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3. Production Process Characterization
3.2. Assumptions / Prerequisites
3.2.3. Analysis of Variance Models (ANOVA)
3.2.3.1. One-Way ANOVA

3.2.3.1.1.

One-Way Value-Splitting

Example Let's use the data from the machining example to illustrate how to use the techniques of value-splitting to break each data value into its component parts. Once we have the component parts, it is then a trivial matter to calculate the sums of squares and form the F-value for the test. 
 
Machine
1 2 3 4 5
.125 .118 .123 .126 .118
.127 .122 .125 .128 .129
.125 .120 .125 .126 .127
.126 .124 .124 .127 .120
.128 .119 .126 .129 .121
Calculate level-means Remember from our model,  \(y_{ij} = m + a_{i} + \epsilon_{ij}\), we say each observation is the sum of a common value, a level effect and a residual value. Value-splitting just breaks each observation into its component parts. The first step in value-splitting is to calculate the mean values (rounding to the nearest thousandth) within each machine to get the level means. 
Machine
1 2 3 4 5
.1262 .1206 .1246 .1272 .123
Sweep level means We can then sweep (subtract the level mean from each associated data value) the means through the original data table to get the residuals:
Machine
1 2 3 4 5
-.0012
-.0026
-.0016
-.0012
-.005
.0008
.0014
.0004
.0008
.006
-.0012
-.0006
.0004
-.0012
.004
-.0002
.0034
-.0006
-.0002
-.003
.0018
-.0016
.0014
.0018
-.002
Calculate the grand mean The next step is to calculate the grand mean from the individual machine means as:
Grand Mean
.12432
Sweep the grand mean through the level means Finally, we can sweep the grand mean through the individual level means to obtain the level effects:
Machine
1
2
3
4
5
.00188
-.00372
.00028
.00288
-.00132
It is easy to verify that the original data table can be constructed by adding the overall mean, the machine effect and the appropriate residual.
Calculate ANOVA values Now that we have the data values split and the overlays created, the next step is to calculate the various values in the One-Way ANOVA table. We have three values to calculate for each overlay. They are the sums of squares, the degrees of freedom, and the mean squares.
Total sum of squares The total sum of squares is calculated by summing the squares of all the data values and subtracting from this number the square of the grand mean times the total number of data values. We usually don't calculate the mean square for the total sum of squares because we don't use this value in any statistical test.
Residual sum of squares, degrees of freedom and mean square The residual sum of squares is calculated by summing the squares of the residual values. This is equal to .000132. The degrees of freedom is the number of unconstrained values. Since the residuals for each level of the factor must sum to zero, once we know four of them, the last one is determined. This means we have four unconstrained values for each level, or 20 degrees of freedom. This gives a mean square of .000007.
Level sum of squares, degrees of freedom and mean square Finally, to obtain the sum of squares for the levels, we sum the squares of each value in the level effect overlay and multiply the sum by the number of observations for each level (in this case 5) to obtain a value of .000137. Since the deviations from the level means must sum to zero, we have only four unconstrained values so the degrees of freedom for level effects is 4. This produces a mean square of .000034.
Calculate F-value The last step is to calculate the F-value and perform the test of equal level means. The F- value is just the level mean square divided by the residual mean square. In this case the F-value=4.86. If we look in an F-table for 4 and 20 degrees of freedom at 95% confidence, we see that the critical value is 2.87, which means that we have a significant result and that there is thus evidence of a strong machine effect. By looking at the level-effect overlay we see that this is driven by machines 2 and 4.
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