o Last time block-treatment models - with and with interaction s=1 s>1 k=sv - worked like 2-way o onto incomple block designs k not equal sv (s not an integer) - EX: beasd size: S,M,L v=3 use k=2 then k can not equal sv - Dean discusses balanced, group divisible and cyclic. We will do balanced - equireplicate -rules of thumb: 1. each treatment appears 0 or 1 time 2. pairs occur lambda times (keeps CI equal length) -Mechanism I. Exp. plan = distribute treatments evenly II. EXp. design = 1. randomly assign blocks 2. randomly assign exp. units to treatments in blocks - EX1: size: S,M,L v=3 k=2 boss says 2days, 2 people per M-AM 12 M-PM 23 T-AM 23 T-PM 12 OK? - EX2: M-AM 12 M-PM 23 T-AM 13 T-PM 12 OK? - EX3: size=XS,S,M,L,XL,XXL v=6 k=2 M-AM 13 M-PM 35 T-AM 13 T-PM 24 M-AM 26 M-PM 46 OK? o balanced incomplete block *binary *lambda pairs *contrasts of treatments are estimable *do not exist for all v,k,r,b because necessary (not sufficient) conditions: a) n=vr=bk b)r(k-1)=lambda*(v-1) c)b>=v - EX4 (some impossible) : size=S,M,L v=3 k=2 try b’s - EX5: size=XS,S,M,L,XL,XXL v=6 3 shifts randomized complete block? complete block? balanced incomplete block? boss says 4 a day try r and b’s (see printout) note: remember to randomize note: SAS program can help o what does the model for incomplete block look like? model: Y(hi)=m+O(k)+t(i)+e(hi) where e(hi)~N(0,sigma-squared) h=1,...,b i=1,....,v ** where (h,i) in design ** - need to check assumptions - not just block-treatment, why? not (Y(.i)- Y(..))+(Y(.h)-Y(..)) Y(..) -method: careful accounting and look at errors..sum of squared errors and minimize, solving equations to get FOR BALANCED INCOMPLETE T'(i)=sum of tau(i)s with B(i) effect in it B'(i)=sum of B(i)s with T(i) effect in it tau(i)=(1/lambda*v)*(k*T(i)-B'(i)) theta(i)=(1/k)*B(i)-m-(1/k)*T'(i) error=obs-(m+theta+tau) SSE=sum of squared errors SStot=sum of squared (obs-m) SSO(unadjusted)=k* sum of squared (B(i)/k - m) SST(adjusted)=SStot-SSE-SSO - let SAS do others - EX: see printout - Source SS df MS F --------------------------------------------------------------- Block SSO b-1 Treat SST(adj) v-1 MST MST(adj)/MSE Err SSE bv- b-v+1 MSE Tot SStot n-1 Look at TypeIII SS - Ho: no treatment effect - Ho: blocks aren’t helpful...need MSO(adj)/MSE -assumptions...look at error column -Followup form: sum (c(i)*tau(i)) contained in sum(c(i)*tau-hat(i)) +- w * sqrt (MSE*sum(c(i)-squared)*(k/lambd*v)) balanced incomplete block example...chapter 11 60.00 50.00 40.00 150.00 B(1) 65.00 54.00 35.00 154.00 B(2) 58.00 38.00 31.00 127.00 B(3) 52.00 44.00 33.00 129.00 B(4) 183.00 156.00 122.00 99.00 560.00 T(1) T(2) T(3) T(4) 46.67 m 14.75 4.38 -5.00 -14.13 tau(1) tau(2) tau(3) tau(4) -1.42 2.95 -2.92 1.20 theta(1) theta(2)theta(3)theta(4) T'(i)=sum of T(i)s with B(i) effect in it B'(i)=sum of B(i)s with T(i) effect in it tau(i)=(1/lambda*v)*(k*T(i)-B'(i)) theta(i)=(1/k)*B(i)-m-(1/k)*T'(i) error=obs-(m+theta+tau) SSE=sum of squared errors SStot=sum of squared (obs-m) SSO(unadjusted)=k*sum of squared (B(i)/k - m) SST(adjusted)=SStot-SSE-SSO b t time theta tau pred err sqerr sqtot 1 1 60 -1.42 14.75 60.00 .004 0.00 177.777 1 2 50 -1.42 4.375 49.62 .379 0.14 11.111 1 3 40 -1.42 -5 40.25 -0.246 0.06 44.445 2 1 65 2.95 145 64.37 0.633 0.40 336.11 2 2 54 2.95 4.375 53.99 0.008 0.00 53.777 2 4 35 2.95 -14.125 35.49 -0.492 0.24 136.112 3 1 58 -2.92 14.75 58.50 -0.498 0.25 128.444 3 3 38 -2.92 -5 38.75 -0.748 0.56 75.112 3 4 31 -2.92 -14.125 29.62 1.377 1.90 245.445 4 2 52 1.20 4.375 52.24 -0.244 0.06 28.444 4 3 44 1.20 -5 42.87 1.131 1.28 7.111 4 4 33 1.20 -14.125 33.74 -0.744 0.55 186.779 5.44 1430.667 proc glm; classes blocktod size; model time=blocktod size ; lmeans blocktod; lmeans size/ pDIFF=all cl adjust=scheffe ALPHA=0.10; General Linear Models Procedure Dependent Variable: TIME Sum of Mean Source DF Squares Square F Value Pr > F Model 6 1425.25 237.54 219.27 0.0001 Error 5 5.42 1.08 Corrected tal 11 1430.67 Source DF Type I SS Mean Square F Value Pr > F BLOCKTOD 3 195.33 65.1111 60.10 0.0002 SIZE 3 1229.92 409.9722 378.44 0.0001 Source DF Type III SS Mean Square F Value Pr > F BLOCKTOD 3 55.25 18.4167 17.00 0.0047 SIZE 3 1229.92 409.9722 378.44 0.0001 General Linear Models Procedure Least Squares Means 90% 90% Lower Upper Confidence Confidence SIZE Limit TIME LSMEAN Limit 1 60.15 61.416667 62.68 2 49.78 51.041667 52.31 3 40.40 41.666667 42.93 4 31.28 32.541667 33.81 Adjustment for multiple comparisons: Scheffe Least Squares Means for effect SIZE 90% Confidence Limits for LSMEAN(i)-LSMEAN(j) Simultaneous Simultaneous Lower Difference Upper Confidence Between Confidence i j Limit Means Limit 1 2 7.404735 10.375000 13.345265 1 3 16.779735 19.750000 22.720265 1 4 25.904735 28.875000 31.845265 2 3 6.404735 9.375000 12.345265 2 4 15.529735 18.500000 21.470265 3 4 6.154735 9.125000 12.095265