# Taguchi Methods: Robust Design Experiments - Signal-to-Noise (S/N) Ratios

Measuring quality loss. Even though you have concluded that the quality loss function is probably quadratic in nature, you still do not know precisely how to measure quality loss. However, you know that whatever measure you decide upon should reflect the quadratic nature of the function.

Signal, noise, and control factors. The product of ideal quality should always respond in exactly the same manner to the signals provided by the user. When you turn the key in the ignition of your car you expect that the starter motor turns and the engine starts. In the ideal-quality car, the starting process would always proceed in exactly the same manner - for example, after three turns of the starter motor the engine comes to life. If, in response to the same signal (turning the ignition key) there is random variability in this process, then you have less than ideal quality. For example, due to such uncontrollable factors as extreme cold, humidity, engine wear, etc. the engine may sometimes start only after turning over 20 times and finally not start at all. This example illustrates the key principle in measuring quality according to Taguchi: You want to minimize the variability in the product's performance in response to noise factors while maximizing the variability in response to signal factors.

Noise factors are those that are not under the control of the operator of a product. In the car example, those factors include temperature changes, different qualities of gasoline, engine wear, etc. Signal factors are those factors that are set or controlled by the operator of the product to make use of its intended functions (turning the ignition key to start the car).

Finally, the goal of your quality improvement effort is to find the best settings of factors under your control that are involved in the production process, in order to maximize the S/N ratio; thus, the factors in the experiment represent control factors.

S/N ratios. The conclusion of the previous paragraph is that quality can be quantified in terms of the respective product's response to noise factors and signal factors. The ideal product will only respond to the operator's signals and will be unaffected by random noise factors (weather, temperature, humidity, etc.). Therefore, the goal of your quality improvement effort can be stated as attempting to maximize the signal-to-noise (S/N) ratio for the respective product. The S/N ratios described in the following paragraphs have been proposed by Taguchi (1987). These S/N ratios can be computed with the Taguchi robust design options in the Experimental Design module. You can compute these S/N ratios for any data with STATISTICA Visual BASIC, and use the resulting values with all designs available in the Experimental Design module (e.g., you could use S/N ratios in the context of a response surface experiment).

Smaller-the-better. In cases where you want to minimize the occurrences of some undesirable product characteristics, you would compute the following S/N ratio:

Eta = -10 * log10 [(1/n) * S(yi2)]     for i = 1 to no. vars     (see outer arrays.)

Here, Eta is the resultant S/N ratio; n is the number of observations on the particular product, and y is the respective characteristic. For example, the number of flaws in the paint on an automobile could be measured as the y variable and analyzed via this S/N ratio. The effect of the signal factors is zero, since zero flaws is the only intended or desired state of the paint on the car. Note how this S/N ratio is an expression of the assumed quadratic nature of the loss function. The factor -10 ensures that this ratio measures the inverse of "bad quality;" the more flaws in the paint, the greater is the sum of the squared number of flaws, and the smaller (i.e., more negative) the S/N ratio. Thus, maximizing this ratio will increase quality.

Nominal-the-best. Here, you have a fixed signal value (nominal value), and the variance around this value can be considered the result of noise factors:

Eta = 10 * log10 (Mean2/Variance)

This signal-to-noise ratio could be used whenever ideal quality is equated with a particular nominal value. For example, the size of piston rings for an automobile engine must be as close to specification as possible to ensure high quality.

Larger-the-better. Examples of this type of engineering problem are fuel economy (miles per gallon) of an automobile, strength of concrete, resistance of shielding materials, etc. The following S/N ratio should be used:

Eta = -10 * log10 [(1/n) * S(1/yi2)]     for i = 1 to no. vars     (see outer arrays.)

Signed target. This type of S/N ratio is appropriate when the quality characteristic of interest has an ideal value of 0 (zero), and both positive and negative values of the quality characteristic may occur. For example, the dc offset voltage of a differential operational amplifier may be positive or negative (see Phadke, 1989). The following S/N ratio should be used for these types of problems:

Eta = -10 * log10(s2)

where s2 stands for the variance of the quality characteristic across the measurements (variables).

Fraction defective. This S/N ratio is useful for minimizing scrap, minimizing the percent of patients who develop side-effects to a drug, etc. Taguchi also refers to the resultant Eta values as Omegas; note that this S/N ratio is identical to the familiar logit transformation (see also Nonlinear Estimation):

Eta = -10 * log10[p/(1-p)]

where p is the proportion defective.

Ordered categories (the accumulation analysis). In some cases, measurements on a quality characteristic can only be obtained in terms of categorical judgments. For example, consumers may rate a product as excellent, good, average, or below average. In that case, you would attempt to maximize the number of excellent or good ratings. Typically, the results of an accumulation analysis are summarized graphically in a stacked bar plot.