# Signal to Noise Ratios

Depending upon the objective of the robust parameter design experiment, Taguchi defined three different statistics called signal to noise ratios. These ratios were defined as the means to measure the variation of the response with respect to the noise factors. Taguchi's approach essentially consists of two models - a location model and a dispersion model.

This section is divided into the following subsections:

### Location Model

The location model is the regression model for the mean value of the response at each treatment combination of the inner array. If ( ) represents the response values obtained at the th treatment combination of the inner array (corresponding to the levels of the noise factors), then the mean response at the th level is:(2)

The location model is obtained by fitting a regression model to all values, by treating these values as the response at each of the th treatments of the inner array. As an example, the location model for an inner array with two factors can be written as:(3)where:

• is the intercept.

• is the coefficient for the first factor.

• is the coefficient for the second factor.

• is the coefficient for the interaction.

• and are respectively the variables for the two factors.

The objective of using the location model is to bring the response to its goal regardless of whether this is a target value, maximum value or minimum value. This is done by identifying significant effects and then using the least square estimates of the corresponding coefficients, s, to fit the location model. The fitted model is used to decide the settings of the variables that bring the response to the goal.

### Dispersion Model

The dispersion model measures the variation of the response due to the noise factors. The standard deviation of the response values at each treatment combination, , is used. Usually, the standard deviation is used as a log function of because are approximately normally distributed. These values can be calculated as follows:(4)

Thus, the dispersion model consists of using as the response and investigating what treatment of the control factors results in the minimum variation of the response. Clearly, the objective of using the dispersion model is to minimize variation in the response. Instead of using standard deviations directly, Taguchi defined three signal to noise ratios (abbreviated ) based on the objective function for the response. If the response is to be maximized, the ratio is defined as follows:(5)

The previous ratio is referred to as the larger-the-better ratio and is defined to decrease variability when maximizing the response.

When the response is to be minimized, the ratio is defined as:(6)

This ratio is referred to as the smaller-the-better ratio and is defined to decrease variability when minimizing the response.

If the objective for the response is to achieve a target or nominal value, then the ratio is defined as follows:(7)

This ratio is referred to as the nominal-the-best ratio and is defined to decrease variability around a target response value.

The dispersion model for any of the three signal to noise ratios can be written as follows for an inner array with two factors:where:

• is the intercept.

• is the coefficient for the first factor.

• is the coefficient for the second factor.

• is the coefficient for the interaction.

• and are respectively the variables for the two factors.

The dispersion model is fit by identifying significant effects and then using the least square estimates of the coefficients s. Once the fitted dispersion model is known, settings for the control factors are found that result in the maximum value of , thereby minimizing the response variation.

### Analysis Strategy

The location and dispersion regression models are usually obtained by using graphical techniques to identify significant effects. This is because the responses used in the two models are such that only one response value is obtained for each treatment of the inner array. Therefore, the experiment design in the case of the two models is an unreplicated design. [Note]

Once the location and dispersion models have been obtained by identification of the significant effects, the following analysis strategy may be used [27]:

• To obtain the best settings of the factors for larger-the-better and smaller-the-better cases:
• the experimenter must first select the settings of the significant control factors in the location model to either maximize or minimize the response.
• then, the experimenter must choose the settings of those significant control factors in the dispersion model, that are not significant in the location model, to maximize the ratio.
• For nominal-the-best cases:
• the experimenter must first select the settings of the significant control factors in the dispersion model to maximize the ratio.
• then the experimenter must choose the levels of the significant control factors in the location model to bring the response on target. At times, the same control factor(s) may show up as significant in both the location and dispersion models. In these cases, the experimenter must use his judgement to obtain the best settings of the control factors based upon the two models.

Factors that do not show up as significant in both the models should be set at levels that result in the greatest economy. Generally, a follow-up experiment is usually carried out with the best settings to verify that the system functions as desired.

Example 10.2

This example illustrates the procedure to obtain the location and dispersion models for the experiment in Example 10.1.

Location Model

The response values used in the location model can be calculated using Eqn. (2). As an example, the response value for the third treatment is:

Response values for the remaining seven treatments can be calculated in a similar manner. These values are shown in Figure 10.7 under the column Y Mean. Once the response values for all the treatments are known, the analysis to fit the location model can be carried out by treating the experiment as a single replicate of the 2 design. The results obtained from DOE++ are shown in Figure 10.8. The results are displayed by selecting Y Mean in the Response drop-down menu. The normal probability plot of effects for this model shows that only the main effect of factor is significant for the location model (see Figure 10.9). Using the corresponding coefficient from Figure 10.9, the location model can be written as:(8)

where is the variable representing factor .

 Figure 10.7: Response values for the location and dispersion models in Example 10.2.

 Figure 10.8: Results for the location model in Example 10.2.

 Figure: 10.9: Normal probability plot of effects for the location model in Example 10.2.

Dispersion Model

For the dispersion model, the applicable signal to noise ratio is given by Eqn. (7) since this example is a case of nominal-the-best:

The response values for the dispersion model can now be calculated. As an example, the response value for the third treatment is:

Other values can be obtained in a similar manner. The values are shown in Figure 10.7 under the column Signal Noise Ratio. As in the case of the location model, the analysis to fit the dispersion model can be carried out by treating the experiment as a single replicate of the 2 design. The results obtained from DOE++ are shown in Figure 10.10. The results are displayed by selecting Signal Noise Ratio in the Response drop-down menu. The normal probability plot of effects for this model shows that the interaction is the only significant effect for this model (see Figure 10.11). Using the corresponding coefficient from Figure 10.10, the dispersion model can be written as:(9)

where is the variable representing factor .

 Figure 10.10: Results for the dispersion model in Example 10.2.

 Figure 10.11: Normal probability plot of effects for the dispersion model in Example 10.2.

Robust Parameter Design

Following the analysis strategy mentioned above in Analysis Strategy, for the nominal-the-best case, the dispersion model should be considered first. Eqn. (9) shows that to maximize , either one of the following options can be used:

• and or
• and

Then, considering the location model of Eqn. (8), to achieve a target response value as close to 7 as possible, the only significanteffect for this model, , should be set at the level of . Therefore, option 1 should be selected as the settings for the dispersion model. The final settings for the three factors, as a result of the robust parameter design, are:

• Factor is set at the low level
• Factor is set at the high level and
• Factor , which is not significant in any of the two models, can be set at the level that is most economical.

With these settings the predicted pH value for the product is:

The predicted signal to noise ratio is:

To make the signal to noise ratio model hierarchical, and have to be included in the model. [Note] Then, the predicted ratio is: