91Ó°ÊÓ

An article in Quality Engineering ["Mean and Variance Modeling with Qualitative Responses: A Case Study" (1998-1999, Vol. 11, pp. 141-148)] studied how three active ingredients of a particular food affect the overall taste of the product. The measure of the overall taste is the overall mean liking score (MLS). The three ingredients are identified by the variables \(x_{1}, x_{2}\), and \(x_{3}\). The data are shown in the following table. $$ \begin{array}{crrrr} \hline \text { Run } & x_{1} & x_{2} & x_{3} & \text { MLS } \\ \hline 1 & 1 & 1 & -1 & 6.3261 \\ 2 & 1 & 1 & 1 & 6.2444 \\ 3 & 0 & 0 & 0 & 6.5909 \\ 4 & 0 & -1 & 0 & 6.3409 \\ 5 & 1 & -1 & 1 & 5.907 \\ 6 & 1 & -1 & -1 & 6.488 \\ 7 & 0 & 0 & -1 & 5.9773 \\ 8 & 0 & 1 & 0 & 6.8605 \\ 9 & -1 & -1 & 1 & 6.0455 \\ 10 & 0 & 0 & 1 & 6.3478 \\ 11 & 1 & 0 & 0 & 6.7609 \\ 12 & -1 & -1 & -1 & 5.7727 \\ 13 & -1 & 1 & -1 & 6.1805 \\ 14 & -1 & 1 & 1 & 6.4894 \\ 15 & -1 & 0 & 0 & 6.8182 \end{array} $$ (a) Fit a second-order response surface model to the data. (b) Construct contour plots and response surface plots for MLS. What are your conclusions? (c) Analyze the residuals from this experiment. Does your analysis indicate any potential problems? (d) This design has only a single center point. Is this a good design in your opinion?

Step by step solution

01

Understand the Response Surface Model

The response surface model is a mathematical model used to represent the relationship between multiple independent variables and one or more dependent variables. For this problem, the response (dependent variable) is the MLS, and the independent variables are \(x_1, x_2, x_3\). We aim to develop a second-order polynomial equation that represents this relationship.

02

Set up the Second-Order Model Equation

The general form for a second-order response surface model is:\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_{11} x_1^2 + \beta_{22} x_2^2 + \beta_{33} x_3^2 + \beta_{12} x_1 x_2 + \beta_{13} x_1 x_3 + \beta_{23} x_2 x_3\]where \(y\) is the MLS, and the \(\beta\) terms are parameters to be estimated using regression.

03

Fit the Model Using Regression

Utilize software such as R, Python, or Minitab to perform regression analysis on the given data to estimate the coefficients \(\beta\). Input the data for \(x_1, x_2, x_3\), and the MLS values to fit the second-order model and extract the coefficients.

04

Create Contour and Surface Plots

Use the estimated model to create contour and response surface plots for MLS by varying \(x_1, x_2,\) and \(x_3\). These plots help visualize how changes in the ingredients influence the MLS, highlighting the interaction effects and optimal conditions.

05

Analyze Residuals

Examine the residuals to identify any patterns or deviations from normality. Residuals should ideally be randomly distributed and exhibit homoscedasticity (constant variance). Plot residuals against predicted values and look for any obvious trends or outliers.

06

Evaluate the Design

Discuss the design's strength considering the use of a single center point. A single center point limits the ability to accurately capture curvature in the response surface, potentially affecting the model's fit. More center points provide better estimation of the error and model fit assessment.

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Key Concepts

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

Second-Order Polynomial Equation

In the world of response surface methodology, a second-order polynomial equation is a powerful tool. This equation effectively models the relationship between several independent variables and one dependent variable. Here, the dependent variable is the Mean Liking Score (MLS) while the independent variables are the ingredients represented by \(x_1, x_2,\) and \(x_3\).

The general form of a second-order polynomial equation includes both linear and quadratic terms of the independent variables, along with their interactions. It can be represented as:

• \( y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \)
• \(+ \beta_{11} x_1^2 + \beta_{22} x_2^2 + \beta_{33} x_3^2 \)
• \(+ \beta_{12} x_1 x_2 + \beta_{13} x_1 x_3 + \beta_{23} x_2 x_3 \)

This model is particularly useful because it captures not just the direct effects of each variable but also how they interact with each other. This interaction could reveal valuable insights about optimizing the taste of the food product by varying ingredient combinations.

Regression Analysis

Regression analysis is beautifully versatile and crucial in estimating the parameters of the second-order polynomial equation mentioned above. By fitting a model to the observed data, we can understand how changes in the independent variables affect the dependent variable, MLS, in this case.

Using software tools like R or Python, we can input our experiment data to determine the \( \beta \) coefficients. These coefficients will guide us in predicting the MLS based on different levels of \( x_1, x_2, \text{and}~x_3 \).

• By analyzing these coefficients, you can assess which ingredient has the most significant impact on taste.
• More importantly, regression allows us to test the statistical significance of these variables and their interactions.

Moreover, understanding the residuals, or differences between observed and predicted values, through regression analysis helps in diagnosing model accuracy and fit.

Contour Plots

Contour plots are a visual treat used in response surface methodology to map the relationships modeled by the second-order polynomial equation. These plots can dramatically enhance your grasp of how ingredient changes affect the MLS by demonstrating regions of response surfaces on a two-dimensional level.

In contour plots:

• Each line represents points where the response (MLS) is constant, giving a clear visual of how variables interact.
• They help pinpoint combinations of ingredients that yield optimal or suboptimal MLS values.

These plots serve as a straightforward method to assess the interaction between two ingredients while keeping the third constant. This way, contour plots provide a pathway to finding the combination of ingredient levels that maximize or minimize our response of interest, such as the MLS.

Residual Analysis

Residual analysis is a vital part of assessing the accuracy and reliability of the regression model. A residual, quite simply, is the difference between an observed value and the value predicted by the model.

Key steps in conducting a residual analysis include:

• Plotting residuals against predicted values to check randomness. Non-random patterns could indicate a poorly fitting model.
• Examining the residuals' distribution, ensuring they are normally distributed without significant deviations.
• Assessing homoscedasticity, meaning residuals should display constant variance across levels of the predicted values.

Through residual analysis, it's possible to detect outliers or influential data points that might skew results. This step further validates the robustness of your regression model and provides confidence in the conclusions drawn from your experiment.