3.
Production
Process Characterization
3.2.
Assumptions / Prerequisites
3.2.3.
Analysis of Variance Models (ANOVA)
3.2.3.1.
OneWay ANOVA
3.2.3.1.1.

OneWay ValueSplitting


Example 
Let's use the data from the machining example to illustrate
how to use the techniques of valuesplitting 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 Fvalue 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 levelmeans 
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. Valuesplitting just breaks each observation
into its component parts. The first step in valuesplitting 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:



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
OneWay
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 Fvalue 
The last step is to calculate the Fvalue 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 Fvalue=4.86. If we look
in an Ftable 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 leveleffect overlay we see that this is driven by
machines 2 and 4.
