Introduction to Randomized block designs Accounting for predicted but random variance Block 1 Block 2 Block 3 Block 4 C A A D B D B A BD BC CA DC Blocking •Aim: – Reduce unexplained variation, without increasing size of experiment. Approach: – Group experimental units (“replicates”) into blocks. – Blocks usually spatial units, one experimental unit from each treatment in each block. 1 Walter & O’Dowd (1992) • Effects of domatia (cavities on leaves) on number of mites • Two treatments (Factor A): – shaving domatia removes domatia from leaves – normal domatia as control • Required 14 leaves for each treatment Completely randomized design: - 28 leaves randomly allocated to each of 2 treatments Control leaves Shaved domatia leaves 2 Completely randomized ANOVA • Factor A with p groups (p = 2 for domatia) • n replicates within each group (n = 14 pairs of leaves) Source general df example df Factor A p-1 1 Residual p(n-1) 26 Total pn-1 27 Walter & O’Dowd (1992) • Required 14 leaves for each treatment • Set up as blocked design – paired leaves (14 pairs) chosen – 1 leaf in each pair shaved, 1 leaf in each pair control 3 1 block Control leaves Shaved domatia leaves Rationale for blocking • Micro-temperature, humidity, leaf age, etc. more similar within block than between blocks • Variation in response variable (mite number) between leaves within block (leaf pair) < variation between leaves between blocks 4 Rationale for blocking • Some of unexplained (residual) variation in response variable from completely randomized design now explained by differences between blocks • More precise estimate of treatment effects than if leaves were chosen completely randomly Null hypotheses • No main effect of Factor A –H0: 1 = 2 = … = i = ... = –H0: 1 = 2 = … = i = ... = 0 (i = i - ) – no effect of shaving domatia, pooling blocks • Factor A usually fixed 5 Null hypotheses • No effect of factor B (blocks): – no difference between blocks (leaf pairs), pooling treatments • Blocks usually random factor: – sample of blocks from population of blocks 2 –H0: = 0 Randomized blocks ANOVA • Factor A with p groups (p = 2 treatments for domatia) • Factor B with q blocks (q = 14 pairs of leaves) Source general example Factor A p-1 1 Notice Factor B (blocks) q-1 13 that this Residual (p-1)(q-1) 13 is not Total pq-1 27 pq(n-1) 6 Randomized block ANOVA • Randomized block ANOVA is 2 factor factorial design – BUT no replicates (n) within each cell (treatment-block combination), i.e. unreplicated 2 factor design – No measure of within-cell variation – No test for treatment by block interaction Example – effect of watering on plant growth • Factor 1: Watering, no watering • Factor 2: Blocks (1-4). One replicate of each treatment (watering, no watering) in each of 4 plots • Replication: 1 plant for each watering/black combination (8 total) No water Water 7 Results 15 Block Treatment Growth 10 1 No Water 6 1 Water 10 2 No Water 4 2 Water 6 Growth 3 No Water 11 3 Water 15 4 No Water 5 5 4 Water 8 0 No Water Water Treatment Results 15.0 Block Treatment Growth 10.0 1 No Water 6 1 Water 10 2 No Water 4 h 2 Water 6 Growt 3 No Water 11 3 Water 15 4 No Water 5 5.0 4 Water 8 0.0 1.0 2.0 3.0 4.0 Block 8 Results 15 Treatment No Water Water Block Treatment Growth 10 1 No Water 6 1 Water 10 2 No Water 4 2 Water 6 Growth 3 No Water 11 3 Water 15 4 No Water 5 5 4 Water 8 0 1.0 2.0 3.0 4.0 Block Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering MSBlocks MSResidual 9 2 2 2 EMSResidual = + : Why is this not simply 15 Treatment No Water Water Block Treatment Growth 10 1 No Water 6 1 Water 10 2 No Water 4 2 Water 6 Growth 3 No Water 11 3 Water 15 4 No Water 5 5 4 Water 8 0 1.0 2.0 3.0 4.0 Block Residual 2 2 • Cannot separately estimate and : – no replicates within each block-treatment combination 2 2 •MSResidual estimates + 10 Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering MSBlocks 2 2 MSResidual + 2 2 EMSBlocks = + n 15.0 Block Treatment Growth 10.0 1 No Water 6 1 Water 10 2 No Water 4 h 2 Water 6 Growt 3 No Water 11 3 Water 15 4 No Water 5 5.0 4 Water 8 0.0 1.0 2.0 3.0 4.0 Block 11 Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering 2 2 MSBlocks + n 2 2 MSResidual + 2 2 2 EMSwatering= + + n (i) /p-1 15 Block Treatment Growth 10 1 No Water 6 1 Water 10 2 No Water 4 2 Water 6 Growth 3 No Water 11 3 Water 15 4 No Water 5 5 4 Water 8 0 No Water Water Treatment 2 Why does the EMS for watering include , which is the effect of the interaction? Block is a random effect, hence there are unsampled combinations of block and watering that could affect the estimates of EMSwatering 12 Expected mean squares If factor A fixed and factor B (Blocks) random: 