experimental designs in r

Experimental Design and Process Optimization with R

Gerhard Krennrich

1 Introduction

The present document is a short and elementary course on the Design of Experiments (DoE) and empirical process optimization with the open-source Software R . The course is self-contained and does not assume any preknowledge in statistics or mathematics beyond high school level. Statistical concepts will be introduced on an elementary level and made tangible with R-code and R-graphics based on simulated and real world data. So, then, what is DoE and why should the reader become familiar with the concepts of DoE? Very briefly, DoE is the science of varying many experimental parameters in a systematic way to gain insight on how to further improve and optimize these parameters. Chapter 2 will show how and why multidimensional DoE techniques are superiour to the classical “one-dimensional” optimization approach. Chapter 6 will demonstrate why and how DoE can be combined with optimization. Finally, the use of DoE and optimization will be practically demonstrated in chapter 7 for improving the performance of a catalytic system. Historically, Experimental Design started as a branch of statistics in the early years of the 20 th century and has meanwhile grown into a mature method with a plethora of applications in the experimental sciences. Consequently, there are many good and comprehensive books available about DoE, some of which we will make frequent reference to in the present text, namely (George E.P. Box, Norman R. Draper 1987 ) , (D.C. Montgomery 2013 ) and (G.E.P. Box, W.G. Hunter, J.S. Hunter 2005 ) . A more recent text with emphasis on the use of R in conjuction with DoE is (John Lawson 2015 ) . Linear models are comprehensively covered, e.g., by the text book (A. Sen, M. Srivastava 1990 ) . A general, however fairly technical text on linear and nonlinear statistical model building is the excellent book (T. Hastie, R. Tibshirani, J. Friedman 2009 ) . (J.G. Kalbfleisch 1985 ) is a smooth introduction into statistics, probability and statistical inference. The present text draws on these books and on many years of experience as a statistical consultant in the chemical industry. Most examples in this course are therefore taken from applications and optimization projects in the chemical sciences. The primarily intended readers of this document are chemists and engineers entrusted with empirical optimization in research and development. However, the presented methods and concepts are fairly generic and scientist working in other areas such as biology or the medical sciences might benefit from the text. As to software, R, probably together with Phyton, is the only open-source software which combines the whole spectrum of DoE and optimization with the flexibility of a powerful script language that allows any kind of data pre- and postprocessing within one software environment. That makes, in my opinion, R superior to many commercial GUI based tools which often buy userfriendlyness at the expense of flexibility.

1.1 How to install R

The R-software can be downloaded free of charge from the R repository CRAN

An IDE ( I ntegrated D evelopment E nvironment) is reqired for smoothly working with R. An IDE allows editing, running and debugging of R code and managing programm in- and output. In principle any IDE can be used but we recommend R-Studio as the de-facto standard.

Get R-Studio IDE

The R-introduction at CRAN is a concise introduction into the R-language. A short R-introduction

1.2 Some remarks on how to read the present text

This document is not an introduction into the R language, rather the document follows the philosophy of “learning by doing”. In this spirit the above mentioned text R-introduction is recommended as a first reference together with the present R examples on DoE and optimization. As it is usually easier to modify existing code than writing code from scratch, it is hoped that the R-examples in this course will help learning both R and DoE more rapidly. The course is divided into seven chapters. There is, however, one stand-alone chapter, chapter 5, which can be skipped by those readers not explicitly dealing with mixture problems. The final chapter 7 is a published, (Siebert M., Krennrich G., Seibicke M., Siegle A.F., Trapp O. 2019 ) , real-world example combining many elements of DoE and optimization for improving the performance of a catalytic system. This application should encourage readers to use these powerful methods for the sake of their own projects.

A. Sen, M. Srivastava. 1990. Regression Analysis, Theory, Methods and Applications . 1st ed. Springer-Verlag, New York.

D.C. Montgomery. 2013. Design and Analysis of Experiments . 8th ed. John Wiley & Sons Inc.

G.E.P. Box, W.G. Hunter, J.S. Hunter. 2005. Statistics for Experimenters: Design, Innovation, and Discovery . 2nd ed. John Wiley & Sons, Hoboken.

George E.P. Box, Norman R. Draper. 1987. Empirical Model-Building and Response Surfaces . 1st ed. John Wiley & Sons.

J.G. Kalbfleisch. 1985. Probability and Statistical Inference, Vol 1&2 . 2nd ed. Springer.

John Lawson. 2015. Design and Analysis of Experiments with R . 1st ed. Chapman & Hall.

Siebert M., Krennrich G., Seibicke M., Siegle A.F., Trapp O. 2019. “Identifying High-Performance Catalytic Conditions for Carbon Dioxide Reduction to Dimethoxymethane by Multivariate Modelling.” Chemical Science 10:45. https://pubs.rsc.org/en/content/articlelanding/2019/sc/c9sc04591k#!divAbstract .

T. Hastie, R. Tibshirani, J. Friedman. 2009. The Elements of Statistical Learning . 2nd ed. Springer-Verlag.

Design of Experiments with R

• First Online: 01 January 2012

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• Emilio L. Cano 4 ,
• Javier M. Moguerza 4 &
• Andrés Redchuk 4  

Part of the book series: Use R! ((USE R,volume 36))

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Design of experiments (DoE) is one of the most important tools in the Six Sigma methodology. It is the essence of the Improve phase and the basis for the design of robust processes. An adequate use of DoE will lead to the improvement of a process, but a bad design can result in wrong conclusions and engender the opposite of the desired effect: inefficiencies, higher costs, and less competitiveness. In this chapter, we introduce the foundations of DoE and describe the essential functions in R to perform it and analyze its results. We will describe two-level factorial designs using a representative example of how DoE should be used to achieve the improvement of a process in a Six Sigma way. The chapter is not intended as a thorough review of DoE. The idea is to introduce a simple model in an intuitive way. For more technical or advance training a number of references are given at the end of the chapter.

