1. Applications of Regression to Water Quality Analysis Unite 5: Module 18, Lecture 1

2. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s2 Statistics  A branch of mathematics dealing with the collection, analysis, interpretation and presentation of masses of numerical data  Descriptive Statistics (Lecture 1)  Basic description of a variable  Hypothesis Testing (Lecture 2)  Asks the question – is X different from Y?  Predictions (Lecture 3)  What will happen if…

3. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s3 Objectives  Introduce the basic concepts and assumptions of regression analysis  Making predictions  Correlation vs. causal relationships  Applications of regression  Basic linear regression  Assumptions  Techniques  What if it is not linear: data transformations  Water quality applications of regression analyses  Survey of regression software

4. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s4 Regression defined  A statistical technique to define the relationship between a response variable and one or more predictor variables  Here, fish length is a predictor variable (also called an “independent” variable.  Fish weight is the response variable

5. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s5 Regression and correlation  Regression:  Identify the relationship between a predictor and response variables  Correlation  Estimate the degree to which two variables vary together  Does not express one variable as a function of the other  No distinction between dependent and independent variables  Do not assume that one is the cause of the other  Do typically assume that the two variable are both effects of a common cause

6. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s6 Basic linear regression  Assumes there is a straight-line relationship between a predictor (or independent) variable X and a response (or dependent) variable Y  Equation for a line: Y = mX + b m – the slope coefficient (increase in Y per unit increase in X)  b – the constant or Y Intercept (value of Y when X=0)

7. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s7 Basic linear regression  Assumes there is a straight-line relationship between a predictor (or independent) variable X and a response (or dependent) variable Y  Regression analysis finds the ‘best fit’ line that describes the dependence of Y on X

8. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s8 Basic linear regression  Assumes there is a straight-line relationship between a predictor (or independent) variable X and a response (or dependent) variable Y  Outputs of regression  Regression model Y = mX + b Weight = 4.48*Length + - 28.722

9. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s9 Basic linear regression  Assumes there is a straight-line relationship between a predictor (or independent) variable X and a response (or dependent) variable Y  Outputs of regression  Regression model Y = mx + b Weight = 4.48*Length + - 28.722  Coefficient of Determination R2 = 0.89

10. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s10 How good is the fit? The Coefficient of Determination  R2: The proportion of the total variation that is explained by the regression  Coefficient of determination  R2 = 0.89  Ranges from 0.00 to 1.00  0.00 – No correlation  1.00 – Perfect correlation  no scatter around line

11. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s11 R 2 = 0.08 R 2 = 0.54 Example coefficients of determination

12. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s12 Four assumptions of linear regression -adapted from Sokal and Rohlf (1981)  The independent variable X is measured without error  Under control of the investigator  X’s are ‘fixed’

13. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s13 Four assumptions of linear regression -adapted from Sokal and Rohlf (1981)  The independent variable X is measured without error  Under control of the investigator  X’s are ‘fixed’  The expected value for Y for a given value of X is described by the linear function Y = mX +b

14. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s14 Four assumptions of linear regression -adapted from Sokal and Rohlf (1981)  The independent variable X is measured without error  Under control of the investigator  X’s are ‘fixed’  The expected value for Y for a given value of X is described by the standard linear function y = mx +b  For any value of X, the Y’s are independently and normally distributed  Scan figure 14.4 from S&R

15. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s15 Four assumptions of linear regression -adapted from Sokal and Rohlf (1981)  The independent variable X is measured without error  Under control of the investigator  X’s are ‘fixed’  The expected value for Y for a given value of X is described by the standard linear function y = mx +b  For any value of X, the Y’s are independently and normally distributed  Scan figure 14.4 from S&R  The variance around the regression line is constant; variability of Y does not depend on value of X  Extra credit word: the samples are homoscedastic

16. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s16  It is often possible to ‘linearize’ data in order to use linear models  This is particularly true of exponential relationships Data transformations: What if data are not linear?

17. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s17 N Applications: Standard curves for lab analyses  A classic use of regression: calibrate a lab instrument to predict some response variable – a “calibration curve”  In this example, absorbance from a spectrophotometer is measured from series of standards with fixed N concentrations.  Once the relationship between absorbance and concentration is established, measuring the absorbance of an unknown sample can be used to predict its N concentration

18. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s18  The USGS has real time water quality monitors installed at several stream gaging sites in Kansas Using regression to estimate stream nutrient and bacteria concentrations in streams

19. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s19 Using regression to estimate stream nutrient and bacteria concentrations in streams: data flow

20. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s20 Using Regression to estimate stream nutrient and bacteria concentrations in streams: Results  USGS developed a series of single or multiple regression models  Total P = 0.000606*Turbidity + 0.186 R2=0.964  Total N = 0.0018*Turbidity + 0.0000940*Discharge + 1.08 R2=0.916  Total N = 0.000325 * Turbidity + 0.0214 * Temperature + 0.0000796*Conductance + 0.515 R2=0.764  Fecal Coliform = 3.14 * Turbidity + 24.2 R2=0.62

21. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s21 Using Regression to estimate stream nutrient and bacteria concentrations in streams: Important Considerations  Explanatory variables were only included if they had a significant physical basis for their inclusion  Water temperature is correlated with season and therefore application of fertilizers  Conductance is inversely related to TN and TP, which tend to be high during high flow  Turbitidy is a measure of particulate matter – TN and TP are related to sediment loads  The USGS needed a separate model for each stream!  The basins were different enough that a general model could not be developed  By using the models with the real-time sensors, USGS can predict events, e.g. when fecal coliform concentrations exceed criteria

22. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s22 Measured and regression estimated density

23. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s23 Using regression to estimate stream nutrient and bacteria concentrations in streams: Important Considerations  Explanatory variables were only included if they had a significant physical basis for their inclusion  Water temperature is correlated with season and therefore application of fertilizers  Conductance is inversely related to TN and TP, which tend to be high during high flow  Turbitidy is a measure of particulate matter – TN and TP are related to sediment loads  The USGS needed a separate model for each stream!  The basins were different enough that a general model could not be developed  By using the models with the real-time sensors, USGS can predict events, e.g. when fecal coliform concentrations exceed criteria  Concentration estimates can be coupled with flow data to estimate nutrient loads  Finally, these regressions can be useful tools for estimating TMDL’s

24. Developed by: Host Updated: Jan. 21, 2003 U5-m18a-s24 Software for regression analyses  Any basic statistical package will do regressions  SigmaStat  Systat  SAS  Excel and other spreadsheets also have regression functions  Excel requires the Analysis Toolpack Add-in  Tools > Add-in > Analysis ToolPack