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SPLIT PLOT DESIGNS
Key Features : More than one type of experimental unit and more than one randomization. Typical Use : When one factor is difficult to change. Example (and terminology): An agricultural researcher is studying the effects of corn variety and irrigation level on corn yields. Four varieties of corn and three irrigation levels are used. It’s easy to plant a specified variety of corn wherever told to do so. (Thus, variety is an easy-to-change factor.) However, irrigation is done by large sprinklers that irrigate a large area of land. (Thus, irrigation is a difficult-to- change factor.) A crossed factorial experiment with just two replications of each of the twelve treatment combinations would require 24 large areas of land. This is beyond resources available. To get around the problem:
- Use just six plots of land, chosen so that each can be irrigated at a set level without affecting the irrigation of the others. (These large plots are called the whole plots. )
- Randomly assign irrigation levels to each whole plot so that each irrigation level is assigned to exactly two plots. (Irrigation is called the whole-plot factor ).
- Divide each whole plot into four subplots. (Each subplot is called a split plot .)
- Within each whole plot, randomly assign the four corn varieties to the four split plots. (Variety is called the split-plot factor .)
- Draw a picture to illustrate the design. 2.What are the two experimental units and the corresponding two randomizations?
- How is this design like a two-way crossed factorial design? How are the two designs different?
- How is this like a randomized complete block design? How are the two designs different?
- Does the split-plot design introduce any possible confounding?
- Second example : An industrial experimenter is studying how the water resistance of wood depends on the pretreatment (two types) and the stain (four types). It turns out to be very difficult to apply the pretreatment to a small wood panel, so instead each type of pretreatment is applied to a whole board, the board is then cut into four smaller wood panels, and one type of stain is applied to each panel. Six whole boards are used.
- Give details to make this into a split-plot design.
- Identify whole and split plots and whole-plot and split- plot factors.
- Is there any confounding? Note : The book starts with the more complex situation where whole plots are within blocks; we will discuss this generalization later. Model for Split-Plot Designs A split-plot experiment can be considered as two experiments superimposed:
- One experiment has the whole-plot factor applied to the large experimental units (whole plots).
- The other experiment has the split-plot factor applied to the smaller experimental units (split plots). The model will reflect this by including two error terms:
- the whole-plot random error !W, and
- the split-plot random error !S.
In our examples (the special case where each level of B is assigned exactly once to each whole plot), the model can be simplified a little. In this special case
- m = al
- if we know i (the level of A) and u (the replication number of the level of A), we have narrowed down to a specific whole plot
- if we also know j (the level of B), we have narrowed down to a specific split plot Thus we do not need t, and can write the model as Yiuj = μ + "i + !iuW
- #j + ("#)ij + !iujS where i = 1, … , a; u = 1, … , l ; j = 1, … , b; !iuW^ ~ N(0, $W^2 ),! (^) iujS^ ~ N(0, $S^2 ), !iuW’s and! (^) iujS’s all mutually independent. Note : The design and model we are working with are actually the simplest form of split-plot designs. There are many variations. Examples:
- The more general form discussed in the book also has blocks containing the whole plots. (More on this later.)
- Random-effects split-plot designs
- Mixed-effects split-plot designs
- Split-plot designs involving more than two factors.
- Split-split-plot designs, where each split-plot is further divided into subplots.
Analysis of Split-Plot Designs For now, we will discuss only the two fixed-factor, no block model described above: Yiujt = μ + "i + !iuW
- #j + ("#)ij + !jt(iu)S where i = 1, … , a u = 1, … , l j = 1, … , b t = 1, … , m; !iuW^ ~ N(0, $W^2 ),! (^) jt(iu)S^ ~ N(0, $S^2 ), !iuW’s and! (^) jt(iu)S’s all mutually independent. The model can be fit by least squares. Model checking : Since there are two error terms in the model, two kinds of residuals need to be checked, the WP (whole plot) and SP (split plot) residuals. To calculate them, first calculate the residuals in the usual way (response values minus fitted values). Then
- WP residuals are obtained by averaging residuals corresponding to each whole plot.
- SP (split plot) residuals are obtained by subtracting WP residuals from residuals. Now make the following model-checking plots: - Normal plot of SP residuals (to check normality of split-plot errors) - Normal plot of WP residuals (to check normality of whole-plot errors) - Plot SP residuals against fitted values (to check constant variance for SP errors) - Plot WP residuals against the average fitted value for the corresponding whole plot (to check constant variance for WP errors) - Plot WP residuals against SP residuals (to check independence of WP and SP errors) - Plot residuals against any other variable available (to check for possible violations of independence).
