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Stratification (Blocking) Grouping similar experimental units together and assigning different treatments within such groups of experimental units A technique used to eliminate the effects of selected confounding variables when comparing the treatment If you anticipate a difference between morning and afternoon measurements: Ensure that within each period, there are equal numbers of subjects in each treatment group. Take account of the difference between periods in your analysis.

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The Blocking Principle Blocking is a technique for dealing with nuisance factors A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter. However, the variability it transmits to the response needs to be minimized Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units Many industrial experiments involve blocking (or should) Failure to block is a common flaw in designing an experiment

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The Blocking Principle If the nuisance variable is known and controllable, we use blocking If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

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4 The comparison of various statistical procedures A measure of the optimality of an estimator, of an experimental design or of an hypothesis testing procedures A more efficient estimator, experiment or hypothesis testing needs fewer samples than a less efficient one to achieve a given performance Efficiency Relative Efficiency A comparative efficiency of the two estimator of the same paramaters The ratio of two efficiency statistics

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5 Relative Efficiency It is difficult to prove that an estimator is the best among all estimators, a relative concept is usually used. The efficiency and the relative efficiency depend theoretically on the sample size available for a given procedures Efficiencies are often defined using the variance or mean square error as the measure of desirability

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6 Relative Efficiency For experimental design, efficiency relates to the ability of a design to achieve the objective with minimal expenditure of resource such as time and money In simple cases, the relative efficiency can be expressed as the ratio of the sample sizes required to achieve a given objective

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Relative Efficiency of RCBD to CRD RE(RCB, CR): the relative efficiency of the randomized complete block design compared to a completely randomized design Did blocking increase the precision for comparing treatment means in a given experiment?

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Relative Efficiency of RCBD to CRD

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Relative Efficiency of LS to CRD RE(LS, CR): the relative efficiency of the Latin square design compared to a completely randomized design Did accounting for row/column sources of variability increase the precision in estimating the treatment means?

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Relative Efficiency of LS to CRD

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Chapter 3 Analysis of Variance

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