLS fitted lines are mean of x and y, slopes of 0 and approaching ∞ geometric average of slopes = 1. Cant test if slope sig > 0
Don’t use if r < 0.6 (Smith 2009)
Smith, R.J., Use and misuse of the reduced major axis for line-fitting. Am J Phys Anthropol 140,

9. Assume variation in both TP and EQR Type II regression
Minimise variation of both X and Y
Several methods
Orthogonal, major axis (MA) regression error variances of Y and X are equal, ratio λ= 1
Geometric mean, standardised major axis(SMA) or reduced major axis regression (don’t confuse with ranged major axis regression)
ratio error variance similarly proportional to variance ratio of observations x and y i.e. λ= s2y / s2x
Calculated as the geometric average of the slopes of the OLS regressions of Y on X and X on Y
Ranged major axis regression (RMA)
An orthogonal (MA) regression, but data standardised by ranging (0 – 1)
Overcomes the limitations of MA (require equal scales of X & Y) and no longer determined simply be ratio s2y / s2x as in geometric regression
Still assumes error variances of Y and X are equal, ratio λ= 1
RMA most appropriate method – implemented in R and in v4 of Excel workbook

10. Summary of regression methods
Unless R2 is high (>0.7) choice of dependent variable will influence slope and thus predicted boundary values
True slope lies between the extremes of OLS of EQR on Nut and Nut on EQR
RMA regression is proposed as the best predictor of boundaries, but range will lie between the values predicted by the 2 OLS approaches
Compare all 3 methods

11. Data Template convenient format for R & Excel workbook
Need to use summary data not spot samples
Spatial and temporal match
Water bodies from same or similar types

12. Data need to extend across the EQR gradient
Reasonable relationship
Extend from High to Poor biological status
Adequate data above and below EQR of 0.6 Good/Moderate boundary 
Not such a clear relationship
Only a single point below an EQR of 0.6 Good/Moderate boundary
More data needed to fit a reliable model 
TP concentration ug/l

13. Is the response linear Select linear portion of the data By eye
Fit GAM model (Tkit_check_data.R)
Use segmented regression
Example data linear in range TP 10 – 100 µgl-1
Consider excluding outliers

14. Excel tool – Data table Enter data columns B:D
Sort data into ascending order of nutrient concentration
Enter the range of records to be used
Enter the Boundary EQR values & record of data used (BQE, Type, Nutrient etc)

15. Excel tool – PR Plot Plots for OLS of EQR v Nutrient,
Nutrient v OLS and
Ranged Major Axis regressions are shown (v4 Geometric mean regression is shown in distributed v2)
Open circles show data not used

16. Excel tool – Results
Summary of the regression models are provided in sheet Results
3 models are shown
OLS EQR v Log10 Nut
OLS Log10 Nut v EQR
Geometric regression (average slope of models 1 & 2) SMA regression
The sheet also calculates the residuals of the model and their interquartile range.
The range is used to determine the where 50% if the observed data lie in comparison to the best fit line
v4 now also includes 4th model
Ranged major axis regression

17. Use of residuals
By adding the 25th and 75th quartiles of the residuals to the regression model lines parallel to the best fit line mark boundaries below/above which are 25% of the data.
These lines can be used to provide estimates of the likely range of TP concentrations at G/M boundary (0.6)
at 35 µgL-1 75% WBs had EQR ≥ 0.6
At 72 µgL-1 25% WBs had EQR ≥ 0.6
The most likely boundary would be 53 µgL-1, the intersection of best fit line. 50% of WBs had EQR ≥ 0.6, 50% < 0.6
For model 1 OLS EQR v TP the GM boundary for TP would fall in the range

18. Summary of results
The most likely boundary value is taken from prediction using the Ranged Major Axis regression model (v4) G/M = 49 µgL-1
The most likely boundary value range is taken from predictions using the two OLS models
G/M = µgL-1
The possible boundary value range is taken from predictions using the upper and lower quantiles of the fit + residuals of the two OLS models
G/M = µgL-1

19. Categorical methods Average adjacent quartiles
Calculate the median and interquartile range of nutrient concentration in each biological class
Average the upper 75th quantile of Good with lower 25th quantile of Moderate

20. Categorical methods Average adjacent medians
Calculate the median and interquartile range of nutrient concentration in each biological class
Average the median of Good with median of Moderate

21. Categorical methods Upper 75th quantile of higher class
Issue with box plots are
Perhaps easier to use for non-linear data, but lower quantiles are influenced by outliers. Better to restrict data to linear region of data, derived using scatter plots.
Need to decide if log transform is appropriate (used in Excel work book)
No estimate of uncertainty
Similar approach to OLS where nutrient is dependent variable.

22. Minimise mis-match of classifications
Binary classifications of biology & nutrients (good or better and moderate or worse)
Use logarithmic series of nutrient concentrations to define nutrient class
Plot rate of mis-classifications
Point of intersection identifies nutrient boundary concentration for minimum mis-classification.

23. Minimisation mis-match class
Benefits of method
No modelling, so simple to understand.
Not sensitive to outliers and non linear data
Directly shows rate of mis-classification
No bias
Disadvantages
No modelling, so not a general case
No uncertainty analysis
(might be possible to explore by Monte Carlo simulation of subsets of data)

24. Multivariate regression models
EQR influenced by N & P
multivariate model (NP.mod0) EQR = a1 log10[TP] + a2 log10[TN] + C0 + Error
univariate model (P.mod0) EQR = a3 log10[TP] + C1 + Error
univariate model (N.mod0) EQR = a4 log10[TN] + C2 + Error
Example using high alkalinity very shallow lakes (N, CB, EC GIGs)
TN and TP not significantly correlated (VIF < 2)
df AIC
N.mod
NP.mod lowest AIC, best model
P.mod

25. Multivariate regression models
Two predictor variables, so difficult to interpret results
Calculate a matrix of TN & TP values that would predict an EQR = 0.6 (G/M boundary)
Overlay as a contour line on scatter plot of TP v TN.
Use intersection of best fit line relating TP to TN (fitted using RMA regression) to the contour to determine pairs of boundary values for TP and TN
(More about situations where N or P are likely to be limiting nutrient later)

26. Summary Tool kit provides range of regression approaches
Propose that Ranged Major Axis regression provides most unbiased model
Range of true slope & thus most likely values defined by predictions from two OLS models
Categorical analysis is an alternative approach
Still influenced by outliers and non linear relationships
Minimisation of mis-match is a potentially useful approach
Data quality is a key factor
Should cover full range of pressures
Nutrient data need to be a summary statistic (e.g. mean)
Need to be checked for linearity and for outliers
Outstanding issues
How to deal with data showing weak relationships

27. Comments Tool kit needs moving to a separate section – Appendix
Specific comment
Clearer definition outliers.
Is type II regression needed ?
Need to consider N & P together
Other variables not included (e.g. turbidity, flushing)
Tool-kit aimed at freshwater
(units in template?)
R scripts need more explanation