The document discusses the 2k factorial design, which is a special case of the general factorial design with k factors at two levels. It provides examples of using 2k factorial designs to investigate how multiple factors affect a response. For an unreplicated 2k design with no replication, there are challenges in statistical testing due to having zero degrees of freedom for error. Various methods are discussed for analyzing the effects in an unreplicated 2k design, such as normal probability plotting, Lenth's method, and conditional inference charts. Transformation of the response may also be needed to meet assumptions of the model such as equal variance.
S1 - Process product optimization using design experiments and response surfa...CAChemE
An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system.
Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.”
Quite often in experimental work, many situations arise where some observations are lost or become
unavailable due to some accidents or cost constraints. When there are missing observations, some
desirable design properties like orthogonality,rotatability and optimality can be adversely affected. Some
attention has been given, in literature, to investigating the prediction capability of response surface
designs; however, little or no effort has been devoted to investigating same for such designs when some
observations are missing. This work therefore investigates the impact of a single missing observation of the
various design points: factorial, axial and center points, on the estimation and predictive capability of
Central Composite Designs (CCDs). It was observed that for each of the designs considered, precision of
model parameter estimates and the design prediction properties were adversely affected by the missing
observations and that the largest loss in precision of parameters corresponds to a missing factorial point.
S1 - Process product optimization using design experiments and response surfa...CAChemE
An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system.
Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.”
Quite often in experimental work, many situations arise where some observations are lost or become
unavailable due to some accidents or cost constraints. When there are missing observations, some
desirable design properties like orthogonality,rotatability and optimality can be adversely affected. Some
attention has been given, in literature, to investigating the prediction capability of response surface
designs; however, little or no effort has been devoted to investigating same for such designs when some
observations are missing. This work therefore investigates the impact of a single missing observation of the
various design points: factorial, axial and center points, on the estimation and predictive capability of
Central Composite Designs (CCDs). It was observed that for each of the designs considered, precision of
model parameter estimates and the design prediction properties were adversely affected by the missing
observations and that the largest loss in precision of parameters corresponds to a missing factorial point.
Introduction & Basics of DoE
Terminologies
Key steps in DOE
Softwares used for DOE
Factorial Designs ( Full and Fractional)
Mixture Designs
Response Surface Methodology
Central Composite Design
Box -Behnken Design
Conclusion
References
Comparative Efficiency of Stepped-Wedge Designs
Alan Girling
Society for Clinical Trials Conference; Arlington, Virginia
May 17th 2015
This presentation was part of the workshop organised by Karla Hemming: Research and reporting methods for the stepped wedge cluster randomised controlled trial
Introduction to Design of Experiments by Teck Nam Ang (University of Malaya)Teck Nam Ang
This set of slides explains in a simple manner the purpose of experiment, various strategies of experiment, how to plan and design experiment, and the handling of experimental data.
Adaptive response surface by kriging using pilot points for structural reliab...IOSR Journals
Structural reliability analysis aims to compute the probability of failure by considering system uncertainties. However, this approach may require very time-consuming computation and becomes impracticable for complex structures especially when complex computer analysis and simulation codes are involved such as finite element method. Approximation methods are widely used to build simplified approximations, or metamodels providing a surrogate model of the original codes. The most popular surrogate model is the response surface methodology, which typically employs second order polynomial approximation using least-squares regression techniques. Several authors have been used response surface methods in reliability analysis. However, another approximation method based on kriging approach has successfully applied in the field of deterministic optimization. Few studies have treated the use of kriging approximation in reliability analysis and reliability-based design optimization. In this paper, the kriging approximation is used an alternative to the traditional response surface method, to approximate the performance function of the reliability analysis. The main objective of this work is to develop an efficient global approximation while controlling the computational cost and accurate prediction. A pilot point method is proposed to the kriging approximation in order to increase the prior predictivity of the approximation, which the pilot points are good candidates for numerical simulation. In other words, the predictive quality of the initial kriging approximation is improved by adding adaptive information called “pilot points” in areas where the kriging variance is maximum. This methodology allows for an efficient modeling of highly non-linear responses, while the number of simulations is reduced compared to Latin Hypercubes approach. Numerical examples show the efficiency and the interest of the proposed method.
An introduction to SigmaXL's Design of Experiments tools.
