Basic
Bloc
king
ENSAI 3 eAnnée 2017-2018V
ariance
Reduction
Design
W e no w b egin a new phase of the course where w e move aw ay from completely randomized designs. P ow er increases and ma rgin of erro r decreases if N is la rger or if σ 2 is smaller. It costs resources to mak e N bigger. This pa rt of the course is ab out making σ 2 smaller.Blo
cking
The pri nc ipal to ol in va riance reduction design is blo cking. A blo ck is a homogeneous su bse t of units. Prio r to running the exp eriment, w e kno w these units are simila r in some w ay that w e exp ect will mak e them lik ely to h ave simila r resp onses. Simila r soil , simila r instrument calib ration, simila r batch of ra w material, simila r op erato rs, simila r genetics, simila r environmental conditions, simila r so cio-economic background, simila r something or other.Blo cking will b e some fo rm of rep eating the exp eriment (o r pa rt of it) sepa rately and indep endently in ea ch blo ck. This restricts the randomization of tr ea tm ents to units. With ap ologies to W o o dw ard, Bernstein, and Deep Throat: F ollo w the randomization! Different designs co rresp ond to differe n t randomizations, and examining the randomization can allo w you to discern the design.
Randomized
Complete
Blo
ck
design
The RCB is the progenito r of all blo ck designs. W e have: g treatments g units p er blo ck r blo cks rg = N total units Within each blo ck, randomly and indep endently assign the g treatments to the g units. It’s lik e r single-replication CRDs glued together. Notes: This is a complete blo ck design b ecause every treatm ent o ccurs in eve ry blo ck. The treatments could have facto rial structure. Consider blo cking when you can identify a source of va riab ilit y prio r to exp erimentation. Blo cking is done at the time of randomization; it is not imp osed later. The randomization in an exp eriment could id en ti fy it as RCB.Mo del: yij = µ = αi + βj + ij i = 1, .. ., g ; j = 1, .. ., r W e think that units in some blo cks might resp ond high, units in others might resp ond lo w, but within a blo ck units are mo re homogeneous (les s va riable) than ra ndomly chosen units from the universe of units. F or a tw o-facto r treatment d esign, w e w ould use the mo del: yijk = µ = αi + βj + α βij + γk + ijk i = 1, .. ., a; j = 1, .. ., b ; k = 1, .. ., r The mo del assumes that treatments have th e same effect in every blo ck, i.e., treatme n ts and blo cks are additive. Assuming additivit y do es not mak e additivit y true; transfo rmation of the resp onse can sometimes imp rove additivit y. Because there is only a single observation fo r each treatment in each blo ck, w e cannot distinguish b et w een random erro r and any p otential interaction b et w een treatments and blo cks. F rom a practical p ersp ective, it do esn’t matter much whether w e think of blo cks as fixed or random. F rom a theo retical p ersp ective, b lo cks are probably ran dom in most situations. Why do es RCB w ork? Here are a couple p oints of view. Mak e compa risons within blo cks (thus small va riance), and then combine across blo cks. T reatment totals all contain the same blo ck totals, so blo ck effects cancel out when compa ring treatment totals (and simila rly fo r treatment averages). Blo ck to blo ck va riabilit y is still in the totalit y of va ria bilit y in the data, but w e contrive to mak e it disapp ea r when compa ring treatments. Do not test blo cks! Note that blo cks are in the nature of the units. There ought to b e big differences b et w een blo cks. W e did not assign blo cks to units. Blo cks w ere not ra ndomly assigned, so there is no randomization test fo r blo cks. The soft w are do es n ot kno w any b etter than to test blo cks, but no w you do! F or unbala nc ed data, alw ays lo ok at treatme nts adjusted fo r blo cks (i.e., blo cks are alw ays in the b ase mo del fo r any treatment facto r). F or balanced data, blo cks and treatments are orthogonal.