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Allen, T. T. (2010). Introduction to engineering statistics and lean Six Sigma—Statistical quality control and design of experiments and systems . New York: Springer.

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Lalanne, C. (2006). R companion to montgomery’s design and analysis of experiments . http://www.aliquote.org/articles/tech/dae/ . Retrieved 19.01.2012.

Lopez-Fidalgo, J. (2009). A critical overview on optimal experimental designs. Boletin de Estadística e Investigación Operativa , 25 (1), 14–21. http://www.seio.es/BEIO/files/BEIOv25n1_ES_J.Lopez-Fidalgo.pdf . Retrieved 19.01.2012.

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Pyzdek, T., & Keller, P. (2009). The Six Sigma handbook: A complete guide for green belts, black belts, and managers at all levels . New York: McGraw-Hill.

Rasch, D., Pilz, J., & Simecek, P. (2010). Optimal experimental design with R . London: Taylor & Francis.

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Cano, E.L., Moguerza, J.M., Redchuk, A. (2012). Design of Experiments with R. In: Six Sigma with R. Use R!, vol 36. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3652-2_11

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Design and Analysis of Experiments with R presents a unified treatment of experimental designs and design concepts commonly used in practice. It connects the objectives of research to the type of experimental design required, describes the process of creating the design and collecting the data, shows how to perform the proper analysis of the data,

TABLE OF CONTENTS

Chapter chapter 1 | 15  pages, introduction, chapter chapter 2 | 37  pages, completely randomized designs with one factor, chapter chapter 3 | 55  pages, factorial designs, chapter chapter 4 | 28  pages, randomized block designs, chapter chapter 5 | 52  pages, designs to study variances, chapter chapter 6 | 67  pages, fractional factorial designs, chapter 7 | 45  pages, incomplete and confounded block designs, chapter 8 | 44  pages, split-plot designs, chapter chapter 9 | 32  pages, crossover and repeated measures designs, chapter chapter 10 | 63  pages, response surface designs, chapter chapter 11 | 56  pages, mixture experiments, chapter chapter 12 | 53  pages, robust parameter design experiments, chapter chapter 13 | 14  pages, experimental strategies for increasing knowledge.

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Introduction to Econometrics with R

13 experiments and quasi-experiments.

This chapter discusses statistical tools that are commonly applied in program evaluation, where interest lies in measuring the causal effects of programs, policies or other interventions. An optimal research design for this purpose is what statisticians call an ideal randomized controlled experiment. The basic idea is to randomly assign subjects to two different groups, one that receives the treatment (the treatment group) and one that does not (the control group) and to compare outcomes for both groups in order to get an estimate of the average treatment effect.

Such experimental data is fundamentally different from observational data. For example, one might use a randomized controlled experiment to measure how much the performance of students in a standardized test differs between two classes where one has a “regular”” student-teacher ratio and the other one has fewer students. The data produced by such an experiment would be different from, e.g., the observed cross-section data on the students’ performance used throughout Chapters 4 to 8 where class sizes are not randomly assigned to students but instead are the results of an economic decision where educational objectives and budgetary aspects were balanced.

For economists, randomized controlled experiments are often difficult or even infeasible to implement. For example, due to ethic, moral and legal reasons it is practically impossible for a business owner to estimate the causal effect on the productivity of workers of setting them under psychological stress using an experiment where workers are randomly assigned either to the treatment group that is under time pressure or to the control group where work is under regular conditions, at best without knowledge of being in an experiment (see the box The Hawthorne Effect on p. 528 of the book).

However, sometimes external circumstances produce what is called a quasi-experiment or natural experiment . This “as if” randomness allows for estimation of causal effects that are of interest for economists using tools which are very similar to those valid for ideal randomized controlled experiments. These tools draw heavily on the theory of multiple regression and also on IV regression (see Chapter 12 ). We will review the core aspects of these methods and demonstrate how to apply them in R using the STAR data set (see the description of the data set).

The following packages and their dependencies are needed for reproduction of the code chunks presented throughout this chapter:

• AER ( Christian Kleiber and Zeileis 2008 ) ,
• dplyr ( Wickham et al. 2023 ) ,
• MASS ( Ripley 2023 ) ,
• mvtnorm ( Genz et al. 2023 ) ,
• rddtools ( Stigler and Quast 2022 ) ,
• scales ( Wickham and Seidel 2022 ) ,
• stargazer ( Hlavac 2022 ) ,
• tidyr ( Wickham, Vaughan, and Girlich 2023 ) .

Make sure the following code chunk runs without any errors.

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How to create a max-diff experimental design in r.

Posted on May 17, 2017 by Tim Bock in R bloggers | 0 Comments

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Creating the experimental design for a max-diff experiment is easy in R. This post describes how to create and check a max-diff experimental design. If you are not sure what this is, it would be best to read A beginner’s guide to max-diff first.

Step 1: Installing the packages

The first step is to install the flipMaxDiff  package and a series of dependent packages. Depending on how your R has been setup, you may need to install none of these (e.g., if using Displayr), or even more packages than are shown below.

Step 2: Creating the design

The MaxDiffDesign function is a wrapper for the optBlock function in the wonderful AlgDesign package. The following snippet can be used to create a design. The arguments used in the code snippet here are described immediately below.

