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An article in the Journal of Radioanalytical and Nuclear Chemistry (2008, Vol. 276(2), pp. 323-328) presented a \(2^{8-4}\) fractional factorial design to identify sources of Pu contamination in the radioactivity material analysis of dried shellfish at the National Institute of Standards and Technology (NIST). The data are shown in the following table. No contamination occurred at runs \(1,4,\) and \(9 .\) The factors and levels are shown in the following table. (Table) $$ \begin{array}{ccccccccc} \hline & & & & & & & & & & \\ 2^{8-4} & \text { Glassware } & \text { Reagent } & \text { Sample Prep } & \text { Tracer } & \text { Dissolution } & \text { Hood } & \text { Chemistry } & \text { Ashing } & \text { mBq } \\ \hline \text { Run } & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & y \\ 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 \\ 2 & +1 & -1 & -1 & -1 & -1 & +1 & +1 & +1 & 3.31 \\ 3 & -1 & +1 & -1 & -1 & +1 & -1 & +1 & +1 & 0.0373 \\ 4 & +1 & +1 & -1 & -1 & +1 & +1 & -1 & -1 & 0 \\ 5 & -1 & -1 & +1 & -1 & +1 & +1 & +1 & -1 & 0.0649 \\ 6 & +1 & -1 & +1 & -1 & +1 & -1 & -1 & +1 & 0.133 \\ 7 & -1 & +1 & +1 & -1 & -1 & +1 & -1 & +1 & 0.0461 \\ 8 & +1 & +1 & +1 & -1 & -1 & -1 & +1 & -1 & 0.0297 \\ 9 & -1 & -1 & -1 & +1 & +1 & +1 & -1 & +1 & 0 \\ 10 & +1 & -1 & -1 & +1 & +1 & -1 & +1 & -1 & 0.287 \\ 11 & -1 & +1 & -1 & +1 & -1 & +1 & +1 & -1 & 0.133 \\ 12 & +1 & +1 & -1 & +1 & -1 & -1 & -1 & +1 & 0.0476 \\ 13 & -1 & -1 & +1 & +1 & -1 & -1 & +1 & +1 & 0.133 \\ 14 & +1 & -1 & +1 & +1 & -1 & +1 & -1 & -1 & 5.75 \\ 15 & -1 & +1 & +1 & +1 & +1 & -1 & -1 & -1 & 0.0153 \\ 16 & +1 & +1 & +1 & +1 & +1 & +1 & +1 & +1 & 2.47 \end{array} $$ (a) Write down the alias relationships. (b) Estimate the main effects. (c) Prepare a normal probability plot for the effects and interpret the results.

Step by step solution

01

Understanding Alias Structure

In a fractional factorial design, we assume some high-order interactions are negligible. The design is represented as a resolution IV, implying main effects are aliased with three-factor interactions, and two-factor interactions are aliased with one another. For a \(2^{8-4}\) design, each factor's effects are represented by a sign alias structure matrix, and this is constructed based on the generators. We need to identify the generators first to construct the confounding pattern.

02

Identify Generators

The given \(2^{8-4}\) design has 8 factors and 4 generators. Typical generators for creating such a design could be \(I = ABCD = ABEF = CEFG = DGH\). These generators indicate which interactions are confounded with each other.

03

Write Alias Relationships

Using the generators, express alias relationships. For example, if \(A\) is one of the factors, it would be aliased with \(BCD, BEF, EFG, GH\), etc. The full aliases for each factor can be detailed by substituting the generators into one another.

04

Calculation of Main Effects

Calculate the effect of each factor. The effect, \(E_i\), for a factor \(X_i\) is computed by taking the average of the response at high levels of \(X_i\) minus the average at low levels, i.e., \[ E_i = \frac{1}{2^k} \sum_{+} y - \frac{1}{2^k} \sum_{-} y \] where \(\sum_{+} y\) and \(\sum_{-} y\) represent the sum of responses at high and low levels of \(X_i\), respectively.

05

Construction of Normal Probability Plot

Compute standardized effects and plot them on a normal probability plot. This will help in determining which factors have statistically significant effects on the response. Often, effects that lie off the line of normality are considered significant.

06

Interpretation of Normal Plot

Analyze the normal probability plot. Significant main effects appear as points deviating sharply from the assumed line, indicating which factors or interactions notably affect the response.

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Key Concepts

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

Alias Relationships

In a fractional factorial design, alias relationships are an essential concept to grasp, especially when making decisions about what can be ignored in the analysis. In our context, a factorial design allows us to examine the effects of multiple factors by varying them simultaneously. However, conducting a full factorial design can be resource-intensive, thus a fractional approach is used to simplify this task.
By designating a subset of the total runs and assuming some higher-order interactions are negligible, we can streamline this process while still extracting valuable insights. The concept of resolution is introduced here, where we discuss a resolution IV design.

• Resolution IV means that main effects are aliased with three-factor interactions.
• Two-factor interactions are aliased with one another.

These aliasing patterns can be deduced based on the design's generators, such as in our given design where we have generators like \(I = ABCD = ABEF = CEFG = DGH\). Each factor is aliased with various higher-order interaction terms, meaning their individual effects can't be distinguished from these interactions using the available data. Identifying these generators helps in constructing the confounding pattern and understanding which interactions are mixed together.

Main Effects Estimation

Estimating main effects is crucial to identify which factors have the most significant impact on the response variable. In the fractional factorial design, the main effects are computed to quantify these impacts precisely.
The mathematical approach involves:

• Calculating the average response at the high levels of a factor.
• Calculating the average response at the low levels.

The difference between these averages gives the main effect, expressed as: \[ E_i = \frac{1}{2^k} \sum_{+} y - \frac{1}{2^k} \sum_{-} y \] where \( \sum_{+} y \) and \( \sum_{-} y \) are the sums of the measured responses at the high and low levels of factor \( i \). This calculation highlights the impact of each factor, aiding in recognizing which elements are more influential in causing changes in the response variable. This estimation is vital for optimizing processes, as it reveals factors that need adjustment for desired outcomes.

Normal Probability Plot

A normal probability plot is a graphical technique used to identify significant effects in a factorial design. Its purpose is to determine which factors have significant effects on the response variable.
The process includes plotting standardized effects against a normal distribution. Effects that deviate prominently from the drawn line of normality are marked as significant. Such deviations suggest that these factors have a statistical influence on the outcome.
To construct this plot:

• Compute the effects of each factor.
• Order these effects and calculate their cumulative probability.
• Plot them against the expected distribution under the null hypothesis of no effect.

This visually intuitive method helps in quickly identifying which factor effects warrant further investigation, making it a powerful tool in experimental design analysis.