Randomized Complete Block Design Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison March 22 2007 Statistics 572 Spring 2007 March 22 2007 The Big Picture 1 13 Randomized Block Designs The Big Picture A blocking variable is a categorical variable that is not the primary variable of interest where observations within each level ought to be homogeneous except for treatment In a randomized complete block design each treatment is applied to individuals selected at random within each block In a randomized incomplete block design treatments are assigned at random within blocks but every treatment may not be represented in every block Statistics 572 Spring 2007 March 22 2007 2 13 Example Seed Germination Experiment Seed Germination Experiment In an experiment four sites were selected where the soil and climate conditions were expected to be very similar within the site Here we will treat each site as a block Within each block five plots were identified The treatment was applying a seed disinfectant to seeds There were four different treatments brands plus a control The researchers planted 100 seeds from a single treatment in each plot The response is the number of seeds that germinated We will also analyze this data using a generalized linear model later in the semester The response variable is better modeled as binomial than as normal Statistics 572 Spring 2007 March 22 2007 Example 3 13 Seed Germination Experiment Data Treatment 1 Control 86 Arasan 98 Spergon 96 Semesan 97 Fermate 91 Lines show Block 2 3 90 88 94 93 90 91 95 91 93 95 treatment Statistics 572 Spring 2007 4 87 89 92 92 95 Treatment 1 Control 86 Arasan 98 Spergon 96 Semesan 97 Fermate 91 Lines show Block 2 3 90 88 94 93 90 91 95 91 93 95 blocking March 22 2007 4 87 89 92 92 95 4 13 Example Seed Germination Experiment Crossed Variables A categorical variable A is crossed with a categorical variable B if there is one or more observations for each combination of levels of A and B Grand Mean Treatment Block Here treatment and blocking are crossed Both treatment and blocking are nested within the grand mean Error Individual error is nested within each other variable Block could be random or fixed Statistics 572 Spring 2007 March 22 2007 Example 5 13 Seed Germination Experiment Model A model is yij i j eij where is an overall mean P i is the effect of treatment i where i i 0 for i 1 5 P j is the effect in block j where j i 0 for j 1 4 and eij iid N 0 e2 is individual random error Note that ij i j Instead of the sum constraints we could instead assume 1 1 0 the default in R Statistics 572 Spring 2007 March 22 2007 6 13 Example Seed Germination Experiment Data seed read table seed txt header T attach seed str seed data frame 20 obs of 3 variables count int 86 90 88 87 98 94 93 89 96 90 treatment Factor w 5 levels AControl Arasan 1 1 1 1 2 2 2 2 5 5 block Factor w 4 levels b1 b2 b3 1 2 3 4 1 2 3 4 1 2 seed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 count treatment block 86 AControl b1 90 AControl b2 88 AControl b3 87 AControl b4 98 Arasan b1 94 Arasan b2 93 Arasan b3 89 Arasan b4 96 Spergon b1 90 Spergon b2 91 Spergon b3 92 Spergon b4 97 Semesan b1 95 Semesan b2 91 Semesan b3 92 Semesan b4 91 Fermate b1 93 Fermate b2 95 Fermate b3 95 Fermate b4 Statistics 572 Spring 2007 March 22 2007 Example 7 13 Seed Germination Experiment Plot of Data AControlArasanFermate Semesan Spergon b3 b4 98 96 94 The block effect seems pretty weak 92 90 88 count library lattice fig1 xyplot count treatment block pch 16 print fig1 86 b1 98 b2 96 94 92 90 88 86 AControlArasanFermate Semesan Spergon treatment Statistics 572 Spring 2007 March 22 2007 8 13 Seed Germination Experiment Example Alternative Plot of Data 98 96 library lattice fig1 xyplot count treatment pch 16 print fig1 94 count It looks like each treatment improves germination rate over the control 92 90 The spread is similar within treatments 88 86 AControl Arasan Fermate Semesan Spergon treatment Statistics 572 Spring 2007 March 22 2007 Example 9 13 Seed Germination Experiment Linear Model Analysis summary seed lm Call lm formula count treatment block Residuals Min 1Q Median 3 950 1 113 0 325 seed lm lm count treatment block anova seed lm Max 3 050 Coefficients Analysis of Variance Table Response count Df Sum Sq Mean Sq treatment 4 102 300 25 575 block 3 18 950 6 317 Residuals 12 85 300 7 108 Signif codes 0 0 001 3Q 1 850 F value Pr F 3 5979 0 03775 0 8886 0 47480 0 01 0 05 0 1 Estimate Std Error t value Pr t Intercept 89 200 1 686 52 899 1 37e 15 treatmentArasan 5 750 1 885 3 050 0 01008 treatmentFermate 5 750 1 885 3 050 0 01008 treatmentSemesan 6 000 1 885 3 183 0 00788 treatmentSpergon 4 500 1 885 2 387 0 03433 blockb2 1 200 1 686 0 712 0 49028 blockb3 2 000 1 686 1 186 0 25854 1 blockb4 2 600 1 686 1 542 0 14904 Signif codes 0 0 001 0 01 0 05 0 1 Residual standard error 2 666 on 12 degrees of freedom Multiple R Squared 0 587 Adjusted R squared 0 3461 F statistic 2 437 on 7 and 12 DF p value 0 08395 Statistics 572 Spring 2007 March 22 2007 10 13 1 Example Seed Germination Experiment Residual Plot 3 plot fitted seed lm residuals seed lm abline h 0 2 0 2 1 3 Normal approximation to binomial is still not too bad as there are sufficient numbers of successes and failures in each treatment group 4 Residual plot looks okay residuals seed lm 1 88 90 92 94 fitted seed lm Statistics 572 Spring 2007 March 22 2007 Example 11 13 Seed Germination Experiment Random Effects Model There are very few differences in a random effects model treating block as random The variance of the block variable is essentially 0 It is more difficult to carry out hypothesis tests Statistics 572 Spring 2007 March 22 2007 12 13 Example Seed Germination Experiment Comparison library lme4 seed lmer lmer count treatment 1 block summary seed lmer Linear mixed effects model fit by REML Formula count treatment 1 block AIC BIC logLik MLdeviance REMLdeviance 90 58 96 56 39 29 89 78 78 58 Random effects Groups Name Variance Std Dev block Intercept 3 475e 09 5 8949e 05 Residual 6 950e 00 2 6363e 00 number of obs 20 groups block 4 Fixed effects Intercept treatmentArasan treatmentFermate treatmentSemesan treatmentSpergon Estimate Std Error t value 87 750 1 318 66 57 5 750 …