o What’s a block? - block looks like a treatment except you do not care about its effect it is a nuisance - call it: block, noise, covariate - which? hard to say * noise=control in experiment, not in real world EX: * covariate=like noise, generally conitnue EX: * block=goal: compare treatment averaged over block? level is a characterization not easily measured. EX: - why block? so large differences between experimental units don’t mask (confound) effect of treatment EX: S M L AM 50 35 30 PM 65 55 45 x-bar= s= later show with block can see effect of size o How To Block - select blocking category - select number of exp units per block * could have 2 in AM, 3 in PM beyond scope of text and class so k = same number of units in each block * so k units in b=# blocks, here 2 * what if let AMers do S,S,M and PMers M,L,L? EX: AM S=50 S=45 M=35 PM M=55 L=45 L=50 show later the confounding makes sense to do each treatment-level (v of these) combo in each block at least once. So k=s*v is called a complete block design when s=1 called (misnomer) randomized complete block if k not equal to s*v then an incomplete block design (ch.11) like: AM has 2 people and PM has 2 people...why? - how many blocks? or how big is s? 1. use (10.5.2) calc b by C.I. width approach or 2. use (10.6.10) calc s by a power of test approach via 1. later talk of multiple comps (follow-ups) Scheffe: (Y(.m) - Y(.p))+- sqrt((v-1)*F(dfn=v-1, dfd=bv-b-v+2)* sqrt(MSE*(2/b)) EX: “boss says only 3 people a day” we want bound=5 seconds trial and error in A6... b=4, b=6? 2. H0: no effect H1: effect nc F(dfn=v-1, dfd= bvs-b-v+1; phi=sqrt(sb/2v)*(delta/sigma)) EX: “boss says all in a day” delta=5 trial and error in A7...s=3, s=4? - despite s.s. above let us use b=2 v=3 s=1 still must randomly assign persons to bead sizes (A1) AM-P1 26 S AM-P2 45 L AM-P3 40 M PM-P1 58 M PM-P2 87 L PM-P3 15 S (rearranged to be SML on printout) trouble? what if P3 is late and lazy? o The Analysis of Randomized Complete Block - see printout model: Y(hi)=m+O(k)+t(i)+e(hi) where e(hi)~N(0,sigma-squared) h=1,...,b i=1,....,v called block-treatment Note: no interaction...not enough df and assume not - check assumptions: normal, equal- constant variance, outliers, independence, form (see printout) OK...talk about zscore, rank,blom,cacee TOD SIZE TIME ZSCORE RANK (R-.375)/6.25 CACEE 1 1 50 .6460 4.55000 .6600 .4125 1 2 35 -1.2920 1.0000 .1000 -1.2816 1 3 30 .6460 4.5000 .6600 .4125 2 1 65 -.6460 2.5000 .3400 - .4125 2 2 55 1.2920 6.0000 .9000 1.2816 2 3 45 - .6460 2.5000 .3400 - .4125 TOD: 1.0000 Variable Mean Std Dev Variance N TIME 38.33 10.41 108.33 3 TOD: 2.0000 Variable Mean Std Dev Variance N TIME 55.00 10.00 100.00 3 SIZE: 3.0000 Variable Mean Std Dev Variance N TIME 37.50 10.61 112.50 2 SIZE: 2.0000 Variable Mean Std Dev Variance N TIME 45.00 14.14 200.00 2 SIZE: 1.0000 Variable Mean Std Dev Variance N TIME 57.50 10.61 112.50 2 Variable Mean Std Dev Variance N TIME 46.67 12.91 166.67 6 - looks like 2-way main effects and is analyzed in this manner, too Source SS df MS F --------------------------------------------------------- Block SSO b-1 Treat SST v-1 MST MST/MSE Err SSE bv- b-v+1 MSE Tot SStot n-1 Note: get SSE = SStot - rest Source SS DF MS F p-val SIZE 408.3 2 204.2 49.0 .020 TOD 416.7 1 416.7 100.0 .010 Error 8.3 2 4.2 Total 833.3 5 166.7 - Proper Questions 1. Is there a block effect -- not like this Rephrase: H0:blocking is not help in discerning between treatments (generally, I do not test) * why is blocking mathematically helpful? Source SS DF MS F p-val SIZE 408.3 2 204.2 1.4 .364 Error 425.0 3 141.7 Total 833.3 5 166.7 SST+SSO+SSE=SStot 2. H0: no treatment effect - note: warning ..if block dles not help do not rerun as one-way. (to illustrate, I did both) - note: the S,S,M v M,L,L has ANOVA 1 1 50 Source SS DF MS F p-val 1 1 45 SIZE 141.7 2 70.8 5.7 .150 1 2 35 TOD 200.0 1 200.0 16.0 .057 2 2 55 Error 25.0 2 12.5 - there is a signficant effect due to size do follow-ups - followups form: sum (c(i)*Y(.i)) contained in sum(c(i)*Y(.i)) +- w * sqrt (MSE*sum(c(i)-squared/b) w is Bonferonni= t(df=same as MSE, alpha/2m) A4 Scheffe= sqrt((v-1)*F(dfn=v-1, dfd=same as MSE, alpha) A6 Tukey= q(v,df=same as MSE,alpha) A8 Hsu & Dunnett1= t(.5)(v-1, df same as MSE, alpha) A9 Dunnett2 = t(.5)(v-1, df same as MSE, alpha) A10 EX: c1=1 c2=0 c3=-1 estimable? yes