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Construct a \(2^{5-1}\) design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?

Short Answer

Expert verified
Two blocks are created, and main effects and two-factor interactions are checked for confounding within the block design.

Step by step solution

01

Understand the Problem

We need to construct a fractional factorial design for a 5-factor experiment. The design should allow us to monitor 16 experimental runs divided into two blocks of 8 observations each.
02

Know the Basics of Fractional Factorial Design

A \(2^{5-1}\ (or 2^4) Fractional Factorial Design\) suggests we're conducting an experiment with 5 factors at two levels each but using half the complete number of runs (i.e., 16 instead of 32). This halves the number of experimental runs, enhancing efficiency.
03

Select Generators to Confound with Blocks

Choose the generators wisely to define the half the design. We'll use the generator \(E = ABCD\), which means in this experimental setup, 'E' is equivalent to the product of the four other factors.
04

Determine Blocks

Blocking is useful to separate out variability in experimental runs. Typically, you can choose one interaction to be confounded with a block. In our case, we choose the interaction \(ABCD\) to confound with blocks, creating two blocks with contrasting high and low levels.
05

Create the Design Table

Use the standard logic by defining levels (+ for high and - for low) for each factor A, B, C, and D, then derive the level of E using \(E = ABCD\). Split your table into two blocks based on the chosen confounded interaction, here indicated by the pattern of the {"+" and "-"} signs in the column labeled \(ABCD\).
06

Analyze for Confounding

Given our generator \(E = ABCD\), check all 2-factor interactions to see if any are aliased/confounded with blocks. Main effects and two-factor interactions are alias with higher-order interactions, typically beyond the primary concern of main and two-factor effects.

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Key Concepts

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

Confounding in Experiments
When conducting experiments, especially those involving multiple factors, confounding can become an unavoidable issue. Confounding occurs when the effects of two or more factors are mixed, making it difficult to distinguish their individual impacts on the outcome. This can skew results if not properly managed.

To handle confounding, a common technique is **fractional factorial design**, which allows researchers to estimate the effects of various factors without conducting every possible combination of experiments. This design intelligently combines factors to reduce the number of runs needed. However, one downside is that it naturally introduces confounding as some effects are deliberately "mixed" with others to achieve fewer runs.

In the exercise with a \(2^{5-1}\) design, for instance, we limit the runs from 32 to 16 by confounding a main factor with interactions. Here, the factor "E" is intentionally confounded with the interaction "ABCD." This means any effect seen as a result of E could also be because of ABCD. Such strategic confounding should be planned so that potentially less crucial interactions are the ones mixed together. This way, researchers can focus on significant findings among main effects and crucial interactions, while accepting less clarity in less important areas.
Blocking in Design of Experiments
Blocking is an essential concept in experimental designs, particularly when external factors could influence the outcome of the experiment. This technique groups sets of experimental runs into blocks where conditions are more controlled, thus minimizing variance that could originate from outside influences.

In the context of our fractional factorial design, blocking was done by confounding certain interactions with blocks. We used the interaction pattern ABCD to decide on two distinct blocks of 8 observations each. During blocking, you aim to introduce homogeneity within each block, shifting major sources of variability between blocks. This provides a more precise estimate of the effects under study by controlling some experimental errors.

When setting up blocks, it's crucial to determine which interactions can be confounded with blocks. Typically, interactions of higher orders, which are often less crucial, are chosen to be confounded. This not only simplifies the analysis but also ensures that only the most critical information is unaffected by the potential "noise" introduced by confounding with blocks.
Experimental Design
Experimental design is all about planning an investigation to effectively explore relationships between variables with efficiency and accuracy. Consider fractional factorial designs, like the \(2^{5-1}\) design in our exercise. Such designs aim to understand multiple factors' effects simultaneously, cutting down the number of trials needed to get meaningful data.

The central idea is to have a controlled way to vary factors. Lower half of experiments (half of all possible combinations) are conducted so researchers can get an idea of the major effects and interactions without exhaustive testing. It's a balance between scope and practicality. By utilizing interactions judiciously as confounding variables, we get this efficient snapshot while controlling complexity.

A good experimental design, including good blocking and confounding practices, seeks to maximize the information gained from any given amount of resources. This helps in drawing insightful conclusions about how factors influence outcomes and in identifying key drivers. The \(2^{5-1}\) design and its 16 runs offer an excellent example of how experimental design remains a key tool in tackling complex problems with a simpler approach.

