Interpret the key results for Analyze Taguchi Design

Complete the following steps to interpret a Taguchi design. Key output includes the delta and rank values, the p-values, the coefficients, and the main effect and interaction plots.

Step 1: Identify the best level for each control factor

Use the signal-to-noise ratio (S/N ratio) to identify the control factor settings that minimize the variability caused by the noise factors. Minitab calculates the S/N ratio for each combination of control factors and then calculates the average S/N ratio for each level of each control factor. Choose from four S/N ratios, based on the experimental goal and an understanding of the desired outcome of the process. For more information, go to What is the signal-to-noise ratio in a Taguchi design?.

Delta is the difference between the highest and lowest average response values for each factor. Minitab assigns ranks based on Delta values; Rank 1 to the highest Delta value, Rank 2 to the second highest, and so on, to indicate the relative effect of each factor on the response.

Response Table for Signal to Noise Ratios Dynamic Response Level Variety Light Fertilizer Water Spraying 1 -1.9266 -0.6911 -4.1399 -0.9870 0.2274 2 2.8068 1.5712 5.0201 1.8672 0.6527 Delta 4.7333 2.2623 9.1600 2.8542 0.4253 Rank 2 4 1 3 5
Response Table for Slopes Level Variety Light Fertilizer Water Spraying 1 0.6867 0.6043 0.5264 0.5437 0.7067 2 0.7440 0.8264 0.9043 0.8870 0.7240 Delta 0.0572 0.2220 0.3778 0.3433 0.0174 Rank 4 3 1 2 5
Response Table for Standard Deviations Level Variety Light Fertilizer Water Spraying 1 0.7794 0.5450 0.7677 0.5222 0.6207 2 0.5042 0.7387 0.5159 0.7614 0.6629 Delta 0.2752 0.1937 0.2518 0.2392 0.0422 Rank 1 4 2 3 5
Key Results: Delta, Rank

In these results, the engineer wants to minimize the standard deviation and maximize the S/N ratio and the slopes.
  • Fertilizer (Delta 9.1600, Rank = 1) has the largest effect on the S/N ratio, followed by Variety (Delta 4.7333, Rank = 2), then followed by Water, Light, and Spraying.
  • Fertilizer (Delta 0.3778, Rank = 1) also has the largest effect on the slopes, followed by Water (Delta 0.3433, Rank = 2), then followed by Light, Variety, and Spraying.
  • Variety (Delta 0.2752, Rank = 1) has the largest effect on the standard deviations, followed by Fertilizer (Delta 0.2518, Rank = 2), then followed by Water, Light, and Spraying.

Note

If you have a static design and do not have a signal factor, you will have a response table for means instead of slopes.

Step 2: Determine which factors have statistically significant effects on the response

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that the term's coefficient is equal to zero, which implies that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.
P-value ≤ α: The association is statistically significant
If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
P-value > α: The association is not statistically significant
If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.
If there are multiple predictors without a statistically significant association with the response, you can reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.
If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:
  • If a coefficient for a factor is significant, you can conclude that not all level means are equal.
  • If a coefficient for an interaction term is significant, the relationship between a factor and the response depends on the other factors in the term. In this case, you should not interpret the main effects without considering the interaction effect.

If a model term is not statistically significant, you can remove it and refit the model. Frequently, a significance level of 0.10 is used for evaluating terms in a model.

The coefficient describes the size and direction of the relationship between a term in the model and the response variable. The absolute value of the coefficient indicates the relative strength of each factor. The number of coefficients Minitab calculates for a factor is the number of levels minus one. If a factor has 3 levels, Minitab provides 2 coefficients, which correspond to factor levels 1 and 2. If a factor has 2 levels, Minitab provides 1 coefficient, which corresponds to factor level 1. Minitab includes the values or text that correspond to the level.

The response tables show the average of each response characteristic for each level of each factor. The tables include ranks based on Delta statistics, which compare the relative magnitude of effects. The Delta statistic is the highest minus the lowest average for each factor. Minitab assigns ranks based on Delta values; rank 1 to the highest Delta value, rank 2 to the second highest, and so on. Use the level averages in the response tables to determine which level of each factor provides the best result.

Linear Model Analysis: SN ratios versus Variety, Light, Fertilizer, Water, ...

Estimated Model Coefficients for SN ratios Term Coef SE Coef T P Constant 0.4401 0.2384 1.846 0.316 Variety 1 -2.3667 0.2384 -9.926 0.064 Light 1 -1.1312 0.2384 -4.744 0.132 Fertiliz 1 -4.5800 0.2384 -19.209 0.033 Water 1 -1.4271 0.2384 -5.985 0.105 Spraying 1 -0.2127 0.2384 -0.892 0.536 Variety*Fertiliz 1 1 -0.6041 0.2384 -2.534 0.239
Model Summary S R-Sq R-Sq(adj) 0.6744 99.81% 98.69%
Analysis of Variance for SN ratios Source DF Seq SS Adj SS Adj MS F P Variety 1 44.809 44.809 44.809 98.52 0.064 Light 1 10.236 10.236 10.236 22.51 0.132 Fertilizer 1 167.811 167.811 167.811 368.97 0.033 Water 1 16.293 16.293 16.293 35.82 0.105 Spraying 1 0.362 0.362 0.362 0.80 0.536 Variety*Fertilizer 1 2.920 2.920 2.920 6.42 0.239 Residual Error 1 0.455 0.455 0.455 Total 7 242.886
Key Results: P-Value, Coefficients

In this example, for the signal-to-noise ratio, Fertilizer has a p-value that is less than 0.05, thus, Fertilizer is statistically significant at a significance level of 0.05.