Established in 1998, SigmaXL Inc. is a leading provider of user friendly Excel Add-ins for Lean Six Sigma graphical and statistical tools and Monte Carlo simulation.
SigmaXL® customers include market leaders like Agilent, Diebold, FedEx, Microsoft, Motorola and Shell. SigmaXL® software is also used by numerous colleges, universities and government agencies.
Our flagship product, SigmaXL®, was designed from the ground up to be a cost-effective, powerful, but easy to use tool that enables users to measure, analyze, improve and control their service, transactional, and manufacturing processes. As an add-in to the already familiar Microsoft Excel, SigmaXL® is ideal for Lean Six Sigma training and application, or use in a college statistics course.
DiscoverSim™ enables you to quantify your risk through Monte Carlo simulation and minimize your risk with global optimization. Business decisions are often based on assumptions with a single point value estimate or an average, resulting in unexpected outcomes.
DiscoverSim™ allows you to model the uncertainty in your inputs so that you know what to expect in your outputs.
Introduction & Basics of DoE
Terminologies
Key steps in DOE
Softwares used for DOE
Factorial Designs ( Full and Fractional)
Mixture Designs
Response Surface Methodology
Central Composite Design
Box -Behnken Design
Conclusion
References
Comparative Efficiency of Stepped-Wedge Designs
Alan Girling
Society for Clinical Trials Conference; Arlington, Virginia
May 17th 2015
This presentation was part of the workshop organised by Karla Hemming: Research and reporting methods for the stepped wedge cluster randomised controlled trial
Introduction to Design of Experiments by Teck Nam Ang (University of Malaya)Teck Nam Ang
This set of slides explains in a simple manner the purpose of experiment, various strategies of experiment, how to plan and design experiment, and the handling of experimental data.
Adaptive response surface by kriging using pilot points for structural reliab...IOSR Journals
Structural reliability analysis aims to compute the probability of failure by considering system uncertainties. However, this approach may require very time-consuming computation and becomes impracticable for complex structures especially when complex computer analysis and simulation codes are involved such as finite element method. Approximation methods are widely used to build simplified approximations, or metamodels providing a surrogate model of the original codes. The most popular surrogate model is the response surface methodology, which typically employs second order polynomial approximation using least-squares regression techniques. Several authors have been used response surface methods in reliability analysis. However, another approximation method based on kriging approach has successfully applied in the field of deterministic optimization. Few studies have treated the use of kriging approximation in reliability analysis and reliability-based design optimization. In this paper, the kriging approximation is used an alternative to the traditional response surface method, to approximate the performance function of the reliability analysis. The main objective of this work is to develop an efficient global approximation while controlling the computational cost and accurate prediction. A pilot point method is proposed to the kriging approximation in order to increase the prior predictivity of the approximation, which the pilot points are good candidates for numerical simulation. In other words, the predictive quality of the initial kriging approximation is improved by adding adaptive information called “pilot points” in areas where the kriging variance is maximum. This methodology allows for an efficient modeling of highly non-linear responses, while the number of simulations is reduced compared to Latin Hypercubes approach. Numerical examples show the efficiency and the interest of the proposed method.
An introduction to SigmaXL's Design of Experiments tools.
Established in 1998, SigmaXL Inc. is a leading provider of user friendly Excel Add-ins for Lean Six Sigma graphical and statistical tools and Monte Carlo simulation.
SigmaXL® customers include market leaders like Agilent, Diebold, FedEx, Microsoft, Motorola and Shell. SigmaXL® software is also used by numerous colleges, universities and government agencies.
Our flagship product, SigmaXL®, was designed from the ground up to be a cost-effective, powerful, but easy to use tool that enables users to measure, analyze, improve and control their service, transactional, and manufacturing processes. As an add-in to the already familiar Microsoft Excel, SigmaXL® is ideal for Lean Six Sigma training and application, or use in a college statistics course.
DiscoverSim™ enables you to quantify your risk through Monte Carlo simulation and minimize your risk with global optimization. Business decisions are often based on assumptions with a single point value estimate or an average, resulting in unexpected outcomes.
DiscoverSim™ allows you to model the uncertainty in your inputs so that you know what to expect in your outputs.
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
S2 - Process product optimization using design experiments and response surfa...CAChemE
An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system.
Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.”
