2. Two distinct error terms (Error for main plot effects > Error for subplot effects)

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Lecture 17 Topic 12: The split-plot design and its relatives (Part I) [ST&D Ch. 16] Definition A split plot design results from a two-stage randomization process of a factorial treatment structure. Because
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Lecture 17 Topic 12: The split-plot design and its relatives (Part I) [ST&D Ch. 16] Definition A split plot design results from a two-stage randomization process of a factorial treatment structure. Because of this two-stage process, one loses sensitivity in detecting differences among main plot treatments (the first level of randomization) but gains sensitivity in detecting differences among subplot treatments (the second level), as well as the significance of the MainPlot*Subplot interaction. The basic split-plot design involves assigning the levels of one factor (A) to main plots arranged in a CRD, RCBD, Latin Square, etc., and then assigning the levels of a second factor (B) to subplots within each main plot. 1. Two-stage randomization process 2. Two distinct error terms (Error for main plot effects Error for subplot effects) Uses of the split-plot design 1. Physical constraints to randomization e.g. one factor requires a larger amount of experimental material than another 2. To increase precision of some effects e.g. split-plots often improve precision for detecting subplot and interaction effects 3. To increase the scope of an experiment. e.g. seed protectants and varieties. In general, the subplot (Factor B): requires smaller amounts of experimental material is of primary importance is expected to exhibit smaller differences 1 Split-plots within various designs Factor A is the main plot factor, with 3 levels Factor B is the subplot factor, with 2 levels. There are 4 replications per main plot. Simple 3x2 factorial (no split), arranged as a CRD a1b1 a2b2 a2b1 a1b2 a3b2 a1b1 a2b2 a2b1 a1b2 a3b2 a1b1 a3b2 a2b2 a3b1 a1b2 a3b1 a1b2 a3b2 a2b1 a1b1 a2b2 a3b1 a2b1 a3b1 Split-plot, with main plots arranged as a CRD Stage 1: Randomize the levels of factor A over the main plots: a2 a3 a2 a1 a2 a3 a2 a3 a1 a3 a1 a1 a2 a3 a2 a1 a2 a3 a2 a3 a1 a3 a1 a1 Stage 2: Randomize the levels of factor B over the subplots: a2b2 a3b2 a2b1 a1b1 a2b1 a3b2 a2b1 a3b2 a1b1 a3b1 a1b1 a1b2 a2b1 a3b1 a2b2 a1b2 a2b2 a3b1 a2b2 a3b1 a1b2 a3b2 a1b2 a1b1 Split-plot, with main plots arranged as an RCBD Stage 1: Randomize the levels of factor A within each block. a2 a1 a3 a1 a2 a3 a1 a3 a2 a3 a2 a1 a2 a1 a3 a1 a2 a3 a1 a3 a2 a3 a2 a1 Stage 2: Randomize the levels of factor B over the subplots. a2b1 a1b1 a3b2 a1b2 a2b1 a3b2 a1b2 a3b1 a2b2 a3b1 a2b1 a1b1 a2b2 a1b2 a3b1 a1b1 a2b2 a3b1 a1b1 a3b2 a2b1 a3b2 a2b2 a1b2 2 Linear models for the split-plot The linear model for the split-plot, with main plots arranged as an RCBD: where Y ijk = µ + α i + β j + (αβ) ij + γ k + (αγ) ik + ε ijk i = 1,...,a j = 1,...,b k = 1,...,r indexes the main plot levels indexes the blocks indexes the subplot levels The variance associated with (αβ) ij (i.e. σ 2 αβ ) is used to test the main plot effects. 2 The variance associated with ε ijk (i.e. σ ε ) is used to test the subplot and interaction effects. 2 Usually, σ αγ σ 2 ε. The linear model for the split-plot, with main plots arranged as a CRD: where now Y ijk = µ + α i + (αβ) ij + γ k + (αγ) ik + ε ijk j = 1,...,r indexes the replications CRD Main plot error = Main plot x Replication RCBD Main plot error = Main plot x Block CRD Subplot error = Subplot x Replication + Main plot x Subplot x Replication RCBD Subplot error = Subplot x Block + Main plot x Subplot x Block 3 Split-plot ANOVA The general ANOVA table for the split-plot CRD: Source df SS MS F Total (subplots) rab - 1 TSS Main plots total ra - 1 SS(MP) Factor A a - 1 SSA MSA MSA/MS(MPE) Main plot error a(r - 1) SS(MPE) MS(MPE) Factor B b - 1 SSB MSB MSB/MS(SPE) A x B (a - 1)(b - 1) SS(AxB) MS(AxB) MS(AxB)/MS(SPE) Subplot error a(r - 1)(b - 1) SS(SPE) MS(SPE) CRD RCBD Latin Square Total