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Most popular questions from this chapter

Carbon anodes used in a smelting process are baked in a ring fumace. An experiment is run in the fumace to determine which factors influence the weight of packing material that is stuck to the anodes after baking. Six variables are of interest, each at two levels: \(A=\) pitch/fines ratio \((0.45,0.55), B=\) packing material type \((1,2), C=\) packing material temperature (ambient, \(\left.325^{\circ} \mathrm{C}\right), D=\) flue location (inside, outside), \(E=\) pit temperature (ambient, \(195^{\circ} \mathrm{C}\) ), and \(F=\) delay time before packing (zero, 24 hours). A \(2^{n-3}\) design is run, and three replicates are obtained at each of the design points. The weight of packing material stuck to the anodes is measured in grams. The data in run order are as follows: \(a b d=(984,826,936) ;\) abedef \(=(1275,976,1457) ; b e=(1217,1201,890) ; a f=(1474,\), \(1164,1541) ;\) def \(=(1320,1156,913) ; c d=(765,705,821) ;\) ace \(=(1338,1254,1294) ;\) and \(b c f=(1325,1299,1253)\). We wish to minimize the amount of stuck packing material. (a) Verify that the eight runs correspond to a \(2^{6-3}\) design. What is the alias structure? (b) Use the average weight as a response. What factors appear to be influential? (c) Use the range of the weights as a response. What factors appear to be influential? (d) What recommendations would you make to the process engineers?

A spin coater is used to apply photoresist to a bare silicon wafer. This operation usually occurs early in the semiconductor manufacturing process, and the average coating thickness and the variability in the coating thickness has an important impact on downstream manufacturing steps. Six variables are used in the experiment. The variables and their high and low levels are as follows: $$ \begin{array}{lll} \hline \multicolumn{1}{c}{\text { Factor }} & \text { Low Level } & \text { High Level } \\ \hline \text { Final spin speed } & 7350 \mathrm{rpm} & 6650 \mathrm{rpm} \\ \text { Acceleration rate } & 5 & 20 \\ \text { Volume of resist applied } & 3 \mathrm{cc} & 5 \mathrm{cc} \\ \text { Time of spin } & 14 \mathrm{~s} & 6 \mathrm{~s} \\ \text { Resist batch variation } & \text { Batch 1 } & \text { Batch 2 } \\ \text { Exhaust pressure } & \text { Cover off } & \text { Cover on } \\ \hline \end{array} $$ The experimenter decides to use a \(2^{6-1}\) design and to make three readings on resist thickness on each test wafer. The data are shown in Table 8-30. (a) Verify that this is a \(2^{6-1}\) design. Discuss the alias relationships in this design. (b) What factors appear to affect average resist thickness? (c) Because the volume of resist applied has little effect on average thickness, does this have any important practical implications for the process engineers? (d) Project this design into a smaller design involving only the significant factors. Graphically display the results. Does this aid in interpretation? (e) Use the range of resist thickness as a response variable. Is there any indication that any of these factors affect the variability in resist thickness? (f) Where would you recommend that the process engineers run the process?

Construct a \(2^{7-2}\) design. Show how the design may be run in four blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?

An article in Industrial and Engineering Chemistry ("More on Planning Experiments to Increase Research Efficiency," 1970, pp. \(60-65)\) uses a \(2^{3-2}\) design to investigate the effect of \(A=\) condensation temperature, \(B=\) amount of material \(1, C=\) solvent volume, \(D=\) condensation time, and \(E=\) amount of material 2 on yield. The results obtained are as follows: $$ \begin{array}{rlrlr} e=23.2 & a d=16.9 & c d=23.8 & b d e=16.8 \\ a b=15.5 & b c=16.2 & a c e=23.4 & \text { abcde }=18.1 \end{array} $$ (a) Verify that the design generators used were \(I=A C E\) and \(I=B D E\). (b) Write down the complete defining relation and the aliases for this design. (c) Estimate the main effects. (d) Prepare an analysis of variance table. Verify that the \(A B\) and \(A D\) interactions are available to use as error. (e) Plot the residuals versus the fitted values. Also construct a normal probability plot of the residuals. Comment on the results.

A 16 -run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced. The six variables and their levels are shown below: Each run was replicated four times, and a camber measurement was taken on the substrate. The data are shown below: (a) What type of design did the experimenters use? (b) What are the alias relationships in this design? (c) Do any of the process variables affect average camber? (d) Do any of the process variables affect the variability in camber measurements? (e) If it is important to reduce camber as much as possible, what recommendations would you make?

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