• number.alternatives:  The total number of alternatives considered in the study. In my technology study , for example, I had 10 brands, so I enter the number of alternatives as 10.
• alternatives.per.question : The number of alternatives shown to the respondents in each individual task. I tend to set this at 5. Where I have studies where the alternatives are wordy, I like to reduce this to 4. Where the alternatives are really easy to understand, I have used 6.  The key trade-off here is cognitive difficulty for the respondent. The harder the questions, the more likely people are to not consider them very carefully.
• number.questions : The number of questions (i.e., tasks or sets) to present to respondents. A rule of thumb provided by the good folks at Sawtooth Software states the ideal number of questions:  3 * number.alternatives/ alternatives.per.question . This would suggest that in the technology study, I should have used 3 * 10 / 5 = 6 questions, which is indeed the number that I used in the study. There are two conflicting factors to trade off when setting the number of questions. The more questions, the more respondent fatigue, and the worse your data becomes. The fewer questions, the less data, and the harder it is to work out the relative appeal of alternatives that have a similar level of overall appeal. I return to this topic in the discussion of checking designs, below.
• n.repeats : The algorithm includes a randomization component. Occasionally, this can lead to poor designs being found (how to check for this is described below). Sometimes this problem can be remedied by increasing n.repeats .

Step 3: Interpreting the design

The design is called the binary.design . Each row represents a question. Each column shows which alternatives are to be shown. Thus, in the first question, the respondent evaluates alternatives 1, 3, 5, 6, and 10. More complicated designs can have additional information (this is discussed below)

I tend to add one additional complication to my max-diff studies. I get the data collection to involve randomization of the order of the alternatives between respondents. One and only one respondent had brands shown in this order: Apple, Google Samsung, Sony, Microsoft, Intel, Dell, Nokia, IBM, and Yahoo. So, whenever Apple appeared it was at the top, whenever Google appeared, it was below Apple if Apple appeared, but at the top otherwise, etc. The next respondent had the brands in a different order, and so on.

If doing randomization like this, I strongly advise having this randomization done in the data collection software.  You can then undo it when creating the data file, enabling you to conduct the analysis as if no randomization ever occurred.

There are many other ways of complicating designs, such as to deal with large numbers of alternatives, and to prevent certain pairs of alternatives appearing together. Click here for more information about this.

Step 4: Checking the design

In an ideal world, a max-diff experimental design has the following characteristics, where each alternative appears:

• At least 3 times.
• The same number of times.
• With each other alternative the same number of times (e.g., each alternative appears with each other alternative twice).

Due to a combination of maths and a desire to avoid respondent fatigue, few max-diff experimental designs satisfy these three requirements (the last one is particularly tough).

Above, I described a design with 10 alternatives, 5 alternatives per question, and 6 questions. Below, I show the outputs where I have changed of alternatives per question from 5 to 4. This small change has made a good design awful. How can we see it is awful? The first thing to note is that 6 warnings are shown at the bottom.

The first warning is telling us that we have ignored the advice about how to compute the number of questions, and we should instead have at least 8 questions. (Or, more alternatives per question.)

The second warning is telling us that we have an alternative that only appears two times, whereas good practice is that we should have each alternative appearing three times.

The third alternative tells us that some alternatives appear more regularly than others. Looking at the frequencies output, we can see that options appeared either 2 or 3 times. Why does this matter? It means we have collected more information about some of the alternatives than others, so may end up with different levels of precision of our estimates of the appeal of different alternatives.

The fourth warning is a bit cryptic. To understand it we need to look at the binary correlations , which are reproduced below. This  correlation matrix  shows the correlations between each of the columns of the experimental design (i.e., binary.design shown above). Looking at row 4 and column 8 we see a big problem. Alternative 4 and 8 are perfectly negatively correlated. That is, whenever alternative 4 appears in the design alternative 8 does not appear, and whenever 8 appears, 4 does not appear. One of the cool things about max-diff is that it can sometimes still work even with such a flaw in the experimental design. It would, however, be foolhardy to rely on this. The basic purpose of max-diff is to work out relative preferences between alternatives, and its ability to do this is clearly compromised if some alternatives are never shown with others.

The 5th warning tells us that there is a large range in the correlations. In most experimental designs, the ideal design results in a correlation of 0 between all the variables. Max-Diff designs differ from this, as, on average, there will always be a negative correlation between the variables. However, the basic idea is the same: we strive for designs where the correlations are close to 0 as possible. Correlations in the range of -0.5 and 0.5 should, in my opinion, cause no concern.

The last warning tells us that some alternatives never appear together.  We already deduced this from the binary correlations.

The first thing to do when you have a poor design is to increase the setting for  n.repeats . Start by setting it to 10. Then, if you have patience, try 100, and then bigger numbers. This only occasionally works. But, when it does work it is a good outcome. If this does not work, you need to change something else. Reducing the number of alternatives and/or increasing the number of questions are usually the best places to start.

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Experimental Design in R (DataCamp)

Ch. 1 - introduction to experimental design, intro to experimental design.

Steps of an Experiment

• dependent variable = outcome
• independent variable(s) = explanatory variables

Example of timeline

• Planning & Design - 6 months
• Conduct Experiment - 18 months
• Analysis - 2 months

Key Components of an Experiment

Randomization

Replication, a basic experiment, hypothesis testing.

Breaking down hypothesis testing:

• there is no change
• no difference between groups
• the mean, median, or observation = a number
• there is a change
• difference between groups
• mean, median, or observation is >, <, or != to a number

Power & Sample Size

• Power: probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true.
• Effect size: standardized measure of the difference you’re trying to detect.
• Sample Size: How many experimental units you need to survey to detect the desired difference at the desired power.

Power & Sample Size Calculations

One sided vs. two-sided tests, pwr package help docs exploration.

The pwr package has been loaded for you. Use the console to look at the documentation for the pwr.t.test() function. The list of arguments are specialized for a t-test and include the ability to specify the alternative hypothesis.

If you’d like, take some time to explore the different pwr package functions and read about their inputs.

What does a call to any pwr.*() function yield?

