The response variable was a measure of the variability of the outer diameter of the machined components. One could use the estimated variance, i.e. s2, for each set of experimental conditions. That is, one would replicate the experiment for each set of experimental conditions and estimate s2. Gijo chose to use -10*ln(s2). He lets the symbol S/N represent the -10*ln(s2). Could S/N mean that the response is a Taguchi signal-to-noise ratio? Montgomery (2005, p. 469) discourages the use of signal-to-noise ratios. He states that a more effective approach is to model the mean and variance separately. Hunter (1987) comes to the same conclusion. Gijo does not justify the use of S/N other than a reference to the 3rd edition of Montgomery’s book.
A response variable that has a constant variance over the set of experimental conditions facilitates regression analyses of the results. Montgomery (2005, p. 83) recommends the use of the logarithmic transformation when the standard deviation of the response is proportional to its mean. Let’s proceed by assuming the team used S/N since they wanted to estimate the contribution of the selected factors to the variance of the outer diameter and the standard deviation was roughly proportional to the mean.
The following table gives the experimental design and the observed response for each experiment. The team replicated the experiment twice for each set of experimental conditions. From the two observed outer diameters, they calculated a variance estimate, i.e., s2, and from that computed the response value S/N. The -1 and +1 symbols represent the lower and higher levels of the respective factors.
Experiment | Feed Rate (A) | Wheel Speed (B) | Work Speed (C) | Wheel Grade (D) | Response (S/N) |
1 | -1 | -1 | -1 | -1 | 53.4692 |
2 | -1 | -1 | +1 | +1 | 50.9704 |
3 | -1 | +1 | -1 | +1 | 49.0298 |
4 | -1 | +1 | +1 | -1 | 56.991 |
5 | +1 | -1 | -1 | -1 | 49.0298 |
6 | +1 | -1 | +1 | +1 | 46.1079 |
7 | +1 | +1 | -1 | +1 | 46.1079 |
8 | +1 | +1 | +1 | -1 | 44.9483 |
Effect | -6.067 | -0.625 | 0.345 | -3.056 |
Montgomery (2005, p208) shows how to calculate the average factor effects using the -1 and +1 coding. For a single factor effect, we sum the products of the factor coding times the experiment response over all experiments. Then we divide the sum by the number of -1, +1 pairs. In this experiment, the number of pairs is 4. The last row in the above table shows the estimated factor effects. For an interaction effect, we multiply the experiment coding for each factor to get a coding for the interaction effect.
Experiment | Feed Rate X Wheel Speed AB | Feed Rate X Work Speed AC |
1 | +1 | +1 |
2 | +1 | -1 |
3 | -1 | +1 |
4 | -1 | -1 |
5 | -1 | -1 |
6 | -1 | +1 |
7 | +1 | -1 |
8 | +1 | +1 |
Effect | -1.416 | -2.386 |
Notice that the estimated AB and AC interaction effects are larger than the single factor B and C effects.
The next posting will examine the properties of the experimental design.
References
- Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
- Hunter, J. S. (1987). "Signal-to-Noise Ratio Debated." Quality Progress 20(5): 7-9.
- Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6th Edition, John Wiley & Sons, Inc.