91Ó°ÊÓ

A two-level factorial experiment in four factors was conducted by Chrysler and described in the article "Sheet Molded Compound Process Improvement" by P. I. Hsieh and D. E. Goodwin (Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, \(1986,\) pp. \(13-21\) ). The purpose was to reduce the number of defects in the finish of sheet-molded grill opening panels. A portion of the experimental design, and the resulting number of defects, \(y_{i}\) observed on each run is shown in the following table. This is a single replicate of the \(2^{4}\) design. (a) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors. (b) Fit an appropriate model using the factors identified in part (a). (c) Plot the residuals from this model versus the predicted number of defects. Also prepare a normal probability plot of the residuals. Comment on the adequacy of these plots. (d) The following table also shows the square root of the number of defects. Repeat parts (a) and (c) of the analysis using the square root of the number of defects as the response. Does this change the conclusions? $$ \begin{array}{ccccccc} \hline & \ {\text { Grill Defects Experiment }} \\ \ \text { Run } & \text { A } & \text { B } & \text { C } & \text { D } & \text { y } & \sqrt{y} \\ \hline 1 & \- & \- & \- & \- & 56 & 7.48 \\ 2 & \+ & \- & \- & \- & 17 & 4.12 \\ 3 & \- & \+ & \- & \- & 2 & 1.41 \\ 4 & \+ & \+ & \- & \- & 4 & 2.00 \\ 5 & \- & \- & \+ & \- & 3 & 1.73 \\ 6 & \+ & \- & \+ & \- & 4 & 2.00 \\ 7 & \- & \+ & \+ & \- & 50 & 7.07 \\ 8 & \+ & \+ & \+ & \- & 2 & 1.41 \\ 9 & \- & \- & \- & \+ & 1 & 1.00 \\ 10 & \+ & \- & \- & \+ & 0 & 0.00 \\ 11 & \- & \+ & \- & \+ & 3 & 1.73 \\ 12 & \+ & \+ & \- & \+ & 12 & 3.46 \\ 13 & \- & \- & \+ & \+ & 3 & 1.73 \\ 14 & \+ & \- & \+ & \+ & 4 & 2.00 \\ 15 & \- & \+ & \+ & \+ & 0 & 0.00 \\ 16 & \+ & \+ & \+ & \+ & 0 & 0.00 \end{array} $$

Step by step solution

01

Estimating Factor Effects

For a two-level factorial experiment, the effect of a factor is the difference in the average responses when the factor is at its higher level compared to when it is at its lower level. For each factor, calculate the average number of defects for the '+' level and the '-' level, then subtract the average of the '-' level from that of the '+' level.

02

Normal Probability Plot of Factor Effects

Plot the estimated effects from Step 1 on a normal probability plot. The effects that appear to deviate significantly from the line in the plot are considered important. Identify these factors.

03

Fitting the Model

Using the significant factors identified in Step 2, fit a linear model. Include only the main effects and interactions of these significant factors in the model.

04

Residual Analysis

Plot the residuals from the model against the predicted values to check for any patterns. Also, create a normal probability plot of the residuals to assess their normality. Comment on the adequacy by checking for randomness and close adherence to a normal distribution.

05

Square Root Transformation

Repeat Steps 1 and 4 using the square root of the number of defects as the response instead of the raw number of defects. This often stabilizes variance and makes the data more normally distributed.

06

Compare Results

Compare the important factors and residuals analysis from the original and transformed data. Determine if the conclusions regarding significant factors and model adequacy change with the transformation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factor Effects

In a factorial experiment, factor effects are central to understanding how different variables influence the outcome. For a two-level factorial design, such as the one described in the grill defects experiment, the effect of a factor is calculated as the difference in the average response when the factor is at its high level compared to its low level.

To estimate factor effects, you'll start by computing the mean number of defects when each factor is at its '+' level and its '-' level. This process is repeated for all factors. The calculation is straightforward but crucial:

• Find the average defect count at each factor level.
• Subtract the '-' level average from the '+' level average for each factor.

The resulting value represents the factor effect, indicating how much changing the factor level affects the number of defects. Positive effects suggest an increase in defects when the factor is at a '+' level, and negative effects suggest a decrease. Understanding which factors have the most significant effects helps focus improvements where they matter most.

Normal Probability Plot

The normal probability plot is a vital tool in analyzing factorial experiments. It helps in distinguishing which factor effects are significant. By plotting the estimated factor effects on the plot, you can visualize how far each effect deviates from what would be expected if they were simply random noise.

In an ideal scenario, if all effects were due to random variation, they would fall approximately along a straight line in this plot. However, deviations from this line indicate significant effects, which means that these factors likely influence the response variable. The farther an effect lies from this line, the more significant it is considered to be.

Using a normal probability plot is straightforward. Plot the effects from least to greatest along one axis against their expected normal scores on the other. The analysis of this plot helps you pinpoint which factors to include in your model-building step.

Residual Analysis

Residual analysis is a critical step after fitting a model to data. It helps determine the adequacy of the model by examining the differences between observed and predicted values. These differences, known as residuals, can reveal whether your model is capturing the data pattern accurately.

Start by plotting residuals against the predicted values. Look for patterns:

• Randomly scattered residuals indicate a good fit.
• Non-random patterns suggest model inadequacies or missing interactions.

Further, construct a normal probability plot of the residuals. In this plot, if residuals align closely with the line, they are normally distributed, suggesting the model is appropriate. Deviations from this line highlight the need for potential improvements, like transformations or additional factor inclusion.

Such thorough scrutiny of residuals ensures that the fitted model is robust and reliable.

Square Root Transformation

Sometimes, raw data may not meet the assumptions required for standard analysis methods. A square root transformation is a technique applied to stabilize variance and achieve normality in the data. By taking the square root of each response variable (in this case, the number of defects), you often make the data more symmetric and closer to a normal distribution.

In the context of the exercise, repeating the analysis steps using the square root of defects ensures that model assumptions are met, thus potentially improving the reliability of results.

• Apply this transformation to the response variable.
• Reassess factor effects and perform residual analysis with the transformed data.

Compare findings from both the original and transformed data. Often, factor effects that were unclear in the original might become more pronounced after transformation, thereby refining which factors are truly significant.

Ultimately, using square root transformation can help reinforce your conclusions, making them more trustworthy.