Randomized Complete Block and Repeated Measures Each Subject Receives Each Treatment Designs KNNL Chapters 21 27 1 2 Block Designs Prior to treatment assignment to experimental units we may have information on unit characteristics When possible we will create blocks of homogeneous units based on the characteristics Within each block we randomize the treatments to the experimental units Complete Block Designs have block size number of treatments or an integer multiple Block Designs allow the removal of block to block variation for more powerful tests When Subjects are blocking variable use Repeated Measures Designs with adjustments made to Block Analysis in many cases the analysis is done the same Randomized Block Design Model Estimates Blocks made based on specified categories age gender day of week etc Fixed Effects Model Yij m r i t j eij i 1 nb j 1 r where m overall mean th r i effect of i block typically row effect with nb r i 0 i 1 th t j effect of j treatment typically column effect with r t j 1 eij N 0 s 2 independent Yij N m r i t j s 2 independent Least Squares Estimators m Y r i Y i Y t j Y j Y Y ij m r i t j Y Y i Y Y j Y Y i Y j Y eij Yij Y ij Yij Y i Y j Y j 0 Analysis of Variance nb nb Block Row Sum of Squares SSBL r Y i Y i 1 2 r r i2 E MSBL s 2 i 1 nb 1 df BL nb 1 r r Treatment Column Sum of Squares SSTR nb Y j Y j 1 nb r 2 dfTR r 1 Error Block Trt Sum of Squares SSBL TR Yij Y i Y j Y i 1 j 1 df BL TR nb 1 r 1 nb t 2j j 1 E MSTR s 2 r 1 2 E MSBL TR s 2 Testing for Treatment Effects Rarely interested in Block Effects Except to reduce Experimental Error H 0 t 1 t r 0 No Treatment Differences H A Not all t j 0 Test Statistic F MSTR MSBL TR Rejection Region F F 0 95 r 1 nb 1 r 1 RBD Non Normal Data Friedman s Test When data are non normal test is based on ranks Procedure to obtain test statistic Rank r treatments within each block 1 smallest r largest adjusting for ties Compute rank sums for treatments R j across blocks H0 The r populations are identical HA Differences exist among the r group means 12 r 2 T S X R 3nb r 1 j 1 j nb r r 1 2 F 2 F R R X 2 r 1 2 2 F P val P X Checking Model Assumptions Strip plots of residuals versus blocks equal variance among blocks all blocks received all treatments Plots of residuals versus fitted values and treatments equal variances Plot of residuals versus time order in many lab experiments blocks are days independent errors Block treatment interactions Tukey s test for additivity Comparing Treatment Effects All Pairs Tukey s Method HSD jj q 1 a r nb 1 r 1 MSBL TR nb Conclude t j t j if Y j Y j HSD jj Simultaneous Confidence Intervals Y j Y j HSD jj Bonferroni s Method r r 1 2 of Pairs of Treatment Means a 2MSBL TR BSD jj t 1 r r 1 2 nb 1 r 1 nb 2 Conclude t j t j if Y j Y j BSD jj Simultaneous Confidence Intervals Y j Y j BSD jj Extensions of RCBD Can have more than one blocking variable Gender Age among Human Subjects Region Size among cities Observer Day among Reviewers Note Observers are really subjects same individual Can have more than one replicate per block but prefer to have equal treatment exposure per block Can have factorial structures run in blocks usual breakdown of treatment SS Problems with many treatments non homogeneous blocks Main Effects Interaction Effects Relative Efficiency Measures the ratio of the experimental error variance for the Completely Randomized Design r2 to that for the Randomized Block Design b2 Computed from the Mean Squares for Blocks and Error Represents how many observations would be needed per treatment in CRD to have comparable precision in estimating means standard errors as the RBD s r2 E 2 sb sr2 nb 1 MSBL nb r 1 MSBL TR E 2 sb nb r 1 MSBL TR Sometimes the efficiency is modified to reflect differences in Error df E df 2 1 df1 3 E df 2 3 df1 2 where df1 df CRD r nb 1 and df 2 df RBD r 1 nb 1 Repeated Measures Design Subjects people cities supermarkets etc are selected at random and assigned to receive each treatment in random order Unlike block effects which were treated as fixed subject effects are random variables since the subjects were selected at random Measurements on subjects are correlated however conditional on a subject being selected they are independent no carry over effects or order effects The analysis is conducted in a similar manner to Randomized Complete Block Design Repeated Measures Design Model Subjects Randomly Selected and Assigned to Each Treatment Random Effects Model Yij m r i t j eij i 1 s j 1 r where m overall mean r i effect of i th subject typically row effect with r i N 0 s r2 independent th t j effect of j treatment typically column effect with r t j 0 j 1 eij N 0 s 2 independent Yij N m t j s r2 s 2 s Yij Yij s r2 j j r e independent s Yij Yi j s r2 i i s 2 Yij Yij s 2 Yij s 2 Yij 2s Yij Yij s r2 s 2 s r2 s 2 2s r2 2s 2 s 2 Y j Y j 2s 2 s s 2 Y j Y j 2 MSTR S s s Y j Y j 2MSTR S s Repeated Measures Design ANOVA s Subjects Sum of Squares SSS r Y i Y i 1 2 df S s 1 E MSS s 2 rs r2 r r Treatment Sum of Squares SSTR s Y j Y j 1 2 dfTR r 1 s r s t 2j j 1 E MSTR s 2 r 1 Error Subject by Treatment Sum of Squares SSTR S Yij Y i Y j Y i 1 j 1 dfTR S r 1 s 1 2 E MSTR S s 2 s r 2 Within Subjects Sum of Squares SSW Yij Y i SSTR SSTR S i 1 j 1 Testing for Treatment Effects H 0 t 1 t r 0 Test Statistic F MSTR MSTR S H A Not all t i 0 Rejection Region F F 1 a r 1 r 1 s 1 s 1 MSS s r 1 MSTR S sr 1 MSTR S Relative Efficiency E Completely Randomized Design needs n s E replicates …