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Chapter 14: Problem 103

Consider a \(2^{2}\) design in two blocks with \(A B\) confounded with blocks. Prove algebraically that \(S S_{\mathrm{AB}}=S S_{\text {Blocks }}\).

Short Answer

Expert verified
When AB is confounded with blocks, SS_AB equals SS_Blocks due to identical calculations.

Step by step solution

01

Understand the Experimental Design

We are dealing with a \(2^2\) factorial design, which means there are four treatment combinations: (1), A, B, and AB. The design is divided into two blocks, and the interaction \(AB\) is confounded with these blocks. This means that the effect of \(AB\) is completely blended with block effects.
02

Set Up the Linear Model

The general linear model for a \(2^2\) factorial design is \(Y_{ijk} = \mu + \tau_i + \beta_j + (\tau\beta)_{ij} + \epsilon_{ijk}\), where \(\mu\) is the overall mean, \(\tau_i\) are treatment effects, \(\beta_j\) are block effects, and \((\tau\beta)_{ij}\) are the interactions. When \(AB\) is confounded with blocks, \(\tau_B\) and \(\beta_{block}\) are equivalent.
03

Calculate Sum of Squares for Blocks

For blocks, the sum of squares \(SS_{Blocks}\) is calculated using the formula \(SS_{Blocks} = \frac{n}{k} \sum(\bar{Y}_{B.}^2) - (TotalEffectiveMean)^2\), where \(n\) is the number of observations per block and \(k\) is the number of blocks.
04

Calculate Sum of Squares for AB

The sum of squares for the interaction \(AB\) is found similarly to that of blocks, due to confounding. It's given by the formula \(SS_{AB} = \frac{n}{k} \sum(\bar{Y}_{AB}^2) - (TotalEffectiveMean)^2\).
05

Show Sum of Squares Equality

Since \(AB\) is confounded with blocks, the observed means for blocks are the same as those for the interaction \(AB\). Therefore, the calculations for \(SS_{AB}\) and \(SS_{Blocks}\) are based on identical data, leading to \(SS_{AB} = SS_{Blocks}\). Thus, we have proved algebraically that \(SS_{AB} = SS_{Blocks}\).

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Key Concepts

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

2^2 Factorial Design
A \(2^2\) factorial design is a simple and powerful experimental framework utilized when there are two factors, each at two levels, creating four possible combinations. These combinations are represented as (1), A, B, and AB. Such a design allows researchers to concurrently study the individual effects and the interaction effects of these factors.
\[\text{Four Combinations:} \quad (1), \ A, \ B, \ ext{and} \ AB\]
The magic of this design is its efficiency; with only four treatment combinations, it gives insights into both main effects and the interaction between factors.
This design is perfect for preliminary investigations in experiments. It cleverly reduces the number of trials while still offering rich data for analysis.
Examples of application span multiple areas, from agriculture, where seed types and fertilizers might be tested, to industry, where machine settings and raw material types are varied.
Confounding in Experimental Design
In experimental design, confounding is a situation where the effect of one variable cannot be distinguished from the effect of another. This often happens when two or more variables are correlated or mixed up.
In the \(2^2\) factorial design scenario provided, the \(AB\) interaction is confounded with the blocks used in the experiment.
  • \(AB\)'s effect blends with the block effects, making it difficult to separate them.
  • This allows us to focus on the primary effects instead of both interaction and block effects.
When confounding, it's essential to remember that the primary aim is not to measure the confounded interaction. Instead, it helps in simplifying complex experiments by balancing design and interpretations.
It is crucial to clearly understand the arrangement of experiments and the potential for confounding to ensure accurate data interpretation and analysis.
Sum of Squares Analysis
Sum of Squares (SS) analysis is a statistical technique used to assess variability in data. It measures the total variance and separates it into various components for different factors.
For the exercise at hand, we focus on the Sum of Squares for blocks \(SS_{Blocks}\) and the interaction \(SS_{AB}\).
  • \(SS_{Blocks}\) calculates how much of the total variability is due to differences between blocks.
  • \(SS_{AB}\) focuses on how much variability is attributable to the interaction.
Both these metrics use a similar structure. They rely on similar data, especially when the interaction is confounded with blocks, making their calculations and results identical.
Sum of Squares is particularly useful in analyzing how different factors contribute to the outcome, enhancing understanding of experimental variability, and determining significance in factorial design studies.

