Generalized Randomized Block Design and ... - People.stat.sfu.ca

Two-way classification: GRBD vs. RBDSTrtBlock1 2 ... t1r units r units r units...2r unitsr units...r units...b... ... ...r units r units......r units

GRBD with equal block sizesbtrbtr experimental unitstr tr ...... trb blocks, tr units per blockr ... tr r ... tr r ... tr5t treatments, r experimental units per treatment

RBDSbtt ...... btbt experimental unitsb blocks, t units per block1 ...... t t1 1 ...... 1one unit per treatment... r ... r ... r ... r6r observational units per experimental unit

Mathematical recently [7]. Modelhas been givenanalysisrandomized block design with either equal or unequalblock sizes is usually described in the literature as theanalysis for the two-way classification with an interactionand either equal or unequal but proportional numbersof observations per classification. Some of thebetter known references are [4], [5], [6] and moreThe two-way classification is almost always introducedby either a two-way table with single or multipleobservations in each cell or by a statement such asGeneralized randomized block design with equal""Assume that we have a two-way classification withinteraction and equal cell frequencies". There thenfollows the mathematical modelblock sizesYijk = t' + /i + ri + (/T) ii + CEjk (1)i = 1,2... b; j = 1,2.... t; k = 1,2....with the terms defined in the usual manner. The analysistechniques for the two-way classification are thenpresented for situations where the interaction effect isor is not significant, aiid when one or both of the factorsare fixed or random.Unfortunately, there are two different experimentaldesigns that will produce the same two-way table withmultiple observations, only one of which results in themodel given by (1). The design that results in model (1) Y=Treatment Block Treatment*Block(R) Plot(Treatment*Block) (R)is the erroaccounts ftional unitIt is evidnot possiblmodel woument wasusing an inmental dealways beThe genequal sizedas was theblock sizesmental unis randomlunequal sizk now rangAn awadesign shodomized bexperimenbe equal towhether orinto that

Mathematical ModelGeneralized randomized block design withunequal block sizesIt may be obtained in the same manner as was generalizedrandomized block design with equal block sizes expect thateach block will contain tri experimental units, i=1,2,...,b.

Why should we use GRBD?– The randomized block design requires that the number of blocksbe equal to the number of replicates of each treatment, whetheror not the experimental units divide naturally into that manyblocks of homogeneous units;– GRBD allows us to test the block-treatment interaction;– GRBD gives the advantage of blocking---increased precision.

Experimental ErrorExactly what is the composition of " experimentalerror " in designs such as the randomized (complete)block design (RBD)?

A definition of experimental errorExperimental error measures the variance, assumedhomogeneous, of the levels of treatment factorsapplied to the experimental units, and thus is theappropriate error term for testing hypothesesconcerning the treatment factor.

Wrong understandingExperimental error is simply the interaction ofblocks with treatments!!!

What is a GRBD?It is an randomized (complete) block design, RBD, in which more thanone experimental unit within blocks is assigned randomly the sametreatment. Example 1RBDGRBD

GRBDSY= B T B *T(R) P(B *T) (R) S(P*B*T) (R)

GRBDS with s=1GRBDY= B T B *T(R) P(B *T) (R)

GRBDS with p=1RBDSY= B T B *T(R) S(B*T)(R)

GRBDS with p=s=1RBDY= B T B *T(R)

ConclusionExperimental error for testing treatments, is comprisedof three sources of variability: block by treatmentinteraction, within block plot-to-plot variability, andwithin experimental plot sampling variation.

Thank you very much!Lu Wang21