• A vector containing the thing to be calculated.
• A data frame of the power, sample size, and other inputs.
• [*] An object of class “power.htest”.
• An object of class “integer”.

Ch. 2- Basic Experiments

Single & multiple factor experiments.

• Used to compare 3+ groups
• won’t know which groups’ means are different without additional post hoc testing
• Two ways to implement in R:

Single Factor Experiments

Multiple Factor Experiments

Exploratory data analysis (eda) lending club, how does loan purpose affect amount funded, which loan purpose mean is different, model validation, pre-modeling eda.

Post-modeling model validation

• Residual plot
• QQ-plot for normality
• Homogeneity of variances
• Try non-parametric alternatives to ANOVA

Post-modeling validation plots + variance

Kruskal-wallis rank sum test, a/b testing, which post-a/b test test.

We’ll be testing the mean loan_amnt, which is the requested amount of money the loan applicants ask for, based on which color header (green or blue) that they saw on the Lending Club website.

Which statistical test should we use after we’ve collected enough data?

• Kruskal-Wallis Rank Sum test
• Chi-Square Test
• Linear Regression

Sample size for A/B test

Basic a/b test, a/b tests vs. multivariable experiments, ch. 3 - randomized complete (& balanced incomplete) block designs, intro to nhanes & sampling.

Intro to Sampling

• Probability Sampling: probability is used to select the sample (in various ways)
• Voluntary response: whoever agrees to respond is the sample
• Convenience sampling: subjects convenient to the researcher are chosen.

Simple Random Sampling (SRS)

Stratified Sampling

Cluster Sampling

Systematic Sampling

Multi-stage Sampling

NHANES dataset construction

Nhanes data cleaning, resampling nhanes data, randomized complete block designs (rcbd), which is not a good blocking factor, drawing rcbds with agricolae, nhanes rcbd, rcbd model validation, balanced incomplete block designs (bibd), is a bibd even possible, drawing bibds with agricolae, bibd - cat’s kidney function, nhanes bibd, ch. 4 - latin squares, graeco-latin squares, & factorial experiments, latin squares, nyc sat scores eda, dealing with missing test scores, drawing latin squares with agricolae, latin square with nyc sat scores, graeco-latin squares, nyc sat scores data viz, drawing graeco-latin squares with agricolae, graeco-latin square with nyc sat scores, factorial experiments, nyc sat scores factorial eda, factorial experiment with nyc sat scores, evaluating the nyc sat scores factorial model, what’s next in experimental design, about michael mallari.

Michael is a hybrid thinker and doer —a byproduct of being a StrengthsFinder “Learner” over time. With 20+ years of engineering, design, and product experience, he helps organizations identify market needs, mobilize internal and external resources, and deliver delightful digital customer experiences that align with business goals. He has been entrusted with problem-solving for brands—ranging from Fortune 500 companies to early-stage startups to not-for-profit organizations.

Michael earned his BS in Computer Science from New York Institute of Technology and his MBA from the University of Maryland, College Park. He is also a candidate to receive his MS in Applied Analytics from Columbia University.

LinkedIn | Twitter | www.michaelmallari.com/data | www.columbia.edu/~mm5470

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Statistics By Jim

Making statistics intuitive

Experimental Design: Definition and Types

By Jim Frost 3 Comments

What is Experimental Design?

An experimental design is a detailed plan for collecting and using data to identify causal relationships. Through careful planning, the design of experiments allows your data collection efforts to have a reasonable chance of detecting effects and testing hypotheses that answer your research questions.

An experiment is a data collection procedure that occurs in controlled conditions to identify and understand causal relationships between variables. Researchers can use many potential designs. The ultimate choice depends on their research question, resources, goals, and constraints. In some fields of study, researchers refer to experimental design as the design of experiments (DOE). Both terms are synonymous.

Ultimately, the design of experiments helps ensure that your procedures and data will evaluate your research question effectively. Without an experimental design, you might waste your efforts in a process that, for many potential reasons, can’t answer your research question. In short, it helps you trust your results.

Learn more about Independent and Dependent Variables .

Design of Experiments: Goals & Settings

Experiments occur in many settings, ranging from psychology, social sciences, medicine, physics, engineering, and industrial and service sectors. Typically, experimental goals are to discover a previously unknown effect , confirm a known effect, or test a hypothesis.

Effects represent causal relationships between variables. For example, in a medical experiment, does the new medicine cause an improvement in health outcomes? If so, the medicine has a causal effect on the outcome.

An experimental design’s focus depends on the subject area and can include the following goals:

• Understanding the relationships between variables.
• Identifying the variables that have the largest impact on the outcomes.
• Finding the input variable settings that produce an optimal result.

For example, psychologists have conducted experiments to understand how conformity affects decision-making. Sociologists have performed experiments to determine whether ethnicity affects the public reaction to staged bike thefts. These experiments map out the causal relationships between variables, and their primary goal is to understand the role of various factors.

Conversely, in a manufacturing environment, the researchers might use an experimental design to find the factors that most effectively improve their product’s strength, identify the optimal manufacturing settings, and do all that while accounting for various constraints. In short, a manufacturer’s goal is often to use experiments to improve their products cost-effectively.

In a medical experiment, the goal might be to quantify the medicine’s effect and find the optimum dosage.

Developing an Experimental Design

Developing an experimental design involves planning that maximizes the potential to collect data that is both trustworthy and able to detect causal relationships. Specifically, these studies aim to see effects when they exist in the population the researchers are studying, preferentially favor causal effects, isolate each factor’s true effect from potential confounders, and produce conclusions that you can generalize to the real world.

To accomplish these goals, experimental designs carefully manage data validity and reliability , and internal and external experimental validity. When your experiment is valid and reliable, you can expect your procedures and data to produce trustworthy results.

