Modeling the effects of restricted randomization

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Johnston, Robert S.
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Because the usual linear model does not convey information pertinent to the randomization process, the linear model and associated expected mean squares are inadequate when randomization is restricted in some way. The linear model is extended to incorporate such information by viewing the realized observations as a fraction of the potential observations, and utilizing the idea of bias or aliasing in a manner similar to fractional designs. In particular, two areas of restricted randomization are examined (1) restricting the run-order of a sequential experiment and (2) restrictions resulting from inherent structure of the experimental material (blocking). In both cases treatment effects and effects associated with the randomization process may be non-additive, in contrast to some of the previous work that has assumed such effects are additive. In addition, the nature of the resulting bias is examined. Rules are specified that allow construction of pseudo expected mean squares that indicate the presence of bias that results from restricting the randomization process.