ra-1 Total ra-1 Total ra-1 Rows a-1 Columns a-1 A a-1 Error A A Error A Factor B A x B Error B Total a-1 a(r-1) b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 Blocks A Error A Factor B A x B Error B Total r-1 a-1 (r-1)(a-1) b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 Factor B A x B Error B Total (a-1)(a-2) b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 CRD lm( Y ~ A + Rep:A + B + A:B ) RCBD lm( Y ~ Block + A + Block:A + B + A:B ) Latin Square lm( Y ~ Row + Col + A + Row:Col:A + B + A:B ) Replicated Latin Square (shared rows and columns) lm( Y ~ Square + Row + Col + A + Square:Row:Col:A + B + A:B ) 4 Example of a split-plot with main plots arranged as an RCBD (Phytopathology 71: ) Experiment conducted to determine the effect of bacterial vascular necrosis on the root yield of sugar beets planted at different in-row spacings. Main plot A: Inoculation (inoculated vs. not inoculated with Erwinia carotovora) Subplot B: In-row spacing between plants (4, 6, 12, and 18 inches) Field Layout Block VI No inoculation Inoculation V Inoculation No inoculation IV No inoculation Inoculation III No inoculation Inoculation II Inoculation No inoculation I Inoculation No inoculation 5 The R code sugar_bad.mod -lm(yield ~ A_inoc + block + A_inoc:block + B_space + A_inoc:B_space, sugar.dat) anova(sugar_bad.mod) The above command follows the linear model specified on the previous page and therefore seems to be a reasonable approach to analyzing this particular dataset. If you run this code, you obtain the following output: Analysis of Variance Table Response: yield A_inoc 2.2e-16 *** --WRONG! block ** --WRONG! A_inoc:block * --MP error B_space e-06 *** A_inoc:B_space e-09 *** Residuals Programmed in this way, all the effects in the model are being tested with the residual error (0.783, df = 30). Unfortunately, this is the incorrect error term for testing the effect of the main plot (A_inoc) and the Block factors, as can be seen in the following table of expected mean squares: Source Block A_Inoc Block*A_Inoc B_Space A_Inoc*B_Space Expected Mean Square Var(Error) + 4 Var(Block*A_Inoc) + 8 Var(Block) Var(Error) + 4 Var(Block*A_Inoc) + Q(A_Inoc,A_Inoc*B_Space) Var(Error) + 4 Var(Block*A_Inoc) Var(Error) + Q(B_Space,A_Inoc*B_Space) Var(Error) + Q(A_Inoc*B_Space) The appropriate error term for A is Block*A The appropriate error term for B and A*B is the residual error 6 To obtain the correct F and p-values for the main plot factor and the blocks, you need to tell R to test those factors with the appropriate error term, namely A*Block. One way to do this is to use the aov() function: sugar_good.mod -aov(yield ~ A_inoc + block + Error(A_inoc:block) + B_space + A_inoc:B_space, sugar.dat) summary(sugar_good.mod) The result of this line of code is: Error: A_inoc:block A_inoc *** block Residuals Error: Within B_space e-06 *** A_inoc:B_space e-09 *** Residuals Using the wrong error term before, we concluded that there were significant yield differences among blocks and highly a significant effect of inoculation (main plot p 2.2e-16). Now, using the correct error term for these effects, we see that the differences among blocks are not significant and the effect of inoculation is twelve orders of magnitude smaller than we thought (p = 1.32e-4)! Putting everything together, we arrive at this final ANOVA table for the sugar beet root rot study: Source df SS MS F Total (subplots) Block NS Inoculation (A) *** Error A (Block * A) Spacing (B) *** Interaction (A x B) *** Error B Notice: MSE A (2.31) MSE B (0.78) 7 Mean comparisons As with all factorial treatment structures, subsequent analysis of the data depends first and foremost on whether or not a significant interaction exists among the factors. In this particular case, the interaction A_inoc*B_space was found to be highly significant (p = 9.84e-09). When the interaction between main plot and subplot is significant, we can make three different kinds of comparisons: 1. Comparisons among subplot levels within a main plot level 2. Comparisons among main plot levels within a subplot level 