An excellent experimental design involves the following:

• Lots of preplanning.
• Developing experimental treatments.
• Determining how to assign subjects to treatment groups.

The remainder of this article focuses on how experimental designs incorporate these essential items to accomplish their research goals.

Learn more about Data Reliability vs. Validity and Internal and External Experimental Validity .

Preplanning, Defining, and Operationalizing for Design of Experiments

This phase of the design of experiments helps you identify critical variables, know how to measure them while ensuring reliability and validity, and understand the relationships between them. The review can also help you find ways to reduce sources of variability, which increases your ability to detect treatment effects. Notably, the literature review allows you to learn how similar studies designed their experiments and the challenges they faced.

Operationalizing a study involves taking your research question, using the background information you gathered, and formulating an actionable plan.

This process should produce a specific and testable hypothesis using data that you can reasonably collect given the resources available to the experiment.

• Null hypothesis : The jumping exercise intervention does not affect bone density.
• Alternative hypothesis : The jumping exercise intervention affects bone density.

To learn more about this early phase, read Five Steps for Conducting Scientific Studies with Statistical Analyses .

Formulating Treatments in Experimental Designs

In an experimental design, treatments are variables that the researchers control. They are the primary independent variables of interest. Researchers administer the treatment to the subjects or items in the experiment and want to know whether it causes changes in the outcome.

As the name implies, a treatment can be medical in nature, such as a new medicine or vaccine. But it’s a general term that applies to other things such as training programs, manufacturing settings, teaching methods, and types of fertilizers. I helped run an experiment where the treatment was a jumping exercise intervention that we hoped would increase bone density. All these treatment examples are things that potentially influence a measurable outcome.

Even when you know your treatment generally, you must carefully consider the amount. How large of a dose? If you’re comparing three different temperatures in a manufacturing process, how far apart are they? For my bone mineral density study, we had to determine how frequently the exercise sessions would occur and how long each lasted.

How you define the treatments in the design of experiments can affect your findings and the generalizability of your results.

Assigning Subjects to Experimental Groups

A crucial decision for all experimental designs is determining how researchers assign subjects to the experimental conditions—the treatment and control groups. The control group is often, but not always, the lack of a treatment. It serves as a basis for comparison by showing outcomes for subjects who don’t receive a treatment. Learn more about Control Groups .

How your experimental design assigns subjects to the groups affects how confident you can be that the findings represent true causal effects rather than mere correlation caused by confounders. Indeed, the assignment method influences how you control for confounding variables. This is the difference between correlation and causation .

Imagine a study finds that vitamin consumption correlates with better health outcomes. As a researcher, you want to be able to say that vitamin consumption causes the improvements. However, with the wrong experimental design, you might only be able to say there is an association. A confounder, and not the vitamins, might actually cause the health benefits.

Let’s explore some of the ways to assign subjects in design of experiments.

Completely Randomized Designs

A completely randomized experimental design randomly assigns all subjects to the treatment and control groups. You simply take each participant and use a random process to determine their group assignment. You can flip coins, roll a die, or use a computer. Randomized experiments must be prospective studies because they need to be able to control group assignment.

Random assignment in the design of experiments helps ensure that the groups are roughly equivalent at the beginning of the study. This equivalence at the start increases your confidence that any differences you see at the end were caused by the treatments. The randomization tends to equalize confounders between the experimental groups and, thereby, cancels out their effects, leaving only the treatment effects.

For example, in a vitamin study, the researchers can randomly assign participants to either the control or vitamin group. Because the groups are approximately equal when the experiment starts, if the health outcomes are different at the end of the study, the researchers can be confident that the vitamins caused those improvements.

Statisticians consider randomized experimental designs to be the best for identifying causal relationships.

If you can’t randomly assign subjects but want to draw causal conclusions about an intervention, consider using a quasi-experimental design .

Learn more about Randomized Controlled Trials and Random Assignment in Experiments .

Randomized Block Designs

Nuisance factors are variables that can affect the outcome, but they are not the researcher’s primary interest. Unfortunately, they can hide or distort the treatment results. When experimenters know about specific nuisance factors, they can use a randomized block design to minimize their impact.

This experimental design takes subjects with a shared “nuisance” characteristic and groups them into blocks. The participants in each block are then randomly assigned to the experimental groups. This process allows the experiment to control for known nuisance factors.

Blocking in the design of experiments reduces the impact of nuisance factors on experimental error. The analysis assesses the effects of the treatment within each block, which removes the variability between blocks. The result is that blocked experimental designs can reduce the impact of nuisance variables, increasing the ability to detect treatment effects accurately.

Suppose you’re testing various teaching methods. Because grade level likely affects educational outcomes, you might use grade level as a blocking factor. To use a randomized block design for this scenario, divide the participants by grade level and then randomly assign the members of each grade level to the experimental groups.

A standard guideline for an experimental design is to “Block what you can, randomize what you cannot.” Use blocking for a few primary nuisance factors. Then use random assignment to distribute the unblocked nuisance factors equally between the experimental conditions.

You can also use covariates to control nuisance factors. Learn about Covariates: Definition and Uses .

Observational Studies

In some experimental designs, randomly assigning subjects to the experimental conditions is impossible or unethical. The researchers simply can’t assign participants to the experimental groups. However, they can observe them in their natural groupings, measure the essential variables, and look for correlations. These observational studies are also known as quasi-experimental designs. Retrospective studies must be observational in nature because they look back at past events.

Imagine you’re studying the effects of depression on an activity. Clearly, you can’t randomly assign participants to the depression and control groups. But you can observe participants with and without depression and see how their task performance differs.

Observational studies let you perform research when you can’t control the treatment. However, quasi-experimental designs increase the problem of confounding variables. For this design of experiments, correlation does not necessarily imply causation. While special procedures can help control confounders in an observational study, you’re ultimately less confident that the results represent causal findings.

