Example of analyzing a two-level split plot design
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A plastic manufacturing company wants to increase the strength of its plastic. The research team identifies additive percentage, agitation rate, and processing time as possible contributors to strength. The temperature at which the batches are baked also impacts plastic strength. They decide to use a split-plot design; all 8 combinations of additive percent, agitation rate and processing time are baked at the same time for each temperature setting. This is repeated so that each temperature level is used twice. This results in 32 observations, run in 4 whole plots of 8 subplots each. The research team has created a split-plot design (see Example of creating a 2-level split-plot design) and collected the strength data. Now they are ready to analyze it.

1    Open the worksheet STRENGTH.MTW. (The design and response data have been saved for you.)

2    Choose Stat > DOE > Factorial > Analyze Factorial Design.

3    In Responses, enter Strength.

4    Click Terms. In Include terms in the model up through order, choose 2. Click OK.

5    Click Graphs. Under Effects Plots, check Pareto, Normal, and Half Normal.

6    In the drop-down, choose Display only model terms, subplot effects only.

7    Click OK in each dialog box.

Session window output

Split-Plot Factorial Regression: Strength versus Temp[HTC], Add, Rate, Time

 

 

Analysis of Variance

 

Source          DF   Seq SS  Contribution   Adj SS  Adj MS  F-Value  P-Value

Temp[HTC]        1   85.478        11.20%   85.478  85.478     1.52    0.343

WP Error         2  112.391        14.73%  112.391  56.195     5.74    0.011

Add              1   45.363         5.95%   45.363  45.363     4.64    0.044

Rate             1   41.178         5.40%   41.178  41.178     4.21    0.054

Time             1   75.953         9.95%   75.953  75.953     7.76    0.012

Temp[HTC]*Add    1    1.088         0.14%    1.088   1.088     0.11    0.742

Temp[HTC]*Rate   1   78.438        10.28%   78.438  78.438     8.02    0.011

Temp[HTC]*Time   1   62.440         8.18%   62.440  62.440     6.38    0.021

Add*Rate         1   27.938         3.66%   27.938  27.938     2.86    0.107

Add*Time         1    2.940         0.39%    2.940   2.940     0.30    0.590

Rate*Time        1   43.945         5.76%   43.945  43.945     4.49    0.047

SP Error        19  185.858        24.36%  185.858   9.782

Total           31  763.010       100.00%

 

 

Model Summary

 

      S  R-sq(SP)    S(WP)  R-sq(WP)

3.12762    67.11%  2.40866    43.20%

 

 

Coded Coefficients

 

Term            Effect   Coef  SE Coef       95% CI       T-Value  P-Value   VIF

Constant                62.00     1.33  ( 59.23,  64.78)    46.79    0.000

Temp[HTC]         3.27   1.63     1.33  ( -1.14,   4.41)     1.23    0.343     *

Add              2.381  1.191    0.553  (-3.713, -1.399)     2.15    0.044  1.00

Rate             2.269  1.134    0.553  (-1.857,  0.457)     2.05    0.054  1.00

Time             3.081  1.541    0.553  ( 0.033,  2.348)     2.79    0.012  1.00

Temp[HTC]*Add    0.369  0.184    0.553  (-0.023,  2.292)     0.33    0.742  1.00

Temp[HTC]*Rate   3.131  1.566    0.553  ( 0.383,  2.698)     2.83    0.011  1.00

Temp[HTC]*Time   2.794  1.397    0.553  (-0.973,  1.342)     2.53    0.021  1.00

Add*Rate         1.869  0.934    0.553  ( 0.408,  2.723)     1.69    0.107  1.00

Add*Time         0.606  0.303    0.553  ( 0.240,  2.554)     0.55    0.590  1.00

Rate*Time        2.344  1.172    0.553  (-0.223,  2.092)     2.12    0.047  1.00

 

 

Regression Equation in Uncoded Units

 

Strength = 62.003 + 1.634 Temp[HTC] + 1.191 Add + 1.134 Rate + 1.541 Time + 0.184 Temp[HTC]

           *Add + 1.566 Temp[HTC]*Rate + 1.397 Temp[HTC]*Time + 0.934 Add*Rate

           + 0.303 Add*Time + 1.172 Rate*Time

 

Equation averaged over whole plots.

 

 

Alias Structure

 

Factor  Name

 

A       Temp[HTC]

B       Add

C       Rate

D       Time

 

 

Aliases

 

I

A

B

C

D

AB

AC

AD

BC

BD

CD

Graph window output

Interpreting results

The analysis of variance table gives a summary of the main effects and interactions. Minitab displays both the sequential sums of squares (Seq SS) and adjusted sums of squares (Adj SS). If the model is orthogonal and does not contain covariates, these will be the same.

The analysis of variance table and the estimated effects and coefficients table show the p-values associated with each individual model term. The p-values for Add (0.044), Time (0.012), Temp * Rate (0.011), Temp * Time (0.021), and Rate * Time (0.047) are significant at the alpha level of 0.05. Because time is part of two significant interactions, you need to understand the nature of the interactions before you can consider the main effect.

WP Error (whole plot error) is the variance between the whole plots. SP error (subplot error) is the variation between subplots not explained by the factors. The variation between the whole plots is large relative to the subplot variation.

The normal, half normal, and Pareto plots of the effects allow you to visually identify the important effects and compare the relative magnitude of the various effects.

You should also plot the residuals versus the run order to check for any time trends or other nonrandom patterns. Residual plots are found in the Graphs subdialog box. See Residual plots choices.

Note

Minitab cannot compute S(WP) in the following cases: WP Error has zero degrees of freedom, SP Error has zero degrees of freedom, or MSE(WP) - MSE(SP) is a negative value.