Regression Modeling Applications in Land use and Transport

The term linear regression implies that  Y|x is linearly related to x by the population regression equation  Y|x =  +  x where the regression coefficients  and  are parameters to be estimated from the sample data. Denoting their estimates by a and b, respectively, we can then estimate  Y|x by  y from the sample regression or the fitted regression line  y = a + bx where the estimates a and b represent the  y intercept and slope, respectively. The symbol  y is used here to distinguish between the estimated or predicted value given by the sample regression line and an actual observed experimental value y for some value of x.

Estimating the Regression Coefficient. Given the sample {(x i, y i ); i = 1, 2, …, n}, the least squares estimates a and b of the regression coefficients  and  are computed from the formulas and

Example: A land use planner observed that in five zones of the city, the number of gas stations (Y) in relation to the population (X) in 1000’s was a.Set up a linear regression equation connecting Y in terms of X. b.Determine R 2. X15324 Y27358 Solution:

SUMMARY OUTPUT Regression Statistics Multiple R R Square0.65 Adjusted R Square Standard Error Observations5 ANOVA dfSSMSF Significance F Regression Residual Total426 Coefficien ts Standar d Errort StatP-valueLower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept (a) X (b) EXCEL Output

xyxyx^ total mean35

Sample coefficient of determination, r 2 xyxyx^ total mean35 S yy = 26 S xy = 13 r 2 = (13) 2 /10(26) = 0.65

t-distribution