# Statistics for Environmental Engineers

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If simplification seems indicated, a simplified version is fitted to the data. We show later how the full model and simplified model are compared to check whether the simplification is justified.

To deal with three categories, two categorical variables are defined:

Category 1:    Z1    =    1    and    Z2 = 0

Category 2:    Z1    =    0    and    Z2 = 1

This implies Z1 = 0 and Z2 = 0 for category 3. The model is:

yt = (a + a1 xt)+ Z 1(A, + в1 xi) + Z 2(70 + Y1 xi) + et

The parameters with subscript 0 estimate the intercept and those with subscript 1 estimate the slopes. This can be rearranged to give:

yt = a0 + e0Z 1 + Yo Z2 + a1 xt + в1 Z1 xt + y{Z 2 xt + et

The six parameters are estimated by fitting the original independent variable xt plus the four created variables Z1, Z2, Z1xi, and Z2xt.

Any of the parameters might be estimated as zero by the regression analysis. A couple of examples explain how the simpler models can be identified. In the simplest possible case, the regression would

estimate во = 0, y0 = 0, в1 = 0, and yi = 0 and the same slope (a-i) and intercept (a0) would apply to all three categories. The fitted simplified model is yi = a0 + a1 xt + e{.

If the intercepts are different for the three categories but the slopes are the same, the regression would estimate в1 = 0 and y1 = 0 and the model becomes:

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