Statistics for Environmental Engineers

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y, = a0 + 0O Z + a1 x, + 01 Zx, + e,


In this last form the regression is done as though there are three independent variables, x, Z, and Zx. The vectors of Z and Zx have to be created from the categorical variables defined above. The four parameters a0, 0о, a1, and 01 are estimated by linear regression.


A model for each category can be obtained by substituting the defined values. For the first category, Z = 0 and:


y, = a0 + a1 x, + e,


Slopes Diffferent


Slopes Equal


Intercepts


Different


Intercepts


Equal


y=(ao+Po) + (ai + Pi)x,-+e,-


y=(ao +Po) + aix + e,


y,-=ao +(ai + Pi)Xi+ei


Уi = ao+aiX,+e,

FIGURE 40.2 Four possible models to fit a straight line to data in two categories.


Complete model y=(aQ+Po)+(a1+p1)x+e


Category 2:


У = (ao+Po)+ai*+e


a0 +eo

FIGURE 40.3 Model with two categories having different intercepts but equal slopes.


For the second category, Z = 1 and:


yt = (a0 + A) ) + (a1 + A) xt + ei


The regression might estimate either P0 or в1 as zero, or both as zero. If во = 0, the two lines have the same intercept. If в1 = 0, the two lines have the same slope. If both в1 and в0 equal zero, a single straight line fits all the data. Figure 40.2 shows the four possible outcomes. Figure 40.3 shows the particular case where the slopes are equal and the intercepts are different.

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