Generalized Randomized Block Design and ... - People.stat.sfu.ca
Generalized Randomized Block Design and ... - People.stat.sfu.ca
Generalized Randomized Block Design and ... - People.stat.sfu.ca
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Two-way classifi<strong>ca</strong>tion: GRBD vs. RBDSTrt<strong>Block</strong>1 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
Mathemati<strong>ca</strong>l recently [7]. Modelhas been givenanalysisr<strong>and</strong>omized block design with either equal or unequalblock sizes is usually described in the literature as theanalysis for the two-way classifi<strong>ca</strong>tion with an interaction<strong>and</strong> either equal or unequal but proportional numbersof observations per classifi<strong>ca</strong>tion. Some of thebetter known references are [4], [5], [6] <strong>and</strong> moreThe two-way classifi<strong>ca</strong>tion is almost always introducedby either a two-way table with single or multipleobservations in each cell or by a <strong>stat</strong>ement such as<strong>Generalized</strong> r<strong>and</strong>omized block design with equal""Assume that we have a two-way classifi<strong>ca</strong>tion withinteraction <strong>and</strong> equal cell frequencies". There thenfollows the mathemati<strong>ca</strong>l 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 classifi<strong>ca</strong>tion are thenpresented for situations where the interaction effect isor is not signifi<strong>ca</strong>nt, aiid when one or both of the factorsare fixed or r<strong>and</strong>om.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 <strong>Block</strong> Treatment*<strong>Block</strong>(R) Plot(Treatment*<strong>Block</strong>) (R)is the erroaccounts ftional unitIt is evidnot possiblmodel woument wasusing an inmental dealways beThe genequal sizedas was theblock sizesmental unis r<strong>and</strong>omlunequal sizk now rangAn awadesign shodomized bexperimenbe equal towhether orinto that
Mathemati<strong>ca</strong>l Model<strong>Generalized</strong> r<strong>and</strong>omized block design withunequal block sizesIt may be obtained in the same manner as was generalizedr<strong>and</strong>omized block design with equal block sizes expect thateach block will contain tri experimental units, i=1,2,...,b.
Why should we use GRBD?– The r<strong>and</strong>omized block design requires that the number of blocksbe equal to the number of repli<strong>ca</strong>tes 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 r<strong>and</strong>omized (complete)block design (RBD)?
A definition of experimental errorExperimental error measures the variance, assumedhomogeneous, of the levels of treatment factorsapplied to the experimental units, <strong>and</strong> thus is theappropriate error term for testing hypothesesconcerning the treatment factor.
Wrong underst<strong>and</strong>ingExperimental error is simply the interaction ofblocks with treatments!!!
What is a GRBD?It is an r<strong>and</strong>omized (complete) block design, RBD, in which more thanone experimental unit within blocks is assigned r<strong>and</strong>omly 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, <strong>and</strong>within experimental plot sampling variation.
Thank you very much!Lu Wang21