Exploring the impact and evolution of Advanced Analytics Tools.pdfStats Statswork
The impact and evolution of advanced analytics tools have transformed how businesses operate, offering unprecedented insights and decision-making capabilities. Statstwork has been at the forefront of this evolution, providing cutting-edge solutions that leverage big data, machine learning, and AI. These tools enable companies to analyze vast amounts of data in real-time, identify trends, and predict future outcomes with high accuracy. As a result, businesses can optimize their operations, enhance customer experiences, and drive innovation. The continuous advancement of these tools promises even greater efficiencies and opportunities, making them indispensable in the modern data-driven landscape.
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Exploring the impact and evolution of Advanced Analytics Tools.pdfStats Statswork
The impact and evolution of advanced analytics tools have transformed how businesses operate, offering unprecedented insights and decision-making capabilities. Statstwork has been at the forefront of this evolution, providing cutting-edge solutions that leverage big data, machine learning, and AI. These tools enable companies to analyze vast amounts of data in real-time, identify trends, and predict future outcomes with high accuracy. As a result, businesses can optimize their operations, enhance customer experiences, and drive innovation. The continuous advancement of these tools promises even greater efficiencies and opportunities, making them indispensable in the modern data-driven landscape.
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Our cutting-edge predictive analytics models and algorithms transform raw data into actionable insights, empowering businesses to stay ahead of the curve.Trust StatsWork to optimize operations, minimize risks, and drive growth through predictive analytics. Experience the transformative impact of our solutions and propel your business towards success today.
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Empowering Business Growth with Predictive Analytic - StatsworkStats Statswork
Our cutting-edge predictive analytics models and algorithms transform raw data into actionable insights, empowering businesses to stay ahead of the curve.Trust StatsWork to optimize operations, minimize risks, and drive growth through predictive analytics. Experience the transformative impact of our solutions and propel your business towards success today.
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How to establish and evaluate clinical prediction models - StatsworkStats Statswork
A clinical prediction model can be used in various clinical contexts, including screening for asymptomatic illness, forecasting future events such as disease, and assisting doctors in their decision-making and health education. Despite the positive effects of clinical prediction models on practice, prediction modelling is a difficult process that necessitates meticulous statistical analysis and sound clinical judgments. Statswork offers statistical services as per the requirements of the customers. When you Order statistical Services at Statswork, we promise you the following always on Time, outstanding customer support, and High-quality Subject Matter Experts.
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How to establish and evaluate clinical prediction models - StatsworkStats Statswork
A clinical prediction model can be used in various clinical contexts, including screening for asymptomatic illness, forecasting future events such as disease, and assisting doctors in their decision-making and health education. Despite the positive effects of clinical prediction models on practice, prediction modeling is a difficult process that necessitates meticulous statistical analysis and sound clinical judgments. Statswork offers statistical services as per the requirements of the customers. When you Order statistical Services at Statswork, we promise you the following always on Time, outstanding customer support, and High-quality Subject Matter Experts.
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7 excellent reasons why statistics are important statsworkStats Statswork
Statistics are used to analyze what's happening within the world around us. In this data-driven world, all activities of ours are monitored by someone else every time. Statistics help us to convert whatever occurs in the past can be used in predicting the future. Statswork Is A Premier Statistics Consulting Company That Spearheaded Online Statistics Consultancy Service With Clientele Ranging From Educational Institutions, Academics, Corporations And Ngos. We Provide End-To-End Service And Assistance For Your Statistical Research And Analytical Needs From Data Collection, Data Mining, Data Analysis To Research Framework And Research Methodology.
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Research Design may be described as the researchers scheme of outlining the flow of his project. It is based on research design, that the researcher goes about gathering data to answer his research question. It enables the researcher to prioritize his work, create better questionnaires and arrive at conclusions with greater clarity. Statswork offers statistical services as per the requirements of the customers. When you Order statistical Services at Statswork, we promise you the following – Always on Time, outstanding customer support, and High-quality Subject Matter Experts.
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The Clustering technique in Statistical Analysis is used to determine the subsets as clusters in the data using the specified distance measure. However, this technique cannot be applied easily for longitudinal or time-series data. In this blog, I will discuss some of the methods used for modeling longitudinal or panel data using the Clustering Analysis technique as explained in Schmatter (2011). Statswork offers statistical services as per the requirements of the customers. When you Order statistical Services at Statswork, we promise you the following – Always on Time, outstanding customer support, and High-quality Subject Matter Experts.