Learn more about Observational Studies .

For a good comparison, learn about the differences and tradeoffs between Observational Studies and Randomized Experiments .

Between-Subjects vs. Within-Subjects Experimental Designs

When you think of the design of experiments, you probably picture a treatment and control group. Researchers assign participants to only one of these groups, so each group contains entirely different subjects than the other groups. Analysts compare the groups at the end of the experiment. Statisticians refer to this method as a between-subjects, or independent measures, experimental design.

In a between-subjects design , you can have more than one treatment group, but each subject is exposed to only one condition, the control group or one of the treatment groups.

A potential downside to this approach is that differences between groups at the beginning can affect the results at the end. As you’ve read earlier, random assignment can reduce those differences, but it is imperfect. There will always be some variability between the groups.

In a  within-subjects experimental design , also known as repeated measures, subjects experience all treatment conditions and are measured for each. Each subject acts as their own control, which reduces variability and increases the statistical power to detect effects.

In this experimental design, you minimize pre-existing differences between the experimental conditions because they all contain the same subjects. However, the order of treatments can affect the results. Beware of practice and fatigue effects. Learn more about Repeated Measures Designs .

Design of Experiments Examples

For example, a bone density study has three experimental groups—a control group, a stretching exercise group, and a jumping exercise group.

In a between-subjects experimental design, scientists randomly assign each participant to one of the three groups.

In a within-subjects design, all subjects experience the three conditions sequentially while the researchers measure bone density repeatedly. The procedure can switch the order of treatments for the participants to help reduce order effects.

Matched Pairs Experimental Design

A matched pairs experimental design is a between-subjects study that uses pairs of similar subjects. Researchers use this approach to reduce pre-existing differences between experimental groups. It’s yet another design of experiments method for reducing sources of variability.

Researchers identify variables likely to affect the outcome, such as demographics. When they pick a subject with a set of characteristics, they try to locate another participant with similar attributes to create a matched pair. Scientists randomly assign one member of a pair to the treatment group and the other to the control group.

On the plus side, this process creates two similar groups, and it doesn’t create treatment order effects. While matched pairs do not produce the perfectly matched groups of a within-subjects design (which uses the same subjects in all conditions), it aims to reduce variability between groups relative to a between-subjects study.

On the downside, finding matched pairs is very time-consuming. Additionally, if one member of a matched pair drops out, the other subject must leave the study too.

Learn more about Matched Pairs Design: Uses & Examples .

Another consideration is whether you’ll use a cross-sectional design (one point in time) or use a longitudinal study to track changes over time .

A case study is a research method that often serves as a precursor to a more rigorous experimental design by identifying research questions, variables, and hypotheses to test. Learn more about What is a Case Study? Definition & Examples .

In conclusion, the design of experiments is extremely sensitive to subject area concerns and the time and resources available to the researchers. Developing a suitable experimental design requires balancing a multitude of considerations. A successful design is necessary to obtain trustworthy answers to your research question and to have a reasonable chance of detecting treatment effects when they exist.

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Methodology

• Guide to Experimental Design | Overview, Steps, & Examples

Guide to Experimental Design | Overview, 5 steps & Examples

Published on December 3, 2019 by Rebecca Bevans . Revised on June 21, 2023.

Experiments are used to study causal relationships . You manipulate one or more independent variables and measure their effect on one or more dependent variables.

Experimental design create a set of procedures to systematically test a hypothesis . A good experimental design requires a strong understanding of the system you are studying.

There are five key steps in designing an experiment:

• Consider your variables and how they are related
• Write a specific, testable hypothesis
• Design experimental treatments to manipulate your independent variable
• Assign subjects to groups, either between-subjects or within-subjects
• Plan how you will measure your dependent variable

For valid conclusions, you also need to select a representative sample and control any  extraneous variables that might influence your results. If random assignment of participants to control and treatment groups is impossible, unethical, or highly difficult, consider an observational study instead. This minimizes several types of research bias, particularly sampling bias , survivorship bias , and attrition bias as time passes.

Table of contents

Step 1: define your variables, step 2: write your hypothesis, step 3: design your experimental treatments, step 4: assign your subjects to treatment groups, step 5: measure your dependent variable, other interesting articles, frequently asked questions about experiments.

You should begin with a specific research question . We will work with two research question examples, one from health sciences and one from ecology:

To translate your research question into an experimental hypothesis, you need to define the main variables and make predictions about how they are related.

Start by simply listing the independent and dependent variables .

Then you need to think about possible extraneous and confounding variables and consider how you might control  them in your experiment.

Finally, you can put these variables together into a diagram. Use arrows to show the possible relationships between variables and include signs to show the expected direction of the relationships.

Here we predict that increasing temperature will increase soil respiration and decrease soil moisture, while decreasing soil moisture will lead to decreased soil respiration.

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Now that you have a strong conceptual understanding of the system you are studying, you should be able to write a specific, testable hypothesis that addresses your research question.

The next steps will describe how to design a controlled experiment . In a controlled experiment, you must be able to:

• Systematically and precisely manipulate the independent variable(s).
• Precisely measure the dependent variable(s).
• Control any potential confounding variables.

If your study system doesn’t match these criteria, there are other types of research you can use to answer your research question.

How you manipulate the independent variable can affect the experiment’s external validity – that is, the extent to which the results can be generalized and applied to the broader world.

First, you may need to decide how widely to vary your independent variable.

• just slightly above the natural range for your study region.
• over a wider range of temperatures to mimic future warming.
• over an extreme range that is beyond any possible natural variation.

Second, you may need to choose how finely to vary your independent variable. Sometimes this choice is made for you by your experimental system, but often you will need to decide, and this will affect how much you can infer from your results.