Relative
efficiency
Ho w w ell did bl o cking w ork? S hould w e use blo cking in our next simila r exp eriment? “T esting” the blo ck effect is not what matters. W h at matters is ho w la rge the er ro r va riance w ould have b een if w e had not blo ck ed. Relative efficiency answ ers the follo wing question: using the same universe of units, by wha t facto r w ould w e nee d to increase our sample size to get the same p ow er in a CRD that w e w ould achieve using the RCB? This is mostly an issue of ho w the erro r va riance changes, but there is also a mino r effect due to the fact that fitting blo cks use s up degrees of freedom fo r erro r. (In CRD, bigger σ 2 hurts, but la rger dferro r helps; usually the first facto r d om ina te s.) W e first estimate wha t erro r va riance w ould have b een if w e had used a CRD instead of RCB, then w e mak e a mino r df adj ustm ent. W e estimate σ 2 RCB by MS E or residual va riance in the RCB analysis. We estimate σ 2 CRD via bσ 2 CRD = dfblo cks MS blo cks + [df T rt + dferro r ]MS E dfblo cks + dfT rt + dferro r = (r − 1) MS blo cks + [( g − 1) + (r − 1)( g − 1)] MS E (r − 1) + (g − 1) + (r − 1)( g − 1) This is an average of MSE and MSBlo ck w eighted by df, bu t w e use df erro r plus df treatments as the w eight fo r MSE. T ypically this estimate is less than the MSE you w ould get if you just left blo cks out of the mo del. The df adj us tment is less obvious. Let νRCB = (r − 1)( g − 1) b e the erro r df in the RCB analysis. Le t νCRD = rg − g b e the err or df if you had not blo ck ed. The estimated relative efficiency of RCB to CRD is ERCB :CRD = νRCB + 1 νRCB + 3 νCRD + 3 νCRD + 1 bσ 2 CRD bσ 2 RCB If this ratio is 1.7, then you w ould need 1.7 times as many units in a CRD to achieve the same p ow er as an RCB.Latin
Squa
res
An RCB is an effective w ay to blo ck on one source of extraneous va riation; what if you ha ve tw o sources of extraneous va riation? Light and dr ainage in ga rden flo w er trials; gender and blo o d pressure in ca rdiac trials; driver and environmental con ditions in MPG trials Think back to why RCB designs w ork; w e w an t to get that cancellation of blo ck effects to happ en simultaneously fo r tw o blo cking facto rs. The Latin Sq ua re design is the classical design fo r blo cking on tw o sources of va riation. There are g 2 units visualized as a squa re. Those units in the same ro w are all in the same blo ck ba sed on the first extraneous source of va riation. Those units in the same column are all in the same blo ck based on the second extra ne ou s source of va riation. The g treatments are randomized so that each treatment o ccurs once in each ro w and once in each column. T reatments are rep resented by Latin letters, thus Latin Squa res. A B C D B A D C C D A B D C B A If you igno re columns, a Latin Squa re is an RCB in ro ws. If you igno re ro ws, a Latin Squa re is an RCB in columns. Randomization is often done mo re lik e this. T ak e a squa re from a table of squa res (back of the b o ok). Randomly p ermute th e ro ws and the columns. Randomly assign treatments to the letters. Not as random as the ”ra ndomize subject to” description, but generally go o d enough and a lot simpler. Mo del: yijk = µ + αi + βj + γk + ijk i, j, and k all run from 1 to g. Note: w e only obse rve g 2 of the g 3 i, j, k combinations. W e are again assuming additivit y in a majo r w ay , and w e might need to transfo rm to achieve additivit y. F or unbalanced data, treatments adjusted fo r all blo cking facto rs.What if w e need mo re data to achieve acceptable p ow er? In