Environmental ModelingEnvironmental Modeling

Basic Testing Methods - Statistics IIIBasic Testing Methods - Statistics III

1. Covariance1. Covariance

► Joint variation of two variables about their common mean

► Covariance

2. Simple Regression2. Simple Regression

► Regression: models relationships between variables

  Yi = 0 + 1Xi + ei, 0 - intercept, i – slope

Y is the dependent variableX is the independent variableSimple regression has one independent variable Multiple regression has more than one indep var

2. Simple Regression..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

2. Simple Regression..2. Simple Regression..

► Least square methods can help identify the best fit, following two conditions

   n

    (Yi - Yi)2 = minimum; ei = 0   (Yi - Yi = ei)    

1 ► Parameters estimated in the process: b0, b

  Y = b0 + bX

b0 – intercept, b – regression coefficient

3. Goodness of Fit3. Goodness of Fit 1

►Total Sum of Squares: SSt = (Yi - Y)2     n

1

►Sum Squares of Regression: SSr = (Yi - Y)2

n

                      1

►Sum Squares of residuals: SSe = (Yi - Yi)2

                         n      

SSt = SSr + SSe

CoefficientsCoefficients

► Coefficient of determination (goodness of fit):

R2 = SSr/SSt

► Coefficient of correlation: R = R2 = SSr/SSt

       r = Cov(x,y)/sxsy

Adjusted R2

                                      (k-1)(1 - R2) ►Adjusted R2: R2

a = R2 -  -----------------

                       N - k N - sample size k - number of independent variables

4. Test of Regression Model4. Test of Regression Model

► General F test: equality of two variances

► Null hypothesis: S12 = S2

2

            S12

        F = ----------             S2

2

 

Test of Regression ModelTest 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

F Test for Regression ModelF 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

t Test for bt Test for b

► t test for individual parameters b ► Null hypothesis: bi = 0

       bi

   t = ------,  Sbi - standard error of bi

       Sbi

5. Multiple Regression5. Multiple Regression

► Yi = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm + ei

► Y = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm

Regression ResultsRegression Results

► Analysis of varianceAnalysis of varianceDF DF Sum of Squares Sum of Squares Mean SquareMean Square

Regression Regression 3 3 97747.0918497747.09184 32583.0306132583.03061ResidualResidual 36 36 7061.68316 7061.68316 196.15787 196.15787

F = 166.10616F = 166.10616 Signif F = 0.0000Signif F = 0.0000

Multiple rMultiple r 0.87328 0.87328R SquareR Square 0.76262 0.76262Adjusted R Square Adjusted R Square 0.75701 0.75701Standard ErrorStandard Error 14.00564 14.00564

Regression ResultsRegression Results

► Variables in the EquationVariables in the EquationVariableVariablebb Se b Se b Beta Beta t t Sig t Sig t

XX11 0.1917 0.1917 0.0017150.001715 0.7259980.725998 6.262 6.262 0.00000.0000

XX22 -0.0829-0.0829 0.0012190.001219 -0.994050-0.994050 -16.161 -16.161 0.00000.0000

XX33 -4.9594-4.9594 11.07978511.079785 -0.052423-0.052423 -0.4841 -0.4841 0.03100.0310

XX44 5.3639 5.3639 7.39087.3908 7.92737.9273 -0.932-0.932 0.09260.0926

Y = 0.1917XY = 0.1917X11 - 0.0829X - 0.0829X22 - 4.9594X - 4.9594X33 + 5.3639X + 5.3639X44