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Most popular questions from this chapter

An experiment to study the effect of machining factors on ceramic strength was described at www.itl.nist.gov/div898/ handbook/. Five factors were considered at two levels each: \(A=\) Table Speed, \(B=\) Down Feed Rate, \(C=\) Wheel Grit, \(D=\) Direction, \(E=\) Batch. The response is the average of the ceramic strength over 15 repetitions. The following data are from a single replicate of a \(2^{5}\) factorial design. \(\begin{array}{rrrrrr}\text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Strength } \\ -1 & -1 & -1 & -1 & -1 & 680.45 \\ 1 & -1 & -1 & -1 & -1 & 722.48 \\ -1 & 1 & -1 & -1 & -1 & 702.14 \\ 1 & 1 & -1 & -1 & -1 & 666.93 \\ -1 & -1 & 1 & -1 & -1 & 703.67 \\ 1 & -1 & 1 & -1 & -1 & 642.14 \\ -1 & 1 & 1 & -1 & -1 & 692.98 \\ 1 & 1 & 1 & -1 & -1 & 669.26 \\\ -1 & -1 & -1 & 1 & -1 & 491.58 \\ 1 & -1 & -1 & 1 & -1 & 475.52 \\ -1 & 1 & -1 & 1 & -1 & 478.76 \\ 1 & 1 & -1 & 1 & -1 & 568.23 \\ -1 & -1 & 1 & 1 & -1 & 444.72\end{array}\) \(\begin{array}{rrrrrr}1 & -1 & 1 & 1 & -1 & 410.37 \\ -1 & 1 & 1 & 1 & -1 & 428.51 \\ 1 & 1 & 1 & 1 & -1 & 491.47 \\ -1 & -1 & -1 & -1 & 1 & 607.34 \\\ 1 & -1 & -1 & -1 & 1 & 620.8 \\ -1 & 1 & -1 & -1 & 1 & 610.55 \\ 1 & 1 & -1 & -1 & 1 & 638.04 \\ -1 & -1 & 1 & -1 & 1 & 585.19 \\ 1 & -1 & 1 & -1 & 1 & 586.17 \\ -1 & 1 & 1 & -1 & 1 & 601.67 \\ 1 & 1 & 1 & -1 & 1 & 608.31 \\ -1 & -1 & -1 & 1 & 1 & 442.9 \\ 1 & -1 & -1 & 1 & 1 & 434.41 \\ -1 & 1 & -1 & 1 & 1 & 417.66 \\ 1 & 1 & -1 & 1 & 1 & 510.84 \\ -1 & -1 & 1 & 1 & 1 & 392.11 \\\ 1 & -1 & 1 & 1 & 1 & 343.22 \\ -1 & 1 & 1 & 1 & 1 & 385.52 \\ 1 & 1 & 1 & 1 & 1 & 446.73\end{array}\) (a) Estimate the factor effects and use a normal probability plot of the effects. Identify which effects appear to be large. (b) Fit an appropriate model using the factors identified in part (a). (c) Prepare a normal probability plot of the residuals. Also, plot the residuals versus the predicted ceramic strength. Comment on the adequacy of these plots. (d) Identify and interpret any significant interactions. (e) What are your recommendations regarding process operating conditions?

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.

Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, John Wiley, 1977\()\) described an experiment conducted to investigate warping of copper plates. The two factors studied were temperature and the copper content of the plates. The response variable is the amount of warping. The data are as follows: $$ \begin{array}{clccc} \hline {\begin{array}{c} \text { Temperature } \\ \left({ }^{\circ} \mathbf{C}\right) \end{array}} && {\text { Copper } \text { Content }(\%)} \\ \ \mathbf{4 0} & \mathbf{6 0} & \mathbf{8 0} & \mathbf{1 0 0} \\ 50 & 17,20 & 16,21 & 24,22 & 28,27 \\ 75 & 12,9 & 18,13 & 17,12 & 27,31 \\ 100 & 16,12 & 18,21 & 25,23 & 30,23 \\ 125 & 21,17 & 23,21 & 23,22 & 29,31 \end{array} $$ (a) Is there any indication that either factor affects the amount of warping? Is there any interaction between the factors? Use \(\alpha=0.05\) (b) Analyze the residuals from this experiment. (c) Plot the average warping at each level of copper content and compare the levels using Fisher's LSD method. Describe the differences in the effects of the different levels of copper content on warping. If low warping is desirable, what level of copper content would you specify? (d) Suppose that temperature cannot be easily controlled in the environment in which the copper plates are to be used. Does this change your answer for part (c)?