addition, if you think of ro ws and columns as fixed effects, w e have (g − 1)( g − 2) degrees of freedom fo r erro r. That might not b e very many . Latin Squa res are often replicated, i.e., w e use mo re than one squa re with the same set of treatments. Ho w ever, w e need to consider ho w the repli ca ti on is done. Supp ose that w e have r squa res to study g treatments. All squa res will have ro w blo cks and column blo cks. The is sue is whether the squa res ha ve the same ro w blo cks or different ro w blo cks; simila rly fo r columns. Example: th ree squa res (r = 3) fo r g = 4 treatments; 48 total units Treatments: gasoline additives Resp onse: pa rtic u late emissions Ro w blo cks: drivers Column blo cks: ca rs Option 1: every squa re uses different drivers and different ca rs. yijk ` = µ + αi + βj + γk (j ) + δ`( j) + ijk ` This is mean plus treatment plus squa re plus ca r-nested-in-squa re plus driver-nested-in-squa re plus erro r. There are (g-1) df fo r treatments, (r-1) df fo r squa res, r(g-1) df fo r ca rs within squa re, r(g-1) df fo r drivers within squa re. Ca rs 1–4 Drivers1–4 A B C D B A D C C D A B D C B A Ca rs 5–8 Drivers5–8 A B C D B C D A C D A B D A B C Ca rs 9–12 Drivers9–12 D C A B A B D C C D B A B A C D
Option 2: every squa re uses the same ca rs but different drivers. yijk ` = µ + αi + βj + γk + δ`( j) + ijk ` This is mean plus treatment plus squa re plus ca r plus driver-nested-in-squa re plus erro r. There ar e (g-1) df fo r treatments, (r-1) df fo r squa res, (g-1) df fo r ca rs, r(g-1) df fo r drivers within squa re. Ca rs 1–4 Drivers1–4 A B C D B A D C C D A B D C B A Drivers5–8 A B C D B C D A C D A B D A B C Drivers9–12 D C A B A B D C C D B A B A C D Option 3: every squa re uses the same drivers but different ca rs. yijk ` = µ + αi + βj + γk (j ) + δ` + ijk ` This is mean plus treatment plus squa re plus ca r nested in squa re plus driver plus erro r. There are (g-1) df fo r treatments, (r-1) df fo r squa res, r(g-1) df fo r ca rs within squa re, (g-1) df fo r drivers. Ca rs 1–4 Drivers1–4 A B C D B A D C C D A B D C B A Ca rs 5–8 A B C D B C D A C D A B D A B C Ca rs 9–12 D C A B A B D C C D B A B A C D
Option 4: every squa re uses the same drivers and the same ca rs. yijk ` = µ + αi + βj + γk + δ` + ijk ` This is mean plus treatment plus squa re plus ca r plus driver plus erro r. There ar e (g-1) df fo r treatments, (r-1) df fo r squa res, (g-1) df fo r ca rs, (g-1) df fo r drivers. Ca rs 1–4 Drivers1–4 A very common example is the cross over design. In a cross over, one of the blo cking facto rs is time p er io d, and the other blo cking facto r is subject. Each subject has each treatment, but some get one tr ea tment first, others have another treatment first, and so on. T o replicate these designs, w e generally get a new set of subjects, but the p erio d effects are assumed to b e the same fo r all squa res. W e can also compute the relative effic ie ncy of a Latin Squa re relative to an RCB should w e consider not using one of the blo cking facto rs. F or example, if w e consider not using ro ws, then W e estimate σ 2 LS by MS E in the Latin Squa re analysis. W e estimate σ 2 RCB via bσ 2 RCB = dfro ws MS ro ws + [df T rt + dferro r ]MS E dfro ws + dfT rt + dferro r
Let νLS b e the erro r df in the LS analysis. Let νRCB b e the erro r df if you had not blo ck ed on ro ws. The estimated relative efficiency of LS to R CB is ELS :CRD = νLS + 1 νLS + 3 νRCB + 3 νRCB + 1 bσ 2 RCB 2 LSbσ If this ratio is 1.7, then you w ould need 1.7 times as many units if you had run a RCB instead of an L S.
