The interest of the Anova method in relation to the Average and Range method is based on the fact that it calculates the interaction between the operators and the parts.
The typical example of this type of interaction is the case of small parts. For example, we want to measure parts of only a few millimeters. We perform a Gage R & R Test via 3 different operators. One of them has some vision problems and sees less of the parts than the others. There is a good bet that during the analysis, it will come out that there is indeed an interaction operator/ Part, since one of them will have measures probably different from the others.
The use of the Anova for the calculation of the precision of a measuring system is based on the comparison of the average of the different measurement groups with the general average. We calculate:
- Intergroup Variance (explained): difference between the average of each group and the general average. In our case, we will remember that we have 3 different groups: operators, parts and interaction operators/parts.
- Variance Intra (residual): difference between the value of each individual and the average of the group. It will correspond to the Variance due to the error of the measuring instrument.
- Total Variance : The difference between the value of each individual and the general average.
The ANOVA is presented as table summarizing all the results of the calculations we present below.
|Source of variability||Sum of squares of deviations||DOF||Mean square||Practical value||Critical value||P-Value|
|Operator||SST Ope||A - 1||MST Ope||MST Ope / MSE||Fisher||Fisher|
|Parts||SST Parts||B - 1||MST Parts||MST parts / MSE|
|Interaction operators / parts||SST Inter||(A - 1) * (B - 1)||MST Inter||MST Inter / MSE|
|Measurement error||SSE||A * B * (n - 1)||MSE||-|
|Total variability||TSS||N - 1||-|
- A: Number of operators
- B: Number of rooms
- n: number of measurements per operator and per Part
- N: Total number of tests
1 – Calculate the square sums of variances – SST
One starts by calculating the square sum of the differences of the different groups. It consists of calculating the difference in the values of each group relative to the average of all the groups. We are looking at whether the average of the different groups is far from the general average. So if this value is large, the variability between the averages is important and takes us to reject the null hypothesis. In other words, the samples are different.
for operators: SST oper = Σ (Ya2/(B * N)) – Σ Y2/n
for parts: SST Part = Σ (Yb2/(A * N)) – Σ Y2/n
for interaction: SST Inter = Σ (Yab2/n) – Σ Y2/n – SSToper -SST Parts
for residues: SSE = TSS – SSTope – SST parts – SST Inter
- Ya : each of the values for the A operators
- Yb : Each of the values for the B parts
- Yab : Each of the values of the A operators for the B parts
- Y: Each of the set of values of all the operators and all the parts
2 – Calculate the sum of the total squares – TSS
It represents the total change in our measurements. It is calculated by comparing the sum of the values of each of the values to the average of all the measurements.
TSS= Σ Y2 – Σ Y2/n
3 – Calculate the sum of the squares of the residues-SSE
The residual variance actually represents the variability due to the measuring instrument. It is deducted from the other values using the following formula:
SSE = TSS-SSTope – TSSparts – TSSInter
4 – Calculating the means squares
The means squares represent the “weight” that can be given to each of the variability. They are calculated by making the connection with the dof. Each of the means squares is calculated by making the ratio between the square sum of the deviations and the corresponding degrees of freedom:
MST op = SSTop/(A – 1)
MST parts = SSTparts/(B – 1)
MST inter = SST Inter/((A – 1) * (B – 1))
MSE = SSE/((A * B * (n – 1))
5 – Perform the hypothesis test
Carry out the hypothesis test Anova is not necessary in the case of a Gage R & R study. Indeed, the interpretation of this type of analysis is carried out by means of the coefficients specific to the Gage R & R.
However, if you wish to calculate the practical, critical and p-value values, the calculation formulas and their interpretation are strictly identical to a classical test.
6 – Computation of repeatability – EV
Repeatability is the ability of the measuring system to measure the same characteristic and achieve the same result. It is calculated in the following way:
EV = √ MSE
7-Calculation of the interaction
The interaction is measured by the difference between the weight of the interaction and the residuals. The formula is as follows:
Interaction = √ (MST Inter – MSE)/n
8 – Calculation of reproducibility – AV
Reproducibility consists of the difference between the operator’s variability and its interaction with the parts. It is calculated in the following way:
Interaction = √ (MSTope – MST Inter)/(B * N) + Interaction2)
9 – Calculating the Gage R & R
The overall dispersion of the equipment is calculated according to the following formula:
Gage R & R = √ (AV2 + EV2)
10 – Calculating the variability of parts – PV
The dispersion due to the variability of the parts, excluding the effect of the measurement error, is calculated according to the following formula:
PV = √ ( MST parts – MSE)/(A * N)
11 – Calculation of total variability – TV
The total Variation corresponds to the variability of all elements, equipment, man, parts and interaction. It is calculated according to the following formula:
TV = √ (Gage R & R2 + PV2)
F. Bergert, S. Mercier (2011) – Statistical control of processes
D. Drain (1996) – Statistical methods for industrial Process Control
C. D. Montgomery (2009) – Statistical Quality Control