VIEWS: 28 PAGES: 8 POSTED ON: 6/23/2011 Public Domain
Chapter XIV Split – Plot Design Part I Split-Plot Design for CRBD This variation on split-plot design introduces a blocking factor . The yields of three different varieties of soybeans are to be compared under two different levels of fertilizer application. If we want to get two observation per treatment (variety of soybean, fertilizer), we need (3 times 2 times 2)=12 equal-sized plots. Blocking factor FARM is included in the experiment. Suppose there are b=2 farms with a=2 whole plots per farm and k=3 (three varieties of soybeans). Blocks (Farms) 1 2 Factor A: Fertilizer 1 Factor A: Fertilizer 2 Factor A: Fertilizer 1 Factor A: Fertilizer 2 Variety 1 (Factor B) Variety 2 (Factor B) Variety 3 (Factor B) Variety 3 (Factor B) Variety 2 (Factor B) Variety 1 (Factor B) Variety 2 (Factor B) Variety 1 (Factor B) Variety 3 (Factor B) Variety 3 (Factor B) Variety 1 (Factor B) Variety 2 (Factor B) The model for this general two-factor split-plot design laid off in b blocks (SPD for RCBD) is as follows: yijkl i j ij k ik ijkl i 1,..., a; j 1,..., b, k 1,..., c, l 1,..., n yijkl = the response of the lth replication ith level of factor A and the kth level of factor B in the jth block. In other words, it is the yield of the lth replication of the kth variety of soybean planted on the whole plot treated with the ith fertilizer of the jth farm. the overall treatment mean, unknown constant. a i = an effect due to ith level of factor A (the effect of the ith fertilizer), unknown constant, i 1 i 0. j =an effect due to jth block, usually the effect is random. So we assume that j has a normal b distribution with mean 0 and variance . If it’s not random, then 2 j 1 j 0 ij is a whole plot random error term of the ith whole plot on the jth farm variable with mean 0 and variance . 2 1|Page Chapter XIV Split – Plot Design k =the effect of the kth level of factor B (the effect of the kth variety of soybean), unknown constant c k 0 . k 1 ( ) ik =the interaction between the ith level of factor A and the jth level of B (interaction between fertilizer and variety of soybean), unknown constant. ijkl is a subplot random error term that has a normal distribution with mean 0 and standard deviation 2 Hypotheses: Whole plot Analysis: H 0 : 1 2 ... a 0 (effects of all levels of factor A are not significantly different from 0) versus H a : not all i ' s are 0 . Subplot Analysis: H 0 : 1 2 ... c 0 (effects of all levels of factor B are not significantly different from 0) versus H a : not all k ' s are 0 . H 0 : ik 0 for all i, k (there is no significant interaction between factors A and B) versus H a : not all ik ' s are 0 ANOVA table for a Split-Plot Design for RCBD Source SS df MS EMS F Between wholeplots A SSA a-1 MSA cn bcn A 2 2 MSA/MSABlocks Blocks SSBlocks (b-1) MSBblocks acn 2 2 ABlocks (Wholeplot SSABlocks (a-1)(b- 1) MSABlocks cn 2 2 Error) Within wholeplots B (factor) SSB c-1 MSB abn B 2 MSB/MSE AB SSAB (a-1)(c-1) MSAB bn AB 2 MSAB/MSE Subplot Error SSE MSE 2 2|Page Chapter XIV Split – Plot Design Total TSS nabc-1 a c nbc i2 nab k 2 nb ik 2 i 1 k 1 where A , B , AB i, k . a 1 c 1 (a 1)(c 1) Example Soybean yields (in bushels per subplot unit) are shown here for a two-factor split-plot design laid of in b=3 blocks (farms). Fertilizers (factor A) were applied at random to the whole plot units within each farm. Soybean varieties (factor B) were randomly allocated to the subplots within each whole plot. There is only one replication per treatment within each whole plot. Conduct the analysis of variance using the sample data in file split_plot.xls. FARMS 1 2 3 Factor A: Factor A: Factor A: Factor A: Factor A: Factor A: Fertilizer 1 Fertilizer 2 Fertilizer 1 Fertilizer 2 Fertilizer 1 Fertilizer 2 1 10.6 10.9 11.9 11.5 9.5 9.8 Variety 2 11.4 11.7 12.6 12.1 8.1 8.2 3 11.8 12.4 11.6 10.8 8.7 9.3 Model: yijk i j ij k ik ijk i 1,2; j 1,2,3, k 1,2,3 yijk = the yield of the kjth variety of soybean planted on the whole plot treated with the ith fertilizer of the jth farm. the overall treatment mean, unknown constant. i = an effect due to ith level of factor A (the effect of the ith fertilizer). j =an effect due to jth farm (block), random; j has a normal distribution with mean 0 and variance . 