3. Production Process Characterization
3.5. Case Studies
3.5.2. Machine Screw Case Study

## Analysis of Variance

Analysis of Variance Using All Factors We can confirm our interpretation of the box plots by running an analysis of variance when all four factors are included.
 Source            DF    Sum of       Mean   F Statistic   Prob > F
Squares     Square
------------------------------------------------------------------
Machine            2  0.000111   0.000055       29.3159    1.3e-11
Day                2  0.000004   0.000002        0.9884       0.37
Time               1  0.000002   0.000002        1.2478       0.27
Sample             9  0.000009   0.000001        0.5205       0.86
Residual         165  0.000312   0.000002
------------------------------------------------------------------
Corrected Total  179  0.000437   0.000002

Interpretation of ANOVA Output We fit the model
$$Y_{ijklm} = \mu + \alpha_i + \beta_j + \tau_k + \phi_l + \epsilon_{ijklm}$$
which has an overall mean, as opposed to the model
$$Y_{ijklm} = A_i + B_j + C_k + D_l + \epsilon_{ijklm}$$
These models are mathematically equivalent. The effect estimates in the first model are relative to the overall mean. The effect estimates for the second model can be obtained by simply adding the overall mean to effect estimates from the first model.

Only the machine factor is statistically significant. This confirms what the box plots in the previous section had indicated graphically.

Analysis of Variance Using Only Machine The previous analysis of variance indicated that only the machine factor was statistically significant. The following table displays the ANOVA results using only the machine factor.
 Source            DF    Sum of       Mean   F Statistic   Prob > F
Squares     Square
------------------------------------------------------------------
Machine            2  0.000111   0.000055       30.0094    6.0E-12
Residual         177  0.000327   0.000002
------------------------------------------------------------------
Corrected Total  179  0.000437   0.000002

Interpretation of ANOVA Output At this stage, we are interested in the level means for the machine variable. These can be summarized in the following table.

 Level Number Mean Standard Error Lower 95% CI Upper 95% CI 1 60 0.124887 0.00018 0.12454 0.12523 2 60 0.122968 0.00018 0.12262 0.12331 3 60 0.124022 0.00018 0.12368 0.12437

Model Validation As a final step, we validate the model by generating a 4-plot of the residuals.

The 4-plot does not indicate any significant problems with the ANOVA model.