SPLIT PLOT DESIGN new.pptx

Explaining split design

1. SPLIT PLOT DESIGN

2. Element of SPLIT PLOT DESIGN Split Plot Experiment: Factorial design with at least two factors, where experimental units with respect to factors differs in “sizes” or “observational points”. Whole Plot: largest experimental unit Whole Plot Factor : Factor that has levels assigned to whole plots. Can be extended to 2 or more factors . Subplot: experimental units that the whole plot is split into (where observations are made) Subplot factor: factor that has level assigned to subplots. Blocks: Aggregates of whole plots that receive all levels of subplot factor

3. The split plot is a multifactor experiment where it is not possible to completely randomize the order of the runs. Consequently , a split-plot design can be viewed as two experiments “combined” or superimposed on each other . One “ experiment ” has the whole-plot factor applied to the large experimental units (or it is a factor whose levels are hard to change) and the other “ experiment ” has the subplot factor applied to the smaller experimental units (or it is a factor whose levels are easy to change).

4. In some field experiments certain factors may require larger plots than for others. For example , experiments on irrigation, requires larger areas. On the other hand experiments on fertilizers, etc may not require larger areas. To accommodate factors which require different sizes of experimental plots in the same experiment, split plot design has been evolved. Larger plots are taken for the factor which requires larger plots. Next each of the larger plots is split into smaller plots to accommodate the other factor. The different treatments are allotted at random to their respective plots. Such arrangement is called SPD . In SPD the larger plots are called main plots and smaller plots within the larger plots are called as sub plots .

5. SPD is considered as a special case of the two-factor randomized block design where one wants to obtain more precise information about one factor and also about the interaction between the two factors, the second factor being of secondary importance to the experimenter. Suppose there are two factors and , having and levels, respectively. That we wish to evaluate the effects of factor and the interaction between the factors and with greater precision than the effects of factor . In this situation, arrange the treatments of factor A in a randomized block design of blocks. Each of the plots (whole plots) can then be divided into subplots so that the treatments of factor can now be allocated at random over each subplot.  

6. Randomization and layout: There are two separate randomization processes in the SPD. one for the Whole plot and another for the sub-plot. The replication is then divided into number of whole plots equivalent to whole plot treatments. Each whole plot is divided into subplots depending on the number of subplot treatments.

7. A possible layout of split plot experiment with three whole plot treatment ( four sub-plot treatment and three replicate may be laid out as below;  

8. Example: Consider a paper manufacturer who is interested in three different pulp preparation methods (the methods differ in the amount of hardwood in the pulp mixture ) and four different cooking temperatures for the pulp and who wishes to study the effect of these two factors on the tensile strength of the paper. Each replicate of a factorial experiment requires 12 observations, and the experimenter has decided to run three replicates . In this example:- Paper manufacturing - Three pulp preparation methods -Four different temperatures -Each replicate requires 12 runs

9. This is considered to be a factorial experiment with three levels of preparation method (factor A) and four levels of temperature (factor B). Then the order of experimentation within each replicate should be completely randomized.

10. The design used in this example is a split-plot design . In this split-plot design we have 9 whole plots , and the preparation methods are called the whole plot or main treatment. Each whole plot is divided into four parts called subplots (or split-plots ), and one temperature is assigned to each. Temperature is called the subplot treatment .

11. ANALYSIS OF VARIANCE FOR SPD: The analysis of variance will have two parts , which correspond to the whole plots and sub-plots. For the whole plot analysis, replication X whole plot treatments table is formed . From this two-way table sum of squares for replication , whole plot treatments and whole plot error [Error (a)] are computed. For the analysis of sub-plot treatments, whole plot X sub-plot treatments table is formed. From this table the sums of squares for sub-plot treatments and interaction between main plot and sub-plot treatments are computed. Error (b) sum of squares is found out by residual method.

12. The linear model for the split-plot design Where , , and represent the whole plot replicates , whole treatments (factor ), and whole-plot error (replicates x ), that is = , respectively; and , , and represent the subplot and correspond, respectively, to the subplot treatment (factor ), the replicates and interactions, and the subplot error (replicates ) where  

13. Where ; is the effect of the block , is the effect of the treatment of factor is the whole-plot error or ; is the effect of the treatment of factor is the interaction between the block and the treatment of is the interaction between the treatment of factor and the treatment of factor and is the subplot error. Note the subplot error is the same as the interaction and the whole plot error is the same as the interaction  

14. SUM OF SQUARES FOR A SPD: The sums of squares for these factors are computed as in the three-way analysis of variance without replication. Note that the main factor in the whole plot is tested against the whole-plot error, whereas the Subtreatment is tested against the replicates X Subtreatment interaction. The interaction is tested against the subplot error . The total sum of squares is partitioned by the identity  

