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Environmental Modeling Basic Testing Methods - Statistics III

Environmental Modeling Basic Testing Methods - Statistics III. 1. Covariance. Joint variation of two variables about their common mean Covariance. 2. Simple Regression. Regression: models relationships between variables

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Environmental Modeling Basic Testing Methods - Statistics III

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  1. Environmental ModelingBasic 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 Yi = b0 + b1Xi + ei, b0 - intercept, bi – 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 = b0 + bX    Y X

  5. 2. Simple Regression.. • Least square methods can help identify the best fit, following two conditions n S (Yi - Yi)2 = minimum; Sei = 0   (Yi - Yi = ei)     1 • Parameters estimated in the process: b0, b   Y = b0 + bX b0 – intercept, b – regression coefficient

  6. 3. Goodness of Fit 1 • Total Sum of Squares: SSt = S (Yi - Y)2n 1 • Sum Squares of Regression: SSr = S(Yi - Y)2 n 1 • Sum Squares of residuals: SSe = S (Yi - Yi)2 n SSt = SSr + SSe

  7. Coefficients • Coefficient of determination (goodness of fit): R2 = SSr/SSt • Coefficient of correlation: R = R2 = SSr/SSt        r = Cov(x,y)/sxsy

  8. Adjusted R2                                      (k-1)(1 - R2) • Adjusted R2: R2a = R2 -  -----------------                        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: S12 = S22             S12         F = ----------             S22

  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<a, reject the null hypothesis

  11. F Test for Regression Model • F test for regression model: • Null hypothesis: SSr = SSe              SSr/k         F = --------------,              SSe/N-k-1      k - number of parameters excluding b0 N - sample size

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

  13. 5. Multiple Regression • Yi = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm + ei • Y = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm

  14. Regression Results • Analysis of variance DF Sum of Squares Mean Square Regression 3 97747.09184 32583.03061 Residual 36 7061.68316 196.15787 F = 166.10616 Signif 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 Variable b Se b Beta t Sig t X1 0.1917 0.001715 0.725998 6.262 0.0000 X2 -0.0829 0.001219 -0.994050 -16.161 0.0000 X3 -4.9594 11.079785 -0.052423 -0.4841 0.0310 X4 5.3639 7.3908 7.9273 -0.932 0.0926 Y = 0.1917X1 - 0.0829X2 - 4.9594X3 + 5.3639X4

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