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?

An article in the Journal of Applied Electrochemistry (May 2008, Vol. \(38(5),\) pp. \(583-590\) ) presented a \(2^{7-3}\) fractional factorial design to perform optimization of polybenzimidazolebased membrane electrode assemblies for \(\mathrm{H}_{2} / \mathrm{O}_{2}\) fuel cells. The design and data are shown in the following table. $$ \begin{array}{ccccccccc} \text { Runs } & A & B & C & D & E & F & G & \begin{array}{l} \text { Current Density } \\ \left(\mathbf{C D} \mathbf{~ m A} \mathbf{c m}^{2}\right) \end{array} \\ \hline 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 160 \\ 2 & +1 & -1 & -1 & -1 & +1 & +1 & +1 & 20 \\ 3 & -1 & +1 & -1 & -1 & +1 & +1 & -1 & 80 \\ 4 & +1 & +1 & -1 & -1 & -1 & -1 & +1 & 317 \\ 5 & -1 & -1 & +1 & -1 & +1 & -1 & +1 & 19 \\ 6 & +1 & -1 & +1 & -1 & -1 & +1 & -1 & 4 \\ 7 & -1 & +1 & +1 & -1 & -1 & +1 & +1 & 20 \\ 8 & +1 & +1 & +1 & -1 & +1 & -1 & -1 & 88 \end{array} $$ $$ \begin{array}{rrrrrrrrr} \hline \text { Runs } & A & B & C & D & E & F & G & \begin{array}{l} \text { Current Density } \\ \left(\mathbf{C D} \mathrm{mA} \mathrm{cm}^{2}\right) \end{array} \\ \hline 9 & -1 & -1 & -1 & +1 & -1 & +1 & +1 & 1100 \\ 10 & +1 & -1 & -1 & +1 & +1 & -1 & -1 & 12 \\ 11 & -1 & +1 & -1 & +1 & +1 & -1 & +1 & 552 \\ 12 & +1 & +1 & -1 & +1 & -1 & +1 & -1 & 880 \\ 13 & -1 & -1 & +1 & +1 & +1 & +1 & -1 & 16 \\ 14 & +1 & -1 & +1 & +1 & -1 & -1 & +1 & 20 \\ 15 & -1 & +1 & +1 & +1 & -1 & -1 & -1 & 8 \\ 16 & +1 & +1 & +1 & +1 & +1 & +1 & +1 & 15 \end{array} $$ The factors and levels are shown in the following table. $$ \begin{array}{llll} \hline & \text { Factor } & \ {-\mathbf{1}} & \ {+\mathbf{1}} \\ \hline \text { A } & \begin{array}{l} \text { Amount of binder in the } \\ \text { catalyst layer } \end{array} & 0.2 \mathrm{mg} \mathrm{cm}^{2} & 1 \mathrm{mg} \mathrm{cm}^{2} \\\ \text { B } & \text { Electrocatalyst loading } & 0.1 \mathrm{mg} \mathrm{cm}^{2} & 1 \mathrm{mg} \mathrm{cm}^{2} \\ \text { C } & \begin{array}{l} \text { Amount of carbon in the } \\ \text { gas diffusion layer } \end{array} & 2 \mathrm{mg} \mathrm{cm}^{2} & 4.5 \mathrm{mg} \mathrm{cm}^{2} \\\ \text { D } & \text { Hot compaction time } & 1 \mathrm{~min} & 10 \mathrm{~min} \\ \mathrm{E} & \text { Compaction temperature } & 100^{\circ} \mathrm{C} & 150^{\circ} \mathrm{C} \\ \mathrm{F} & \text { Hot compaction load } & 0.04 \text { ton } \mathrm{cm}^{2} & 0.2 \text { ton } \mathrm{cm}^{2} \\ \mathrm{G} & \text { Amount of PTFE in the } & 0.1 \mathrm{mg} \mathrm{cm}^{2} & 1 \mathrm{mg} \mathrm{cm}^{2} \\ & \text { gas diffusion layer } & & \end{array} $$ (a) Write down the alias relationships. (b) Estimate the main effects. (c) Prepare a normal probability plot for the effects and interpret the results. (d) Calculate the sum of squares for the alias set that contains the ABG interaction from the corresponding effect estimate
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