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

3.5.2.3.

Analysis of Variance

Analysis of Variance using All Factors

We can confirm our interpretation of the box plots by running an analysis of variance. Dataplot generated the following analysis of variance output when all four factors were included.

********************************** ********************************** ** 4-WAY ANALYSIS OF VARIANCE ** ********************************** ********************************** NUMBER OF OBSERVATIONS = 180 NUMBER OF FACTORS = 4 NUMBER OF LEVELS FOR FACTOR 1 = 3 NUMBER OF LEVELS FOR FACTOR 2 = 3 NUMBER OF LEVELS FOR FACTOR 3 = 2 NUMBER OF LEVELS FOR FACTOR 4 = 10 BALANCED CASE RESIDUAL STANDARD DEVIATION = 0.13743976597E-02 RESIDUAL DEGREES OF FREEDOM = 165 NO REPLICATION CASE NUMBER OF DISTINCT CELLS = 180 ***************** * ANOVA TABLE * ***************** SOURCE DF SUM OF SQUARES MEAN SQUARE F STATISTIC F CDF SIG ------------------------------------------------------------------------------- TOTAL (CORRECTED) 179 0.000437 0.000002 ------------------------------------------------------------------------------- FACTOR 1 2 0.000111 0.000055 29.3159 100.000% ** FACTOR 2 2 0.000004 0.000002 0.9884 62.565% FACTOR 3 1 0.000002 0.000002 1.2478 73.441% FACTOR 4 9 0.000009 0.000001 0.5205 14.172% ------------------------------------------------------------------------------- RESIDUAL 165 0.000312 0.000002 RESIDUAL STANDARD DEVIATION = 0.00137439766 RESIDUAL DEGREES OF FREEDOM = 165 **************** * ESTIMATION * **************** GRAND MEAN = 0.12395893037E+00 GRAND STANDARD DEVIATION = 0.15631503193E-02 LEVEL-ID NI MEAN EFFECT SD(EFFECT) -------------------------------------------------------------------- FACTOR 1-- 1.00000 60. 0.12489 0.00093 0.00014 -- 2.00000 60. 0.12297 -0.00099 0.00014 -- 3.00000 60. 0.12402 0.00006 0.00014 FACTOR 2-- 1.00000 60. 0.12409 0.00013 0.00014 -- 2.00000 60. 0.12403 0.00007 0.00014 -- 3.00000 60. 0.12376 -0.00020 0.00014 FACTOR 3-- 1.00000 90. 0.12384 -0.00011 0.00010 -- 2.00000 90. 0.12407 0.00011 0.00010 FACTOR 4-- 1.00000 18. 0.12371 -0.00025 0.00031 -- 2.00000 18. 0.12405 0.00009 0.00031 -- 3.00000 18. 0.12398 0.00002 0.00031 -- 4.00000 18. 0.12382 -0.00014 0.00031 -- 5.00000 18. 0.12426 0.00030 0.00031 -- 6.00000 18. 0.12379 -0.00016 0.00031 -- 7.00000 18. 0.12406 0.00010 0.00031 -- 8.00000 18. 0.12376 -0.00020 0.00031 -- 9.00000 18. 0.12376 -0.00020 0.00031 -- 10.00000 18. 0.12440 0.00044 0.00031 MODEL RESIDUAL STANDARD DEVIATION ------------------------------------------------------- CONSTANT ONLY-- 0.0015631503 CONSTANT & FACTOR 1 ONLY-- 0.0013584237 CONSTANT & FACTOR 2 ONLY-- 0.0015652323 CONSTANT & FACTOR 3 ONLY-- 0.0015633047 CONSTANT & FACTOR 4 ONLY-- 0.0015876852 CONSTANT & ALL 4 FACTORS -- 0.0013743977

Interpretation of ANOVA Output

The first thing to note is that Dataplot fits an overall mean when performing the ANOVA. That is, it fits the model as opposed to the model 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. We are primarily interested in identifying the significant factors. The last column of the ANOVA table prints a "**" for statistically significant factors. Only factor 1 (the machine) 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 shows the ANOVA output using only the machine factor.

********************************** ********************************** ** 1-WAY ANALYSIS OF VARIANCE ** ********************************** ********************************** NUMBER OF OBSERVATIONS = 180 NUMBER OF FACTORS = 1 NUMBER OF LEVELS FOR FACTOR 1 = 3 BALANCED CASE RESIDUAL STANDARD DEVIATION = 0.13584237313E-02 RESIDUAL DEGREES OF FREEDOM = 177 REPLICATION CASE REPLICATION STANDARD DEVIATION = 0.13584237313E-02 REPLICATION DEGREES OF FREEDOM = 177 NUMBER OF DISTINCT CELLS = 3 ***************** * ANOVA TABLE * ***************** SOURCE DF SUM OF SQUARES MEAN SQUARE F STATISTIC F CDF SIG ------------------------------------------------------------------------------- TOTAL (CORRECTED) 179 0.000437 0.000002 ------------------------------------------------------------------------------- FACTOR 1 2 0.000111 0.000055 30.0094 100.000% ** ------------------------------------------------------------------------------- RESIDUAL 177 0.000327 0.000002 RESIDUAL STANDARD DEVIATION = 0.00135842373 RESIDUAL DEGREES OF FREEDOM = 177 REPLICATION STANDARD DEVIATION = 0.00135842373 REPLICATION DEGREES OF FREEDOM = 177 **************** * ESTIMATION * **************** GRAND MEAN = 0.12395893037E+00 GRAND STANDARD DEVIATION = 0.15631503193E-02 LEVEL-ID NI MEAN EFFECT SD(EFFECT) -------------------------------------------------------------------- FACTOR 1-- 1.00000 60. 0.12489 0.00093 0.00014 -- 2.00000 60. 0.12297 -0.00099 0.00014 -- 3.00000 60. 0.12402 0.00006 0.00014 MODEL RESIDUAL STANDARD DEVIATION ------------------------------------------------------- CONSTANT ONLY-- 0.0015631503 CONSTANT & FACTOR 1 ONLY-- 0.0013584237

Interpretation of ANOVA Output

At this stage, we are interested in the effect estimates for the machine variable. These can be summarized in the following table. Means for Oneway Anova 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 The Dataplot macro file shows the computations required to go from the Dataplot ANOVA output to the numbers in the above table.

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.