Example : In the experiment studying the effect of pretreatment and stain on water resistance, the data (including W) are as shown: pretreat stain resist W 2 2 53.5 4 2 4 32.5 4 2 1 46.6 4 2 3 35.4 4 2 4 44.6 5 2 1 52.2 5 2 3 45.9 5 2 2 48.3 5 1 3 40.8 1 1 1 43.0 1 1 2 51.8 1 1 4 45.5 1
In Minitab, use General Linear Model. Response: resist Model: pretreat W( pretreat) stain pretreat* stain Random: W The output is: General Linear Model: resist versus pretreat, stain, W Factor Type Levels Values pretreat fixed 2 1 2 W(pretreat) random 6 1 2 3 4 5 6 stain fixed 4 1 2 3 4 Analysis of Variance for resist, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P pretreat 1 782.04 782.04 782.04 4.03 0. W(pretreat) 4 775.36 775.36 193.84 15.25 0. stain 3 266.00 266.00 88.67 6.98 0. pretreat*stain 3 62.79 62.79 20.93 1.65 0. Error 12 152.52 152.52 12. Total 23 2038. Note :
- Ignore the P-value for W.
- This does not work with Minitab 10.
- ssEW is in the line W(pretreat).
- ssES is in the line Error
- Check that the sums of squares add as indicated above.
- Check that the test ratios are as they should be.
- Note that ssEW is much larger than ssES. This is typical. Why?
Interpret the output: Note : If we don't designate W as random, we get different output: General Linear Model: resist versus pretreat, stain, W Factor Type Levels Values pretreat fixed 2 1 2 W(pretreat) fixed 6 1 2 3 4 5 6 stain fixed 4 1 2 3 4 Analysis of Variance for resist, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P pretreat 1 782.04 782.04 782.04 61.53 0. W(pretreat) 4 775.36 775.36 193.84 15.25 0. stain 3 266.00 266.00 88.67 6.98 0. pretreat*stain 3 62.79 62.79 20.93 1.65 0. Error 12 152.52 152.52 12. Total 23 2038. What is different? How do we know the first method is the one we want? Split Plot Designs with Blocks The split plot model we have discussed is a special case (namely, just one block) of a more general split plot design, where the whole plots are themselves nested within blocks. If the randomization is such that each level of A appears exactly once per block and each level of B appears exactly once per whole plot, the model for this design can be expressed as (Equation 19.2.2 in Dean-Voss) Yhij = μ + %h + "i + !i(h)W
- #j + ("#)ij + !j(hi)S where h = 1, … , s; i = 1, … , a; j = 1, … , b !i(h)W^ ~ N(0, $W^2 ),! (^) j(hi)S^ ~ N(0, $S^2 ), !i(h)W’s and! (^) j(hi)S’s all mutually independent. (%h is the effect of the hth^ block.)
General Linear Model: Y versus BLOCK, A, B, WP Factor Type Levels Values BLOCK random 6 1 2 3 4 5 6 A fixed 3 0 1 2 WP(BLOCK) random 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 B fixed 4 0 1 2 3 Analysis of Variance for Y, using Adjusted SS for Tests Source Model DF Reduced DF Seq SS BLOCK 5 5 15875. A 2 2 1786. WP(BLOCK) 12 10+ 6013. B 3 3 20020. A*B 6 6 321. Error 43 45 7968. Total 71 71 51985.
- Rank deficiency due to empty cells, unbalanced nesting,collinearity, or an undeclared covariate. No storage of results or further analysis will be done. The last comment essentially means that we can’t make residual plots and need to do the remaining calculations by hand. The “reduced degrees of freedom” are what we need. Calculating test statistics: For interaction: msAB/msES = (ssAB/6)/(ssES/45) = (321.8/6)/(7968.8/45) = 53.63/177.08 = 0. For an F(6, 45) distribution, this gives p-value 1 – 0.0664 = .9336. This is consistent with no interaction. For level of manure (factor B): msB/ msES = (ssAB/3)/(ssES/45) = (20020.5/3)/(7968.8/45) = 6673.5/177.08 = 37. For an F(3,45) distribution, this gives p-value 1 – 1.0000 = 0.0000 (to four decimal places) This gives strong evidence that the level of manure makes a difference.
For variety of oats (factor A): msA/msEW = (ssA/2)/(ssWP(BLOCK)/10) = (1786.4/2)/(6013.3/10) = 893.18/601.33 = 1.49. For an F(2,10) distribution, this gives p-value 1 – 0.7286 = .2714. This is consistent with no effect of variety. A further analysis would involve contrasts; see pp. 683 – 684. Note that msES = 177.08 (the estimate of error variance for whole plots) is noticeably smaller than msEW = 601.33 (the estimate of error variance for split plots), as is typical of split plot designs.