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Time series analysis is widely applied to forecast the pattern/trends in the financial and market data. The main objective of a time series analysis is to develop a suitable model to describe the pattern or trend in data with more accuracy. Statswork offers statistical services as per the requirements of the customers. When you Order statistical Services at Statswork, we promise you the following – Always on Time, outstanding customer support, and High-quality Subject Matter Experts.
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The present article helps the USA, the UK and the Australian students pursuing their business and marketing postgraduate degree to identify right topic in the area of marketing in business. These topics are researched in-depth at the University of Columbia, brandies, Coventry, Idaho, and many more. Stats work offers UK Dissertation stats work Topics Services in business. When you Order stats work Dissertation Services at Tutors India, we promise you the following – Plagiarism free, Always on Time, outstanding customer support, written to Standard, Unlimited Revisions support and High-quality Subject Matter Experts.
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CHOOSING A QUALITATIVE DATA ANALYSIS (QDA) PLAN
Data Analysis should change what you do, not just how you do it. - Matin Movassate
If you are to choose the right data analysis plan for your study, it is first pertinent to collect qualitative data. Since Qualitative analysis is more about the meaning of the analysis, it is too confusing with unstructured and huge data. For conducting Data Analysis for any research, it is also important to have the right methodology. If the data and methods of data analysis plan are right, it will have numerous benefits, including taking the right decisions.
But before that, there are certain fundamental details to know before choosing the right data analysis plan, which includes:
What is a qualitative data analysis?
QDA is based on interpretative policy to examine the symbolic and meaningful content of data. In other words, it is interpreting the qualitative data by many processes and procedures to transform them into great insights for taking dynamic decisions.
What is qualitative data?
Descriptive data that are non-numerical and capturing concepts and opinions, values and behaviors of people in a social context is called Qualitative data Collection. It is the data from observation of audio and video recordings and also reading the transcripts of interviews and copies of documents.
What purpose does the qualitative data analysis plan perform?
Unlike Quantitative data analysis, which is more of numbers and statistics, qualitative analysis is analysis of subjective and non-numerical qualitative data. Hence it performs many functions including:
• Organizing data
• Interpreting data
• Identifying patterns
• Forming the basis for informed and verifiable conclusions
• Ties research objectives to data
After knowing the above fundamentals of Qualitative data analysis, it is time to choose the right data analysis plan. The plans can be selected for specific research design and can also be applied for a variety of research designs.
Data plays an important role in any research or study conducted. It aids in bringing about a breakthrough in the respective field as well as for future researches. The collection of data is carried out in two forms viz: Qualitative Data and Quantitative Data which includes further bifurcation under it.
What is Qualitative Data?
Qualitative research can be defined as the method of research which focuses on gaining relevant information through observational, open-ended and communication method. They are more exploratory which concentrates on gaining insights about the situation and dig a bit deeper to find the underlying reason. The central idea behind using this method is to find the answer to Why and How rather than How many. Data gathered during a qualitative research is what is termed as qualitative data.
What is the purpose?
A qualitative data is non-numerical and more textual which comprises mostly of images, written texts, recorded audios and spoken words by people. Moreover, one can conduct qualitative research online as well as offline too. Apart from this, the varied purpose of qualitative research is as follows:
- To examine the purpose or reason for the situation
- Gain an understanding of the experience of people
- Understanding of relations and meaning
- Varied norms including social and political as well as contextual and cultural practice which impact the cause.
Data is a source of great information which can enable informed decision making for businesses. Data is divided into Quantitative Data and Qualitative Data. Qualitative data refers to those non-numerical, explanatory data. Herein, we will have a detailed look into the various methods of qualitative data analysis.
Module 6: Outlier Detection for Two Sample CaseStats Statswork
Two sample plot, also known as youden’s plot, is a scatter plot with a confidence region. Youden used it for detecting labs with unusual testing results when two samples are tested in n different lab. Youden plot is a special case of the bivariate control chart, and the idea behind is the principal component analysis.
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SPSS Step-by-Step Tutorial and Statistical Guides by StatsworkStats Statswork
Statswork help to analyse your data, before our step-by-step SPSS Statistics guides show you how to carry out these statistical tests using SPSS Statistics, as well as interpret and analysis the document.