15. HYPOTHESIS TESTS : Whole Plot treatment effects: Hypothesis: Subplot treatment effects: Hypothesis: Whole PlotXSubplot interaction effect Hypothesis:  

16. Statistical analysis of the SPD. Whole plot analysis: This part of analysis is based on comparison of whole plot totals. The levels of are assigned to the whole plots within blocks, and so the sum of squares for needs no block adjustment. There are degrees of freedom for . The sum of squares is given by [the ‘dot’ notation means “add over all values of the subscript replaced with a dot”] There are degrees of freedom for blocks (replicates), giving a block sum of squares of -  

17. There are whole-plots nested with each of the blocks (replicates), so there are, in total, whole-plot degrees of freedom. Thus, the degrees of freedom for whole-plot error is obtained by subtracting the block and A degrees of freedom from the whole-plot total. i.e . Thus, the whole plot error sum of squares, is obtained as = - - The whole plot error mean square = , is used as the error estimate to test the significance of whole plot factor (A).  

18. Sub-plot analysis: This part of analysis is based on the observations arising from the split-plots within whole plots: T here are total degrees of freedom, and the total sum of squares is Due to the fact that all levels of B are observed in every whole plot as in a randomized complete block design, the sum of squares for B needs no adjustment for whole plots, and is given by; corresponding to degrees of freedom.  

19. The interaction between the factors and is also calculated as part of the split-plot analysis. Again, due to the complete block structure of both the whole-plot design and the split-plot design, the interaction sum of squares needs no adjustment for blocks. The number of interaction d egrees of freedom is and the sum of squares is Since there are split plots nested within the whole plots, there are, in total, split-plot degrees o f freedom. Of these, are used to measure the main effect of B, and (a-1)(b-1) are used to measure the AB interaction, leaving, degrees of freedom for error. Can be obtained by subtraction of the whole plot, B, and AB degrees of freedom from the total i.e  

20. The split-plot error sum of squares (subplot ss ) can be calculated by subtraction: = The split-plot error mean square is used as the error estimate in testing the significance of split-plot factor and interaction The analysis the variance table is outlined as follows:  

21. Example: In study carried by agronomists to determine if major differences in the yield response to N fertilization exist among different varieties of jowar , the main plot treatments were three varieties of Jowar ( and and the sub-plot treatments were N rates of 0, 30, and 60Kg/ha. The study was replicated four times, and the data gathered for the experiment are shown in the table.  

22. Steps of analysis: Calculate the replication totals (R), and the grand total (G) by first constructing a table for the replication X Variety totals , and then a second table for the Variety X nitrogen totals. Table 1: Replication X Variety (RA)

23. Table 2: Variety X Nitrogen (AB) Compute various sum of squares for the Whole plot analysis. Total sum of squares From SST= - = =637.97  

24. Replication sum of squares: Recall: - = =190.08 Sum of squares due to variety (SSA): Recall: = =90.08  

25. Whole plot error sum of squares (SSE1) Recall: = - - = =174.103 Sub-plot analysis: Sum of squares due to Nitrogen (SSB): Recall : = - 11342.25 =92.435  

26. Sum of Squares due to interaction ( AxB ): Recall: = -11342.25-90.487-92.435 =9.533 Sub-plot error S.S (SSE2): Recall: = =637.97 - 190.08 - 90.487 - 174.103 - 92.435 - 9.533 =81.332  

27. ANOVA table

28. STANDARD ERRORS AND CRITICAL DIFFERENCES Estimate of S.E of difference between two whole plot treatment means = Estimate of S.E of difference between two sub-plot treatment means = Estimate of S.E of difference between two sub plot treatments means at the same level of whole plot treatment = Estimate of S.E of difference between two whole plot treatment means at the same or different levels of sub-plot treatment=  

29. Try this! The following data are taken from an experiment where the amount of dry matter was measured on wheat plants grown in different levels of moisture and different amounts of fertilizer. There where 48 different peat pots and 12 plastic trays; 4 pots could be put into each tray. The moisture treatment consisted of adding 10, 20, 30, or 40 ml of water per pot per day to the tray, where the water was absorbed by the peat pots. The levels of moisture were randomly assigned to the trays. The levels of fertilizer were 2, 4, 6, or 8 mg per pot. The four levels of fertilizer were randomly assigned to the four pots in each tray so that each fertilizer occurred once in each tray. The wheat seeds were planted in each pot and after 30 days the dry matter of each pot was measured.

30. Workout this Example: Consider a paper manufacturer who is interested in three different pulp preparation methods (the methods differ in the amount of hardwood in the pulp mixture ) and four different cooking temperatures for the pulp and who wishes to study the effect of these two factors on the tensile strength of the paper. The data are shown as;