• a categorical variable : either as binary (yes/no) or as levels of a factor (no phone use, low phone use, high phone use).
• a continuous variable (minutes of phone use measured every night).

How you apply your experimental treatments to your test subjects is crucial for obtaining valid and reliable results.

First, you need to consider the study size : how many individuals will be included in the experiment? In general, the more subjects you include, the greater your experiment’s statistical power , which determines how much confidence you can have in your results.

Then you need to randomly assign your subjects to treatment groups . Each group receives a different level of the treatment (e.g. no phone use, low phone use, high phone use).

You should also include a control group , which receives no treatment. The control group tells us what would have happened to your test subjects without any experimental intervention.

When assigning your subjects to groups, there are two main choices you need to make:

• A completely randomized design vs a randomized block design .
• A between-subjects design vs a within-subjects design .

Randomization

An experiment can be completely randomized or randomized within blocks (aka strata):

• In a completely randomized design , every subject is assigned to a treatment group at random.
• In a randomized block design (aka stratified random design), subjects are first grouped according to a characteristic they share, and then randomly assigned to treatments within those groups.

Sometimes randomization isn’t practical or ethical , so researchers create partially-random or even non-random designs. An experimental design where treatments aren’t randomly assigned is called a quasi-experimental design .

Between-subjects vs. within-subjects

In a between-subjects design (also known as an independent measures design or classic ANOVA design), individuals receive only one of the possible levels of an experimental treatment.

In medical or social research, you might also use matched pairs within your between-subjects design to make sure that each treatment group contains the same variety of test subjects in the same proportions.

In a within-subjects design (also known as a repeated measures design), every individual receives each of the experimental treatments consecutively, and their responses to each treatment are measured.

Within-subjects or repeated measures can also refer to an experimental design where an effect emerges over time, and individual responses are measured over time in order to measure this effect as it emerges.

Counterbalancing (randomizing or reversing the order of treatments among subjects) is often used in within-subjects designs to ensure that the order of treatment application doesn’t influence the results of the experiment.

Finally, you need to decide how you’ll collect data on your dependent variable outcomes. You should aim for reliable and valid measurements that minimize research bias or error.

Some variables, like temperature, can be objectively measured with scientific instruments. Others may need to be operationalized to turn them into measurable observations.

• Ask participants to record what time they go to sleep and get up each day.
• Ask participants to wear a sleep tracker.

How precisely you measure your dependent variable also affects the kinds of statistical analysis you can use on your data.

Experiments are always context-dependent, and a good experimental design will take into account all of the unique considerations of your study system to produce information that is both valid and relevant to your research question.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic

Experimental design means planning a set of procedures to investigate a relationship between variables . To design a controlled experiment, you need:

• A testable hypothesis
• At least one independent variable that can be precisely manipulated
• At least one dependent variable that can be precisely measured

When designing the experiment, you decide:

• How you will manipulate the variable(s)
• How you will control for any potential confounding variables
• How many subjects or samples will be included in the study
• How subjects will be assigned to treatment levels

Experimental design is essential to the internal and external validity of your experiment.

The key difference between observational studies and experimental designs is that a well-done observational study does not influence the responses of participants, while experiments do have some sort of treatment condition applied to at least some participants by random assignment .

A confounding variable , also called a confounder or confounding factor, is a third variable in a study examining a potential cause-and-effect relationship.

A confounding variable is related to both the supposed cause and the supposed effect of the study. It can be difficult to separate the true effect of the independent variable from the effect of the confounding variable.

In your research design , it’s important to identify potential confounding variables and plan how you will reduce their impact.

In a between-subjects design , every participant experiences only one condition, and researchers assess group differences between participants in various conditions.

In a within-subjects design , each participant experiences all conditions, and researchers test the same participants repeatedly for differences between conditions.

The word “between” means that you’re comparing different conditions between groups, while the word “within” means you’re comparing different conditions within the same group.

An experimental group, also known as a treatment group, receives the treatment whose effect researchers wish to study, whereas a control group does not. They should be identical in all other ways.

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How to create a nearly orthogonal experimental design in R?

Does anyone know an R package for nearly orthogonal designs?

I would like to create an experimental design, using 12 runs, up to 10 factors, and with mixed levels (e.g. a combination of 2 and 3 level factors). I would like to explore some nearly-orthogonal designs.

There are lots of packages for orthogonal fractional designs. For example, I have looked at the documentation for AlgDesign, planor, FrF2, support.CEs, DoE.Base.

In SAS, there exist a set of macros for creating orthogonal and nearly orthogonal fractional factorial experimental designs: http://support.sas.com/techsup/technote/ts723.html

Does anyone know if something similar exists in R? The CRAN task view does not mention nearly orthogonal designs. http://cran.r-project.org/web/views/ExperimentalDesign.html

Many thanks Tim

Update: The Federov algorithm implemented in AlgDesign optFederov lets you create nearly orthogonal designs for mixed factors, as shown in the documentation

• experiment-design
• mixed-model

• 1 $\begingroup$ Note, I asked this on stackoverflow and was voted off topic, for asking for a recommendation. Please let me know if the same holds here stackoverflow.com/questions/25056315/… $\endgroup$ –  psychonomics Commented Jul 31, 2014 at 12:51
• 2 $\begingroup$ Did you take a look at the task view for experimental design . If so, can you specify how this does not provide the desired information? $\endgroup$ –  Henrik Commented Jul 31, 2014 at 12:59
• $\begingroup$ Yes, asking for packages / code is off-topic here. You might ask on the r-help listserv. $\endgroup$ –  gung - Reinstate Monica Commented Jul 31, 2014 at 12:59
• 1 $\begingroup$ This question appears to be off-topic because it is about asking for r packages / if r can do certain things. $\endgroup$ –  gung - Reinstate Monica Commented Jul 31, 2014 at 13:01
• 1 $\begingroup$ With 12 runs and 10 factors, you can have at most one 3-level factor involved before you start confounding main effects. $\endgroup$ –  Russ Lenth Commented Jul 31, 2014 at 13:26

2 Answers 2

The example in the post below is a non-orthogonal fractional factorial design https://stackoverflow.com/questions/5044876/how-to-create-a-fractional-factorial-design-in-r

• $\begingroup$ D-optimal designs tend to be close to orthogonal, yes. $\endgroup$ –  kjetil b halvorsen ♦ Commented Mar 25, 2017 at 18:12

by going to suggested link i created below code and though it might be useful.