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Statistical Meta Analysis Sample Work - StatsworkStats Statswork
Statswork is a pioneer statistical consulting company providing full statistical assistance including academic, educational institutions and non-government organizations across the globe. We provide end-to-end solutions for all your analytical needs that include creating hypothetical framework to power point presentation. The objective is to provide prompt, reliable, and understandable information about data analysis to our clients.
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Primary, Secondary and Mixed data collection using online surveys, direct interview or Skype Interview or using sources available through peer reviewed journals or other online sources.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
2. • Text reference, Chapter 6
• Special case of the general factorial design; k factors, all at two levels
• The two levels are usually called low and high (they could be either quantitative or
qualitative)
• Very widely used in industrial experimentation
• Form a basic “building block” for other very useful experimental designs (DNA)
• Special (short-cut) methods for analysis
• We will make use of Design-Expert
6. Analysis Procedure for a Factorial Design
Estimate factor effects
•With replication, use full model
•With an unreplicated design, use normal probability plots
Formulate model
Statistical testing (ANOVA)
Refine the model
Analyze residuals (graphical)
Interpret results
7. Estimation of Factor Effects
1
2
1
2
1
2
(1)
2 2
[ (1)]
(1)
2 2
[ (1)]
(1)
2 2
[ (1) ]
A A
n
B B
n
n
A y y
ab a b
n n
ab a b
B y y
ab b a
n n
ab b a
ab a b
AB
n n
ab a b
See textbook, pg. 209-210 For
manual calculations
The effect estimates are: A
= 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
8. Estimation of Factor Effects Form Tentative Model
Term Effect SumSqr % Contribution
Model Intercept
Model A 8.33333 208.333 64.4995
Model B -5 75 23.2198
Model AB 1.66667 8.33333 2.57998
Error Lack Of Fit 0 0
Error P Error 31.3333 9.70072
Lenth's ME 6.15809
Lenth's SME 7.95671
9. Statistical Testing - ANOVA
The F-test for the “model” source is testing the significance of the
overall model; that is, is either A, B, or AB or some combination of
these effects important?
15. etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
Effects in The 23 Factorial Design
Analysis
done via
computer
16. 16
An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
17. Table of – and + Signs for the 23 Factorial Design (pg. 218)
18. • Except for column I, every column has an equal number of + and –
signs
• The sum of the product of signs in any two columns is zero
• Multiplying any column by I leaves that column unchanged (identity
element)
• The product of any two columns yields a column in the table:
• Orthogonal design
• Orthogonality is an important property shared by all factorial designs
18
Properties of the Table
2
A B AB
AB BC AB C AC
24. • R2 and adjusted R2
• R2 for prediction (based on PRESS)
24
Model Summary Statistics for Reduced Model
5
2
5
2
5
5.106 10
0.9608
5.314 10
/ 20857.75/12
1 1 0.9509
/ 5.314 10 /15
Model
T
E E
Adj
T T
SS
R
SS
SS df
R
SS df
2
Pred 5
37080.44
1 1 0.9302
5.314 10T
PRESS
R
SS
25. Model Summary Statistics
• Standard error of model coefficients (full
model)
• Confidence interval on model coefficients
25
2
2252.56ˆ ˆ( ) ( ) 11.87
2 2 2(8)
E
k k
MS
se V
n n
/2, /2,
ˆ ˆ ˆ ˆ( ) ( )E Edf dft se t se
30. • Section 6-4, pg. 227, Table 6-9, pg. 228
• There will be k main effects, and
The General 2k Factorial Design
two-factor interactions
2
three-factor interactions
3
1 factor interaction
k
k
k
31. • These are 2k factorial designs with one
observation at each corner of the “cube”
• An unreplicated 2k factorial design is also
sometimes called a “single replicate” of the 2k
• These designs are very widely used
• Risks…if there is only one observation at each
corner, is there a chance of unusual response
observations spoiling the results?
• Modeling “noise”?
