Statistics for Environmental Engineers

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We will develop the least squares estimation procedure using matrix algebra. The matrix algebra is general for all linear regression problems (Draper and Smith, 1998). What is special for the balanced two-level factorial designs is the ease with which the matrix operations can be done (i.e., almost by inspection). Readers who are not familiar with matrix operations will still find the calculations in the solution section easy to follow.


The model written in matrix form is:


у = XP + e


where X is the matri* of independent variables, в is a vector of the coefficients, and y is the vector of observed values.


The least squares estimates of the coefficients are:


b = (X ‘X )-1X’y


The variance of the coefficients is:


Var( b) = (X’X)V


Ideally, replicate measurements are made to estimate a .


X is formed by augmenting the design matrix. The first column of+1 ’s is associated with the coefficient во, which is the grand mean when coded variables are used. Additional columns are added based on the form of the mathematical model. For the model shown above, three columns are added for the two-factor interactions. For example, column 5 represents *1*2 and is the product of the columns for and *2. Column 8 represents the three-factor interaction.


The matrix of independent variables is:


-1


-1


-1


+1


1


1


-1


1


-1


-1


-1


-1


1

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