Step 3: Examine factor effects graphically

Minitab provides main effects plots and interaction plots for each response characteristic that you choose. If you have a dynamic design, you can also choose to display a scatterplot of the responses versus the signal.
Note

Although you can use these plots to display the effects, be sure to evaluate the statistical significance of the effects in the analysis that fit the model. If the interaction effects are statistically significant in that analysis, you cannot interpret the main effects without considering the interaction effects.

Main effects plot

Main effects plots show how each factor affects the response characteristic (S/N ratio, means, slopes, standard deviations). A main effect exists when different levels of a factor affect the characteristic differently. For a factor with two levels, you may discover that one level increases the mean compared to the other level. This difference is a main effect.

Minitab creates the main effects plot by plotting the characteristic average for each factor level. These averages are the same as those displayed in the response table. A line connects the points for each factor. Look at the line to determine whether or not a main effect for a factor exists.
  • When the line is horizontal, then there is no main effect. Each level of the factor affects the characteristic in the same way and the characteristic average is the same across all factor levels.
  • When the line is not horizontal, then there is a main effect. Different levels of the factor affect the characteristic differently. The larger the difference in the vertical position of the plotted points (the more the line is not parallel to the X-axis), the larger the magnitude of the main effect.
By comparing the slopes of the lines, you can compare the relative magnitude of the factor effects.

In these results, the main effects plot for S/N ratio indicates that Fertilizer has the largest effect on the signal-to-noise ratio. On average, experimental runs with Fertilizer 2 had much higher signal-to-noise ratios than experimental runs with Fertilizer 1. Spraying had a small effect or no effect on the signal-to-noise ratio.

Interaction plot

The interaction plots show interactions between factors. Minitab creates an interaction plot by plotting for two factors the characteristic average for each combination of factor levels. So, for two factors with two levels each, Minitab plots four points that represent the four possible combinations. The levels of one factor are indicated on the horizontal axis, whereas the levels of the other factor are indicated by different colored lines and symbols.
  • If the lines are parallel to each other, then there is no interaction between the two factors.
  • If the lines are not parallel to each other, then there is an interaction between the two factors.

In these results, for the S/N ratios, the lines are close to parallel. Variety 2 has a higher S/N ratio than Variety 1 using both Fertilizer 1 and 2.

In addition to the interaction plots, examine the linear model analysis to determine whether the interaction is significant.

Scatterplot

The scatterplots for dynamic response experiments show the responses plotted against the signal. Each plot shows all the data for one setting of the control factors in the experiment. The following items are shown on the plot:
  • The least squares regression line through the reference point.
  • The row number at the top of each plot, which refers to the first row in which the factor settings for that plot appear.
  • The signal-to-noise ratio, slope, and standard deviation for the factor settings, which are located at the bottom of the plot.

The graphs are arranged in decreasing order of the signal-to-noise ratio, so that the experimental runs with the highest ratios are plotted first. If the experiment has more than nine combinations of factor settings, Minitab displays more than one graph window of scatterplots.

In this plot, there is a substantial difference in the spread of the data between the best and worst fits. For example, in the plot in the first cell, for row 21, the data are very close to the line. In the plot in the lower left corner, for row 9, the data vary much more widely. The standard deviation for row 21 is 0.4089, but is much larger in row 9. The standard deviation in row 9 is 1.1718.

Step 4: Determine whether your model meets the assumptions of the analysis

Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results.

In these results, the residual plots show that there is only one degree of freedom for error, and only two distinct values of the residuals. It is likely that the model is overfit, with too many terms. In such a case, consider reducing the model, and reexamining the residual plots.

Normal probability plot of the residuals

Use the normal probability plot of the residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.

The patterns in the following table may indicate that the model does not meet the model assumptions.

Pattern What the pattern may indicate
Not a straight line Nonnormality
A point that is far away from the line An outlier
Changing slope An unidentified variable

Residuals versus fits plot

The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
Fanning or uneven spreading of residuals across fitted values Nonconstant variance
Curvilinear A missing higher-order term
A point that is far away from zero An outlier
A point that is far away from the other points in the x-direction An influential point

Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

Histogram of residuals

Use the histogram of the residuals to determine whether the data are skewed or include outliers. The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
A long tail in one direction Skewness
A bar that is far away from the other bars An outlier

Because the appearance of a histogram depends on the number of intervals used to group the data, don't use a histogram to assess the normality of the residuals.

A histogram is most effective when you have approximately 20 or more observations. If the sample is too small, then each bar on the histogram does not contain enough observations to reliably show skewness or outliers.

Residuals versus order plot

Use the residuals versus order plot to verify the assumption that the residuals are independent. Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent. Ideally, the residuals on the plot should fall randomly around the center line:
If you see a pattern, investigate the cause. The following types of patterns may indicate that the residuals are dependent.
Trend
Shift
Cycle
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