3. Comparisons between subplot levels across different main plot levels Block VI No inoculation Inoculation V Inoculation No inoculation IV No inoculation Inoculation III No inoculation Inoculation II Inoculation No inoculation I Inoculation No inoculation 8 1. Comparisons among subplot levels within a common main plot level no_inoc.dat -subset(sugar.dat, A_inoc == 0) no_inoc.mod -lm(yield ~ block + B_space, no_inoc.dat) MP_comp1 -LSD.test(no_inoc.mod, B_space ) MP_comp1 inoc.dat -subset(sugar.dat, A_inoc == 1) inoc.mod -lm(yield ~ block + B_space, inoc.dat) MP_comp2 -LSD.test(inoc.mod, B_space ) MP_comp2 The output: A_inoc = 0 alpha: 0.05 ; Df Error: 15 Mean Square Error: Critical Value of t: Least Significant Difference Groups, Treatments and means a a a b A_inoc = 1 alpha: 0.05 ; Df Error: 15 Mean Square Error: Critical Value of t: Least Significant Difference Groups, Treatments and means a ab b c /2( ) = , the subplot error used in the original analysis! 9 2. Comparisons among main plot levels within a common subplot level sp_4.dat -subset(sugar.dat, B_space == 4) sp_4.mod -lm(yield ~ block + A_inoc, sp_4.dat) SP_comp1 -LSD.test(sp_4.mod, A_inoc ) SP_comp1 sp_6.dat -subset(sugar.dat, B_space == 6) sp_6.mod -lm(yield ~ block + A_inoc, sp_6.dat) SP_comp2 -LSD.test(sp_6.mod, A_inoc ) SP_comp2 sp_12.dat -subset(sugar.dat, B_space == 12) sp_12.mod -lm(yield ~ block + A_inoc, sp_12.dat) SP_comp3 -LSD.test(sp_12.mod, A_inoc ) SP_comp3 sp_18.dat -subset(sugar.dat, B_space == 18) sp_18.mod -lm(yield ~ block + A_inoc, sp_18.dat) SP_comp4 -LSD.test(sp_18.mod, A_inoc ) SP_comp4 The residual error here is automatically the mean square of the Block*A interaction! A summary of the output: B_space MSE LSD No Inoc a a a a Inoc b b b b 10 Mixed Comparisons: Comparisons between subplot levels across different main plot levels The appropriate error (MSE Mix ) is a weighted average of MSE A and MSE B, with emphasis on MSE B. Such comparisons require hand computations. The appropriate weighted error is: MSE Mix = (b 1)* MSE B + MSE A b = (4 1)* = Similarly, the appropriate critical value is an intermediate value between the critical value for the main plot (t A, 5 df = 2.571) and that for the subplot (t B, 30 df = 2.042). The formula: t Mix = (b 1)*t B * MSE B + t A * MSE A 3* * * = = (b 1)MSE B + MSE A 3* Sticking with the LSD method: LSD α=0.05 = t Mix 2MSE Mix r = 2.304* 2( ) 6 =1.435 Example: To compare the mean of inoculated / spacing 4 = with the mean of not inoculated / spacing 6 = 20.82: = 4.32 Since 4.32 This difference is significant 11 If the interaction between main plot * subplot is not significant, there are two possibilities for subsequent analysis: 1. Comparisons among main plot levels 2. Comparisons among subplot levels 1. Main plot comparisons in the absence of an interaction A valid comparison among the means of the main plot levels requires the appropriate error variance (MS Block*A = 2.307). MP_comparison -LSD.test(sugar.dat$yield, sugar.dat$a_inoc, DFerror = 5, MSerror = 2.307) MP_comparison The output: Mean CV MSerror LSD trt means M a b Incidentally, just like the means comparison tests, any orthogonal contrasts must also specify the correct error term! Example: # Contrast No_Inoc vs. Inoc 1,-1 contrastmatrix -cbind(c(1,-1)) contrastmatrix contrasts(sugar.dat$a_inoc) -contrastmatrix sugar.dat$a_inoc sugar_contrast.mod -aov(yield ~ A_inoc + block + Error(A_inoc:block) + B_space + A_inoc:B_space, sugar.dat) summary(sugar_contrast.mod, split = list(a_inoc = list( no_inoc vs. Inoc = 1))) 12 2. Subplot comparisons in the absence of an interaction For comparisons among subplots, the residual error is the correct error term to use: SP_comparison -LSD.test(sugar.dat$yield, sugar.dat$b_space, DFerror = 30, MSerror = ) SP_comparison The output: Mean CV MSerror LSD trt means M a a b c 13 Split-split plot design The concept of the split-plot design extends logically from two to three factors: 1. Split-plot with factorial main plot: Combinations of levels of Factors A and B are assigned to main plots, levels of Factor C to subplots within each mainplot. 2. Split-plot with factorial subplot: Levels of Factor A are assigned to main plots, combinations of levels of Factors B and C are assigned to subplots. 3. Split-split plot: Levels of Factor A are assigned to main plots, levels of Factor B to subplots within each mainplot, and levels of Factor C to sub-subplots within each subplot. In the last case, the additional stage of randomization introduces an additional (third) error term, required for testing the main effects of Factor C and all interactions involving Factor C. Example: (Little & Hills) A split-split plot study to evaluate the effects of dates of planting (A), aphid control (B), and date of harvest (C) on the control of an aphid-borne sugar beet virus. Block I Block II A 1 A 3 A 2 Block III Block IV A 3 B 1 A 1 B 1 A 2 B 2 A 2 B 1 C 1 A 2 B 1 C 3 A 2 B 1 C 2 A 1 B 2 C 3 A 1 B 2 C 1 A 1 B 2 C 2 A 3 B 2 C 1 A 3 B 2 C 3 A 3 B 2 C 2 A 3 B 2 A 1 B 2 A 2 B 1 A 2 B 2 C 3 A 2 B 2 C 2 A 2 B 2 C 1 A 1 B 1 C 1 A 1 B 1 C 3 A 1 B 1 C 2 A 3 B 1 C 3 A 3 B 1 C 1 A 3 B 1 C 2 14 ANOVA for the split-split plot design is an extension of the split-plot case. The various error terms are constructed by extracting different sources of variation from the residual error and pooling them together: Level One: Block A Tested using (Block*A) = Error A Level Two: B A*B Tested using (Block*B + Block*A*B) = Error B Level Three: C A*C B*C A*B*C Tested using (Block*C + Block*A*C + Block*B*C + Block*A*B*C) = resid error = Error C What this means is that a full analysis requires specifying TWO special error terms for custom F tests, in addition to the model's residual error. Unfortunately, the aov() function only permits the specification of ONE custom error term. There are several options as to how to proceed. One strategy is to run a couple of different aov() models and combine the results into a final ANOVA table: 15 Model 1: Testing the main plot effects split_split_mp.mod -aov(y ~ A + Block + Error(A:Block) + B + A:B + C + A:C + B:C + A:B:C, split_split.dat) summary(split_split_mp.mod) The resulting ANOVA table: Error: A:Block A ** Block ** Residuals Error: Within B e-09 *** ß WRONG!! C e-11 *** ß WRONG!! A:B ß WRONG!! A:C ß WRONG!! B:C ß WRONG!! A:B:C ß WRONG!! Residuals Model 2: Testing the subplot effects split_split_sp.mod -aov(y ~ A + Block + A:Block + B + A:B + Error(A:B:Block) + C + A:C + B:C + A:B:C, split_split.dat) summary(split_split_sp.mod) The resulting ANOVA table: Error: A:B:Block A e-05 *** ß WRONG!! Block e-06 *** ß WRONG!! B e-06 *** A:Block * ß WRONG!! A:B Residuals Error: Within C e-10 *** A:C B:C A:B:C Residuals Combining the results into a complete table: A ** Block ** A:Block B e-06 *** A:B A:B:Block C e-10 *** A:C B:C A:B:C Error Another way to do it is to obtain the SS for all the factors in the linear model and then carry out the custom F-tests manually. For this, one would specify the full model: split_split.mod -aov(y ~ A + Block + A:Block + B + A:B + B:Block + A:Block:B + C + A:C + B:C + A:B:C, split_split.dat) summary(split_split.mod) The resulting ANOVA table: A e-07 *** Block e-08 *** B e-07 *** C e-10 *** A:Block A:B Block:B A:C B:C A:Block:B A:B:C Residuals To test the main plot effect (A): F = 83.83/6.70 = p(12.50,3,6) = To test the subplot effect (B): F = /[( )/(2+6)] = p( ,1,8) = 4.00e-06 How would you test the A:B interaction? 18 Strip-plot (or split-block) design In the strip-plot or split-block design, the subunit treatments are applied in strips across a complete set (replication) of main plot levels. A comparison of the layouts for a 5x4 split-plot design and a 5x4 strip-plot design (only one replication, or block, is shown). Split-plot Strip-plot (or split-block) A3 A2 A1 A5 A4 A3 A2 A1 A5 A4 B2 B1 B2 B3 B4 B2 B2 B2 B2 B2 B1 B3 B1 B2 B3 B4 B4 B4 B4 B4 B3 B2 B4 B4 B1 B1 B1 B1 B1 B1 B4 B4 B3 B1 B2 B3 B3 B3 B3 B3 In the strip-plot, we retain the terms main plot and subplot. From a theoretical perspective, however, there is no longer any difference between the two. Reasons for arranging an experiment as a strip-plot design 1. Physical operations (e.g. tractor manipulation, irrigation, harvesting) may be easier. 