I have a question for you all though. I got below values for design efficiency D =0.2519353; A = 5.462121; Ge = 0.748; Dea = 0.714. Which value should we be looking at for D-Eficiency? I assume it's D and how much it should be in order for this design to be usable in a choice experiment as alternatives? is current D value of 0.2519353 good enough for use?

• 2 $\begingroup$ Thank you for sharing your solution. Your question, though, must be posted in a new thread for it to be answered. $\endgroup$ –  whuber ♦ Commented Feb 14, 2015 at 15:10
• $\begingroup$ Well, never mind. I found answer myself after posting this question. stats.stackexchange.com/questions/137695/… $\endgroup$ –  StatguyUser Commented Apr 10, 2015 at 21:02

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Design and Analysis of Experiments with R (Chapman & Hall/CRC Texts in Statistical Science) 1st Edition

Design and Analysis of Experiments with R presents a unified treatment of experimental designs and design concepts commonly used in practice. It connects the objectives of research to the type of experimental design required, describes the process of creating the design and collecting the data, shows how to perform the proper analysis of the data, and illustrates the interpretation of results.

Drawing on his many years of working in the pharmaceutical, agricultural, industrial chemicals, and machinery industries, the author teaches students how to:

• Make an appropriate design choice based on the objectives of a research project
• Create a design and perform an experiment
• Interpret the results of computer data analysis

The book emphasizes the connection among the experimental units, the way treatments are randomized to experimental units, and the proper error term for data analysis. R code is used to create and analyze all the example experiments. The code examples from the text are available for download on the author’s website, enabling students to duplicate all the designs and data analysis.

Intended for a one-semester or two-quarter course on experimental design, this text covers classical ideas in experimental design as well as the latest research topics. It gives students practical guidance on using R to analyze experimental data.

• ISBN-10 9781439868133
• ISBN-13 978-1439868133
• Edition 1st
• Publisher Chapman and Hall/CRC
• Publication date December 17, 2014
• Part of series Chapman & Hall/CRC Texts in Statistical Science
• Language English
• Dimensions 6.75 x 1.25 x 9.75 inches
• Print length 628 pages
• See all details

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Editorial Reviews

"This is an excellent but demanding text. … This book should be mandatory reading for anyone teaching a course in the statistical design of experiments. … reading this text is likely to influence their course for the better." ― MAA Reviews , March 2015

"Thank you for writing your phenomenal book "Design and Analysis of Experiments with R". I'm teaching a new course this spring on experimental design and reinforcement learning. The students are graduate bioengineers, so I was having difficulty finding a text that blends theory, practice, and computation. Your book excels at all three. The first chapter I read clarified several topics and improved both my teaching and research. After testing a dozen DOE and RSM books, yours is the clear winner. I understand the enormous time that goes into a well-constructed textbook. I hope this message conveys my deep appreciation for your effort." ― Paul Jensen , Ph.D., Assistant Professor , Department of Bioengineering and Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign

"In my opinion, this is a very valuable book. It covers the topics that I judge should be in such a book including what might be called the standard designs and more … it has become my go to text on experimental design."

David E. Booth, Technometrics

About the Author

John Lawson is a professor in the Department of Statistics at Brigham Young University.

Product details

• ASIN ‏ : ‎ 1439868131
• Publisher ‏ : ‎ Chapman and Hall/CRC; 1st edition (December 17, 2014)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 628 pages
• ISBN-10 ‏ : ‎ 9781439868133
• ISBN-13 ‏ : ‎ 978-1439868133
• Item Weight ‏ : ‎ 2.25 pounds
• Dimensions ‏ : ‎ 6.75 x 1.25 x 9.75 inches
• #93 in Statistics (Books)
• #327 in Probability & Statistics (Books)

About the author

John lawson.

John Lawson is a Professor Emeritus from the Statistics Department at Brigham Young University where he taught from 1986 to 2019. He is an ASQ-CQE and he has a Masters Degree in Statistics from Rutgers University and a PhD in Applied Statistics from the Polytechnic Institute of N.Y. He worked as a statistician for Johnson & Johnson Corporation from 1971 to 1976, and he worked at FMC Corporation Chemical Division from 1976 to 1986 where he was the Manager of Statistical Services. In industry he used designed experiments and statistical analysis to help engineers and chemists on product development and manufacturing process improvements. At BYU he taught courses on experimental design and quality control and consults with faculty and graduate students involved in research projects through the BYU Center for Statistical Consultation and Collaborative Research. He is the the co-author (with John Erjavec) of Basic Experimental Strategies and Data Analysis for Science and Engineering, CRC Press, the author of Design and Analysis of Experiments with R, CRC Press, and the author of An Introduction to Acceptance Sampling and SPC with R, CRC Press. Additional resources for these books, such as electronic versions of computer code in the books, lecture slides, etc. can be downloaded from: https://lawsonjsl7.netlify.app/webbook/. Additional resources for an earlier book Design and Analysis of Experiments with SAS, CRC Press can be downloaded from: https://sasbook.netlify.com

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