1. Environmental Modeling Basic Testing Methods - Statistics III

2. 1. Covariance ► ► Joint variation of two variables about their common mean ► ► Covariance

3. 2. Simple Regression ► ► Regression: models relationships between variables Y i =  0 +  1 X i + e i,  0 - intercept,  i – slope Y is the dependent variable X is the independent variable Simple regression has one independent variable Multiple regression has more than one indep var

4. 2. Simple Regression.. ► ► We can fit a line through the cloud of dots ► ► Only one position is the best fit Y = b 0 + bX Y X

5. 2. Simple Regression.. ► ► Least square methods can help identify the best fit, following two conditions n  (Y i - Y i ) 2 = minimum;  e i = 0 (Y i - Y i = e i ) 1 ► ► Parameters estimated in the process: b 0, b Y = b 0 + bX b 0 – intercept, b – regression coefficient

6. 3. Goodness of Fit 1 ► ► Total Sum of Squares: SS t =  (Y i - Y) 2 n 1 ► ► Sum Squares of Regression: SS r =  (Y i - Y) 2 n 1 ► ► Sum Squares of residuals: SS e =  (Y i - Y i ) 2 n SS t = SS r + SS e

7. Coefficients ► ► Coefficient of determination (goodness of fit): R 2 = SS r /SS t ► ► Coefficient of correlation: R = R 2 = SS r /SS t r = Cov(x,y)/s x s y

8. Adjusted R 2 (k-1)(1 - R 2 ) ► ► Adjusted R 2 : R 2 a = R 2 - ----------------- N - k N - sample size k - number of independent variables

9. 4. Test of Regression Model ► ► General F test: equality of two variances ► ► Null hypothesis: S 1 2 = S 2 2 S 1 2 F = ---------- S 2 2

10. Test of Regression Model ► ► Compare the computed F value to the critical F value for specified degrees of freedom for both variances and level of significance ► ► If the computed F>critical F, reject the null, accept otherwise ► ► Check the p value, if p< , reject the null hypothesis

11. F Test for Regression Model ► ► F test for regression model: ► ► Null hypothesis: SS r = SS e SS r /k F = --------------, SS e /N-k-1 k - number of parameters excluding b 0 N - sample size

12. t Test for b ► ► t test for individual parameters b ► ► Null hypothesis: b i = 0 b i t = ------, S bi - standard error of b i S bi

13. 5. Multiple Regression ► ► Y i = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 3 +... + b m X m + e i ► ► Y = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 3 +... + b m X m

14. Regression Results ► Analysis of variance DF Sum of Squares Mean Square Regression 3 97747.0918432583.03061 Residual36 7061.68316 196.15787 F = 166.10616Signif F = 0.0000 Multiple r 0.87328 R Square 0.76262 Adjusted R Square 0.75701 Standard Error 14.00564

15. Regression Results ► Variables in the Equation Variableb Se b Beta t Sig t X 1 0.1917 0.001715 0.725998 6.262 0.0000 X 2 -0.08290.001219 -0.994050-16.161 0.0000 X 3 -4.959411.079785 -0.052423-0.4841 0.0310 X 4 5.36397.3908 7.9273-0.932 0.0926 Y = 0.1917X 1 - 0.0829X 2 - 4.9594X 3 + 5.3639X 4