Generalizations
There are many p ossible generalizations of these blo cking designs. The Generalized Randomized Complete Blo ck design is analogous to an RCB except each blo ck has 2g or 3g etc. units and each treatment is assigned to 2 or 3 etc. units w ithin each blo ck. In this case, the standa rd app roach is to mo del blo cks as random and include a random blo ck by treatment interaction term. The ca rry over design, or design balanced fo r residual effects, is a Latin Squa re where, in addition to the usual re q uireme nts, w e also have that each treatment follo ws each other treatment exactly once. This is useful when one of the blo cking facto rs is time p erio d, and the effe ct of one treatment could ca rry over into the next time p erio d. F or example, a to xic drug might not only supp res s the resp onse in the p erio d where it is given, it could also supp ress the re sp onse in the follo wing p erio d. The mo del conta ins an additional facto r with g+1 levels “follo ws treatment 1” up to “follo ws treatment g” and the final le vel of “used first.” If you have three blo cking facto rs, then you can use a Graeco-Latin squa re. Latin letters are treatments, Greek letters indicate third blo cking facto r. Each treatment o ccurs on ce in each ro w, once in each column, and once with each Greek letter. A α B γ C δ D β B β A δ D γ C α C γ D α A β B δ D δ C β B α A γ No 6 by 6 GL squa re. Mo del has additive treatment and (three) blo cking facto rs.Incomplete
Blo
cks
Complete blo ck designs lik e RCB and LS ar e set up with every treatment o ccurring in every blo ck. Sometimes, there are only k units in a blo ck, and k < g . Then w e must use an incomplete blo ck design. Three different ey e drops to study relief from irritation. There is la rge subj ect to subject va riabilit y, so blo ck on subject, but only tw o ey es p er subject. Six different pro cesses fo r extracting avo cado oil. There is la rge fruit to frui t va riabilit y, so blo ck on fruit, but each fruit is only la rge enough to test four pro cesses. Incomplete blo ck designs are inherently less efficient than complete blo ck designs on a p er unit basis with equal va ria nc es. Example: A B A C C B versus A B C A B C In the complete blo ck, unit 1 -unit 2 and unit 4 -unit 5 b oth estimate A -B and b oth have va riance 2σ 2 comp . In the incomplete blo ck, unit 1 -unit 2 and unit 3 -unit 4 + unit 5 -unit 6 b oth estim ate A -B and have va ria nce s 2σ 2 incomp and 4σ 2 incomp W e prefer complete blo cks if σ 2 comp = σ 2 incomp , but often σ 2 comp > σ 2 incomp , and th at can b e where incomplete blo cks are preferred. F or example, supp ose fruit to fruit va riance in oil concentration is 10, but qua rter to qua rter within a fruit va riance is 1. The relative efficiency of a BIBD with six treatments in blo cks of size four is .9. Without blo cking, va rian ce of a pairwise difference in means is 10( 1 n+ 1 n) = 20 n . With a BIBD, va riance of a pairwise difference in means is 1( 1 .9n + 1 .9) n = 2 .22 n . Here the reduced va riance achievable with the BIBD overcomes the loss due to relative effic ie ncy of the BI BD to the RCB.Balanced
Incomplete
Blo
ck
design
The basic protot yp e of all incomplete blo ck designs is the BIBD. Here w e have: g treatments b blo cks k units p er blo ck each treatment used r times bk total uni ts bk = rg In addition, ea ch pair of treatments o ccurs together in the same numb er of blo cks. λ = r( k − 1) / (g − 1).A B A C C B versus A B A B C C Both have g=3, k=2, r=2, b=3, but left side is BIBD and right side is not. Notes: If λ = r( k − 1) /( g − 1) is not a whole numb er, then no BIBD fo r that set of pa rameters. T reatments could have facto rial structure. Also balanced in the sense that va riance of ˆαi − ˆαj do es not dep end on i,j. A BIBD al w ays exists fo r any g > k pair; simply tak e all combinations. There are tables fo r smaller values of b and r. Randomize by randomly assigning the treatments to th e treatment “numb ers,” and then randomly assigning treatment numb ers to units within their blo cks. Mo del: yij = µ + αi + βj + ij F or RCB, it didn’t really matter if blo cks w ere fixed or random. F or BIBD, it do es matter. If w e assume blo cks are fixed, w e get the intrablo ck analysis. All estimates are based on differences from within blo cks. If w e assume blo cks are random, then there is some info rmation ab out the treatments in blo ck totals; this leads to the interblo ck recovery analysis. Interblo ck recovery provides slightly mo re precise estimates in case s where it is app rop riate (i.e., when blo cks are random), but la rge blo ck va riance relative to units-within-blo cks va riance means the imp rovement is often negligible. Interblo ck recovery used to b e a complicated pro cess (old scho ol), but with lmer(), interblo ck recovery is not much extra effo rt.
Intrablo ck is just treatments adjusted fo r fixed blo cks; let R do the w or k. If you could do RCB with same va riance as BIBD, then EBIBD :RCB = g (k − 1) (g − 1) k is the relative efficiency . The effective sample size is rE V ar ( X i wi ˆαi ) = σ 2 X i w 2 i rE SS T rt = X i rE ˆα 2 i
