Coding of randomized complete block designs

Because of tradition, the model of Equation 15.32 is seldom used for the randomized complete block design. Instead, a somewhat different but essentially equivalent model is used  

For the example used in Section 15.5, there would be three y s and four x s (or, perhaps, four y s and three t s). If y is associated with the qualitative factor univalent cation , then y would be the average block difference in response between the experiments involving Li and the overall mean y and y, would be the corresponding differences for experiments involving Na and K. Similarly, x g, Xj-, x, and Xgj would be the average treatment differences for experiments involving the divalent cations Mg, Ca, Sr, and Ba. Thus, the full model would be 

certain terms are turned on or turned off to correspond to a particular factor combination ij. Notice that Equation 15.39 has eight parameters Equation 15.32 has only six parameters. It would appear that the two models are not equivalent 

The relationships between the parameters of Equations 15.32 and 15.39 are easily discovered. 

The remaining parameters of Equation 15.36 (y and x ) are calculated from Equations 15.40 and 15.41. Thus, if the model of Equation 15.32 has been used to treat the data from a randomized complete block design, the results may be readily converted to the form of Equation 15.36. 

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