2 ij =a whole plot random error, the result of application of the ith fertilizer on a whole plot of the jth farm, random variable with mean 0 and variance . 2 3|Page Chapter XIV Split – Plot Design k =the effect of the kth variety of soybean, unknown constant. ( ) ik =the interaction between the ith fertilizer and the jth variety of soybeans, unknown constant. Hypotheses: Wholeplot Analysis: H 0 : 1 2 0 (mean yields for the two fertilizers are not significantly different) versus H a : not all i ' s are 0 . Subplot Analysis: H 0 : 1 2 3 0 (there is no significant difference in mean yields across the three soybeans varieties) versus H a : not all k ' s are 0 . H 0 : ik 0 for all i, k (there is no significant interaction between fertilizer and variety of soybeans) versus H a : not all ik ' s are 0 SAS code: proc glm data=Soy; class Block Fertilizer Variety; model Yield=Fertilizer Block Fertilizer(Block) Variety Variety*Fertilizer; random Fertilizer(Block)/test; means Fertilizer/ Tukey e=Fertilizer(Block); means Variety/ Tukey ; output out=Assumptions P=fitval R=residvar STUDENT=studresid; run; proc glm data=Soy; class Block Fertilizer Variety; model Yield=Fertilizer Block Fertilizer*Block Variety Variety*Fertilizer; random Block Fertilizer*Block/test; means Fertilizer/ Tukey e=Fertilizer*Block; means Variety/ Tukey ; output out=Assumptions P=fitval R=residvar STUDENT=studresid; run; SAS output: 4|Page Chapter XIV Split – Plot Design Source DF Type III SS Mean Square F Value Pr > F * Fertilizer 1 0.845000 0.845000 39.00 0.0247 Block 2 28.863333 14.431667 666.08 0.0015 Error 2 0.043333 0.021667 Error: MS(Block*Fertilizer) * This test assumes one or more other fixed effects are zero. Source DF Type III SS Mean Square F Value Pr > F Block*Fertilizer 2 0.043333 0.021667 0.76 0.4967 * Variety 2 5.343333 2.671667 94.29 <.0001 Fertilizer*Variety 2 0.003333 0.001667 0.06 0.9433 Error: MS(Error) 8 0.226667 0.028333 * This test assumes one or more other fixed effects are zero. Conclusion 1: there is no significant interaction between Fertilizer and Variety of Soybeans. Conclusion 2: There is significant difference in mean yields due to the variety of soybeans. Alpha 0.05 Error Degrees of Freedom 8 Error Mean Square 0.028333 Critical Value of Studentized Range 4.04101 Minimum Significant Difference 0.2777 5|Page Chapter XIV Split – Plot Design Means with the same letter are not significantly different. Tukey Grouping Mean N Variety A 11.36667 6 3 B 10.75000 6 2 C 10.03333 6 1 Conclusion 2: there is significant difference in mean yields due to a fertilizer applied. Alpha 0.05 Error Degrees of Freedom 2 Error Mean Square 0.021667 Critical Value of Studentized Range 6.08486 Minimum Significant Difference 0.2986 Means with the same letter are not significantly different. Tukey Grouping Mean N Fertilizer A 10.93333 9 2 B 10.50000 9 1 Conclusion 4: To get the highest yield we choose fertilizer 2 and variety 3 Model Assumptions: Normality: QQ-plot shows some deviation from normality. 6|Page Chapter XIV Split – Plot Design 3 2 s t u 1 d r e s 0 i d -1 -2 -2 -1 0 1 2 Normal Quantiles Tests for Normality Test Statistic p Value Shapiro-Wilk W 0.959964 Pr < W 0.6009 Kolmogorov-Smirnov D 0.166667 Pr > D >0.1500 Cramer-von Mises W-Sq 0.076539 Pr > W-Sq 0.2226 Anderson-Darling A-Sq 0.390677 Pr > A-Sq >0.2500 According to the normality tests there are not enough evidence to suggest a non-normal distribution. The residual plot does not contain any serious indication of non-constant variance. 7|Page Chapter XIV Split – Plot Design studresid 3 2 1 0 -1 -2 8 9 10 11 12 13 fitval Final Remark The distinction between the SPD for RCBD and two-factor experiments with blocks lies in the randomization. In SPD, there are two stages of the randomization process: first levels of factor A are randomized to the whole plots within blocks, and then levels of factor B are randomized to the subplot unit within each whole plot of every block. In contrast, for a two-factor factorial experiment laid off in RCBD, the randomization is one-step procedure: treatments are randomized to experimental units in each block. 8|Page