Split_plot_RCBD by keralaguest

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									                              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




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                                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



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                                          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



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                                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:



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                            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




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                             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.




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                             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.




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                            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.




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