2. The design tends to reduce precision in testing the main effects but improves precision in detecting interaction effects. Linear model for the strip-plot design (RCBD case) where Y ijk = µ + γ k + α i + β j + (αγ) ik + (βγ) jk + (αβ) ij + ε ijk k = 1,...,r indexes the blocks i = 1,...,a indexes the main plot levels j = 1,...,b indexes the subplot levels The extra term (βγ) jk represents the interaction effect of blocks with subplot levels. 19 ANOVA for the split-block design CRD: To test the main effects of Factor B, the proper error term is the stripplot error MS(StPE) = B*Rep. RCBD: To test the main effects of Factor B, the proper error term is the stripplot error MS(StPE) = B*Block. The general ANOVA table for the RCBD strip-plot design: Source df SS MS F Total rab - 1 TSS Block r - 1 SS(Block) Factor A a - 1 SSA MSA MSA / MS(MPE) MSE A = A*Block (a - 1)(r - 1) SS(MPE) MS(MPE) Factor B b - 1 SSB MSB MSB / MS(StPE) MSE B = B*Block (b - 1)(r - 1) SS(StPE) MS(StPE) A x B (a - 1)(b - 1) SS(AxB) MS(AxB) MS(AxB) / MS(SPE) MSE AB =A*B*Block (a-1)(r-1)(b-1) SS(SPE) MS(SPE) Note the improved precision in the tests for interaction effects. Example of a split-block (modified from Little and Hills, Chapter 10) The root yield in tons per acre for each subplot are given in the diagram below. Main plot A: Nitrogen rates, levels N0, 80, 160, and 320 Subplot B: Harvest times, levels 1 5 Block I Block II H4 H5 H1 H3 H2 H4 H2 H3 H5 H1 N N N N N N N N Again, we are in situation where the results from two separate models must be combined to produce a complete and correct ANOVA table. #The ANOVA to test A split_blocka.mod -aov(yield ~ A_nitrogen + block + Error(A_nitrogen:block) + B_harvest + B_harvest:block + A_nitrogen:B_harvest, split_block.dat) #The ANOVA to test B split_blockb.mod -aov(yield ~ A_nitrogen + block + A_nitrogen:block + B_harvest + Error(B_harvest:block) + A_nitrogen:B_harvest, split_block.dat) The resulting ANOVA tables: Error: A_nitrogen:block A_nitrogen block Residuals Error: Within B_harvest e-12 *** block:b_harvest ** A_nitrogen:B_harvest *** Residuals Error: B_harvest:block block B_harvest ** Residuals Error: Within A_nitrogen e-11 *** A_nitrogen:block e-06 *** A_nitrogen:B_harvest *** Residuals Combining results into a final ANOVA table: A_nitrogen A:Block B_harvest ** B:Block A:B *** Error What is the next step in this analysis? How would you do it? 22 Trend analysis of the main effects of Nitrogen For the sake of covering one more concept, let's assume that the A:B interaction was found to be non-significant, thereby justifying an analysis of the main effects. In this scenario, notice that the previous F test for A_nitrogen was almost significant (p = ). Because the nitrogen levels are not evenly spaced, it is not very convenient to perform a trend analysis using contrasts. Instead, we rely on multiple regression, using a very simple model to obtain the correct sums of squares: #To perform a trend analysis of A_nitrogen using a multiple regression #approach,first re-load the dataset and maintain A_nitrogen as an #integer (not a factor); then: A_nit -split_block.dat$A_nitrogen A_nit2 -A_nit^2 A_nit3 -A_nit^3 A_nit4 -A_nit^4 anova(lm(yield ~ A_nit + A_nit2 + A_nit3 + A_nit4, split_block.dat)) The output: Response: yield A_nit ** A_nit * A_nit Residuals While these are the correct sums of squares for the linear, quadratic, and cubic components of the trend analysis (notice they sum to ), the F-tests are incorrect because they use the wrong error term. As we saw from the previous analysis, the correct error term for the main plot effect A_nitrogen is the A_nitrogen:Block interaction (SS = 111.7, df = 3). The correct F and p-values can be obtained using the customf() function shown on the next page: #Linear A: customf(c(508.21,1,111.7,3)) [1] * #Quadratic A: customf(c(290.19,1,111.7,3)) [1] NS #Cubic A: customf(c(39.90,1,111.7,3)) [1] NS 23 The following function makes custom F tests
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