Statistics for Environmental Engineers

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6.5-



pH



6.0



°8



О



О



°Со


в



о


о



5.5-


0


100′ 2О0 300′ 400′ 500′ 600′ 7О0


Weak Acidity (pg/L)


FIGURE 40.1 The relation of pH and weak acidity data of Cosby Creek after three storms.


Begin by considering data from a single category. The quantitative predictor variable is x1 which can predict the independent variable y1 using the linear model:


Ун = во + 01 xu + e,


where 0O and в1 are parameters to be estimated by least squares.


If there are data from two categories (e.g., data produced at two different laboratories), one approach would be to model the two sets of data separately as:


У ii = a + ai xu + ei


and


У 2i = во + 01 X2i + e,


and then to compare the estimated intercepts (a0 and 0о) and the estimated slopes (a1 and 01) using confidence intervals or t-tests.


A second, and often better, method is to simultaneously fit a single augmented model to all the data. To construct this model, define a categorical variable Z as follows:


Z = 0    if the data are in the first category


Z = 1    if the data are in the second category


The augmented model is:


y, = a0 + a1 x, + Z (0о + 01 x,) + e,


With some rearrangement:

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