2 2 2 MSwatering + + n (i) /p-1 2 2 MSBlocks + n 2 2 MSResidual + Testing null hypotheses • Factor A fixed and blocks random • If H0 no effects of factor A is true: – then F-ratio MSA / MSResidual 1 • If H0 no variance among blocks is true: – no F-ratio for test unless no interaction assumed – if blocks fixed, then F-ratio MSB / MSResidual 1 13 Walter & O’Dowd (1992) • Factor A (treatment - shaved and unshaved domatia) - fixed • Blocks (14 pairs of leaves) - random Source df MS FP Treatment 1 31.34 11.32 0.005 Block 13 1.77 0.64 0.784 ?? Residual 13 2.77 Should this be reported?? Explanation Blocks Treatment 12345678910111213 Shaved1111111111111 Control1111111111111 Cells represent the possible effect of the Block by Treatment interaction but: 1) There is only one replicate per cell, therefore 2) No way to estimate variance term for each cell, therefore 3) No way to estimate the variance associated with the interaction, therefore 2 2 4) The residual term estimates + 14 Randomized Block vs Completely Randomized designs • Total number of experimental units same in both designs – 28 leaves in total for domatia experiment • Test of factor A (treatments) has fewer df in block design: – reduced power of test RCB vs CR designs •MSResidual smaller in block design if blocks explain some of variation in Y: – increased power of test • If decrease in MSResidual (unexplained variation) outweighs loss of df, then block design is better: – when blocks explain much of variation in Y 15 Assumptions • Normality of response variable – boxplots etc. • No interaction between blocks and factor A, otherwise –MSResidual increase proportionally more than MSA with reduced power of F-ratio test for A (treatments) – interpretation of main effects may be difficult, just like replicated factorial ANOVA Checks for interaction • No real test because no within-cell variation measured • Tukey’s test for non-additivity: – detect some forms of interaction • Plot treatment values against block (“interaction plot”) 16 Interaction plots Y No interaction Y Interaction Block Growth of Plantago 17 Growth of Plantago • Growth of five genotypes (3 fast, 2 slow) of Plantago major (ribwort) • Poorter et al. (1990) • One replicate seedling of each genotype placed in each of 7 plastic containers in growth chamber – Genotypes (1, 2, 3, 4, 5) are factor A – Containers (1 to 7) are blocks – Response variable is total plant weight (g) after 12 days Poorter et al. (1990) 3 1 4 2 2 5 1 5 4 3 Container 1 Container 2 Similarly for containers 3, 4, 5, 6 and 7 18 Source df MS FP Genotype 4 0.125 3.81 0.016 Block 6 0.118 Residual 24 0.033 Total 34 Conclusions: • Large variation between containers (= blocks) so block design probably better than completely randomized design • Significant difference in growth between genotypes Mussel recruitment and seastars 19 Mussel recruitment and seastars • Effect of increased mussel (Mytilus spp.) recruitment on seastar numbers • Robles et al. (1995) – Two treatments: 30-40L of Mytilus (0.5-3.5cm long) added, no Mytilus added – Four matched pairs (blocks) of mussel beds chosen – Treatments randomly assigned to mussel beds within pair – Response variable % change in seastar numbers - + +- -+ - + 1 block (pair of mussel beds) + mussel bed with added mussels - mussel bed without added mussels 20 Source df MS FP Blocks 3 62.82 Treatment 1 5237.21 45.50 0.007 Residual 3 115.09 Conclusions: • Relatively little variation between blocks so a completely randomized design probably better because treatments would have 1,6 df • Significant treatment effect - more seastars where mussels added Worked Example – seastar colors • Comparison of numbers of purple vs orange seastars along the CA coast • Data number of purple and orange seastars collected at 7 random locations • Compare models (block vs completely random vs paired t test) sea star colors all sites two sample 21 Number of Seastars as a function of color and site 600 1000 500 800 400 600 300 400 200 100 200 0 0 Orange Purple Color Each error bar is constructed using 1 standard error from the mean. Site Each error bar is constructed using 1 standard error from the mean. Any obvious problem with the data?? 0.91 0.88 Diagnostics: Log transform helps normality and 0.84 0.8 homogeneity of variance assumptions 0.7 0.6 0.5 0.4 Normal Probability 0.3 0.2 0.16 0.12 0.09 0.07 Log Number 0 200 400 600 800 1000 Number 0.91 0.88 0.84 0.8 0.7 Number 0.6 0.5 0.4 Normal Probability 0.3 0.2 0.16 0.12 0.09 0.07 11.522.53 Log Number 22 Model 1: One factor ANOVA Why the difference? SE=0.181 2.5 SE=0.184 2 1.5 1 0.5 0 Orange Purple Color Each error bar is constructed using 1 standard error from the mean.