Unreplicated 2k Factorial Designs
32. Spacing of Factor Levels in the Unreplicated 2k Factorial Designs
If the factors are spaced too closely, it increases the chances
that the noise will overwhelm the signal in the data
More aggressive spacing is usually best
33. • Lack of replication causes potential problems in
statistical testing
– Replication admits an estimate of “pure error” (a
better phrase is an internal estimate of error)
– With no replication, fitting the full model results in zero
degrees of freedom for error
• Potential solutions to this problem
– Pooling high-order interactions to estimate error
– Normal probability plotting of effects (Daniels, 1959)
– Other methods…see text
Unreplicated 2k Factorial Designs
34. • A 24 factorial was used to investigate the
effects of four factors on the filtration rate of
a resin
• The factors are A = temperature, B = pressure,
C = mole ratio, D= stirring rate
• Experiment was performed in a pilot plant
Example of an Unreplicated 2k Design
44. Model Interpretation – Response Surface Plots
With concentration at either the low or high level, high temperature and high
stirring rate results in high filtration rates
46. Dealing with Outliers
• Replace with an estimate
• Make the highest-order interaction zero
• In this case, estimate cd such that ABCD = 0
• Analyze only the data you have
• Now the design isn’t orthogonal
• Consequences?
47.
48. 48
The Drilling Experiment Example 6.3
A = drill load, B = flow, C = speed, D = type of mud,
y = advance rate of the drill
51. • The residual plots indicate that there are problems with the
equality of variance assumption
• The usual approach to this problem is to employ a transformation
on the response
• Power family transformations are widely used
• Transformations are typically performed to
– Stabilize variance
– Induce at least approximate normality
– Simplify the model
Residual Plots
*
y y
52. • Empirical selection of lambda
• Prior (theoretical) knowledge or experience can often suggest the
form of a transformation
• Analytical selection of lambda…the Box-Cox (1964) method
(simultaneously estimates the model parameters and the
transformation parameter lambda)
• Box-Cox method implemented in Design-Expert
52
Selecting a Transformation
53.
54. 54
The Box-Cox Method
A log transformation is
recommended
The procedure provides a
confidence interval on
the transformation
parameter lambda
If unity is included in the
confidence interval, no
transformation would be
needed
55. Effect Estimates Following the Log Transformation
Three main effects are
large
No indication of large
interaction effects
What happened to the
interactions?
58. The Log Advance Rate Model
• Is the log model “better”?
• We would generally prefer a simpler model in a transformed
scale to a more complicated model in the original metric
• What happened to the interactions?
• Sometimes transformations provide insight into the underlying
mechanism
59. Other Examples of Unreplicated 2k Designs
• The sidewall panel experiment (Example 6.4, pg. 245)
– Two factors affect the mean number of defects
– A third factor affects variability
– Residual plots were useful in identifying the dispersion effect
• The oxidation furnace experiment (Example 6.5, pg. 245)
– Replicates versus repeat (or duplicate) observations?
– Modeling within-run variability
60. Other Analysis Methods for Unreplicated 2k Designs
• Lenth’s method (see text, pg. 235)
– Analytical method for testing effects, uses an estimate of error formed by
pooling small contrasts
– Some adjustment to the critical values in the original method can be helpful
– Probably most useful as a supplement to the normal probability plot
• Conditional inference charts (pg. 236)
61. Overview of Lenth’s method
For an individual contrast, compare to the margin of error
62.
63. Adjusted multipliers for Lenth’s method
Suggested because the original method makes too many type I errors, especially for small
designs (few contrasts)
Simulation was used to find these adjusted multipliers
Lenth’s method is a nice supplement to the normal probability plot of effects
JMP has an excellent implementation of Lenth’s method in the screening platform
64.
