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.

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

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

Efficiency

  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

Relative Efficiency

 A comparative efficiency of the two estimator of the same paramaters  The ratio of two efficiency statistics 4

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 Relative Efficiency  Variance Variance of of first second estimator estimator 5

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 6

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?

Relative Efficiency of RCBD to CRD

RE

(

RCB

,

CR

) 

MSE CR MSE RCB



b

(  )1

MSB



b t

(  (

bt

 )1

MSE

)1

MSE

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?

Relative Efficiency of LS to CRD



MSE

LS CR