Example of analyzing a full factorial design with replicates and blocks
main topic     interpreting results     session command     see also  

You are an engineer investigating how processing conditions affect the yield of a chemical reaction. You believe that three processing conditions, or factors—reaction time, reaction temperature, and type of catalyst—affect the yield. You have enough resources for 16 runs, but you can perform only 8 in a day. Therefore, you create a full factorial design, with two replicates, and two blocks.

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

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

3    In Responses, enter Yield.

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

5    Click OK in each dialog box.

Session window output

 

Factorial Regression: Yield versus Blocks, Time, Temp, Catalyst 

 

Analysis of Variance

 

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

Model                    8  68.7449   8.5931    58.93    0.000

  Blocks                 1   0.0374   0.0374     0.26    0.628

  Linear                 3  65.6780  21.8927   150.15    0.000

    Time                 1  35.0328  35.0328   240.27    0.000

    Temp                 1  30.5405  30.5405   209.46    0.000

    Catalyst             1   0.1047   0.1047     0.72    0.425

  2-Way Interactions     3   3.0273   1.0091     6.92    0.017

    Time*Temp            1   2.9751   2.9751    20.40    0.003

    Time*Catalyst        1   0.0222   0.0222     0.15    0.708

    Temp*Catalyst        1   0.0301   0.0301     0.21    0.663

  3-Way Interactions     1   0.0021   0.0021     0.01    0.907

    Time*Temp*Catalyst   1   0.0021   0.0021     0.01    0.907

Error                    7   1.0206   0.1458

Total                   15  69.7656

 

 

Model Summary

 

       S    R-sq  R-sq(adj)  R-sq(pred)

0.381847  98.54%     96.87%      92.36%

 

 

Coded Coefficients

 

Term                 Effect     Coef  SE Coef  T-Value  P-Value   VIF

Constant                     45.5592   0.0955   477.25    0.000

Blocks

  1                          -0.0484   0.0955    -0.51    0.628  1.00

Time                 2.9594   1.4797   0.0955    15.50    0.000  1.00

Temp                 2.7632   1.3816   0.0955    14.47    0.000  1.00

Catalyst             0.1618   0.0809   0.0955     0.85    0.425  1.00

Time*Temp            0.8624   0.4312   0.0955     4.52    0.003  1.00

Time*Catalyst        0.0744   0.0372   0.0955     0.39    0.708  1.00

Temp*Catalyst       -0.0867  -0.0434   0.0955    -0.45    0.663  1.00

Time*Temp*Catalyst   0.0230   0.0115   0.0955     0.12    0.907  1.00

 

 

Regression Equation in Uncoded Units

 

Yield = 39.48 - 0.1026 Time + 0.01502 Temp + 0.49 Catalyst + 0.001150 Time*Temp

        - 0.0029 Time*Catalyst - 0.00281 Temp*Catalyst + 0.000031 Time*Temp*Catalyst

 

Equation averaged over blocks.

 

 

Alias Structure

 

Factor  Name

 

A       Time

B       Temp

C       Catalyst

 

 

Aliases

 

I

Block 1

A

B

C

AB

AC

BC

ABC

Graph window output

Interpreting the 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. See Adjusted versus Sequential Sums of Squares for a description of when the values are different.

Look at the p-values to determine whether you have any significant effects. The nonsignificant block effect indicates that the data do not show a statistically significant difference between the data from the two different days. The p-values for the main (linear) effects (0.000) and two-way interactions (0.017) are significant at alpha = 0.05 significance level.

The analysis of variance table and the estimated effects and coefficients table show the p-values for each individual model term. The p-values indicate that only one two-way interaction, Time * Temp (p = 0.003), and two main (linear) effects, Time (p = 0.000) and Temp (p = 0.000), are significant. See Example of factorial plots for an experiment with three factors for a discussion of this interaction.

The residual error that is shown in the ANOVA table can be made up of three parts: (1) curvature, if there are center points in the data, (2) lack of fit, if a reduced model was fit, and (3) pure error, if there are any replicates. If the residual error is due only to lack of fit, Minitab does not print this breakdown. In all other cases, it does.

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 in the Graphs subdialog box. See Residual plots choices.