65. The 2k design and design optimality
The model parameter estimates in a 2k design (and the effect estimates)
are least squares estimates. For example, for a 22 design the model is
0 1 1 2 2 12 1 2
0 1 2 12 1
0 1 2 12 2
0 1 2 12 3
0 1 2 12 4
(1) ( 1) ( 1) ( 1)( 1)
(1) ( 1) (1)( 1)
( 1) (1) ( 1)(1)
(1) (1) (1)(1)
(1) 1 1 1 1
1 1 1 1
, ,
1 1 1
y x x x x
a
b
ab
a
b
ab
y = Xβ + ε y X
0 1
1 2
2 3
12 4
, ,
1
1 1 1 1
β ε
The four
observations from
a 22 design
66. The least squares estimate of β is
1
0
1
4
2
12
ˆ
4 0 0 0 (1)
0 4 0 0 (1)
0 0 4 0 (1)
0 0 0 4 (1)
(1)
4ˆ
(1) (
ˆ (1)1
ˆ (1)4
(1)ˆ
a b ab
a ab b
b ab a
a b ab
a b ab
a b ab a ab b
a ab b
b ab a
a b ab
-1
β = (X X) X y
I
1)
4
(1)
4
(1)
4
b ab a
a b ab
The matrix is
diagonal –
consequences of an
orthogonal design
XX
The regression
coefficient estimates
are exactly half of the
‘usual” effect estimates
The “usual” contrasts
67. The matrix has interesting and useful properties:XX
2 1
2
ˆ( ) (diagonal element of ( ) )
4
V
X X
Minimum possible value for a four-run
design
|( ) | 256 X X
Maximum possible value for a four-run
design
Notice that these results depend on both the design that you have chosen and the model
What about predicting the response?
68. 2
1 2
1 2 1 2
2
2 2 2 2
1 2 1 2 1 2
1 2
2
1 2
1 2
2
1 2
ˆ[ ( , )]
[1, , , ]
ˆ[ ( , )] (1 )
4
The maximum prediction variance occurs when 1, 1
ˆ[ ( , )]
The prediction variance when 0 is
ˆ[ ( , )]
V y x x
x x x x
V y x x x x x x
x x
V y x x
x x
V y x x
-1
x (X X) x
x
4
What about prediction variance over the design space?average
69. Averageprediction variance
1 1
2
1 2 1 2
1 1
1 1
2 2 2 2 2
1 2 1 2 1 2
1 1
2
1
ˆ[ ( , ) = area of design space = 2 4
1 1
(1 )
4 4
4
9
I V y x x dx dx A
A
x x x x dx dx
70. Design-Expert® Software
Min StdErr Mean: 0.500
Max StdErr Mean: 1.000
Cuboidal
radius = 1
Points = 10000
FDS Graph
Fraction of Design Space
StdErrMean
0.00 0.25 0.50 0.75 1.00
0.000
0.250
0.500
0.750
1.000
71. 71
For the 22 and in general the 2k
• The design produces regression model coefficients that have
the smallest variances (D-optimal design)
• The design results in minimizing the maximum variance of the
predicted response over the design space (G-optimal design)
• The design results in minimizing the average variance of the
predicted response over the design space (I-optimal design)
72. 72
Optimal Designs
• These results give us some assurance that these designs are
“good” designs in some general ways
• Factorial designs typically share some (most) of these properties
• There are excellent computer routines for finding optimal
designs (JMP is outstanding)
73. • Based on the idea of replicating some of the runs in a factorial design
• Runs at the center provide an estimate of error and allow the
experimenter to distinguish between two possible models:
Addition of Center Points to a 2k Designs
0
1 1
2
0
1 1 1
First-order model (interaction)
Second-order model
k k k
i i ij i j
i i j i
k k k k
i i ij i j ii i
i i j i i
y x x x
y x x x x
74.
75. 75
no "curvature"F Cy y
The hypotheses are:
0
1
1
1
: 0
: 0
k
ii
i
k
ii
i
H
H
2
Pure Quad
( )F C F C
F C
n n y y
SS
n n
This sum of squares has a
single degree of freedom
76. 76
Example 6.6, Pg. 248
4Cn
Usually between 3 and 6
center points will work
well
Design-Expert provides
the analysis, including the
F-test for pure quadratic
curvature
Refer to the original experiment shown in Table 6.10.
Suppose that four center points are added to this
experiment, and at the points x1=x2 =x3=x4=0 the
four observed filtration rates were 73, 75, 66, and 69.
The average of these four center points is 70.75, and
the average of the 16 factorial runs is 70.06. Since
are very similar, we suspect that there is no strong
curvature present.
79. If curvature is significant, augment the design with axial runs to create a
central composite design. The CCD is a very effective design for fitting a
second-order response surface model
80. Practical Use of Center Points (pg. 260)
• Use current operating conditions as the center point
• Check for “abnormal” conditions during the time the
experiment was conducted
• Check for time trends
• Use center points as the first few runs when there is little or no
information available about the magnitude of error
• Center points and qualitative factors?
82. 82
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