design and analysis of experiments 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.

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Design and Analysis of Experiments with R

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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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Some basic concepts about design of experiments and how to perform their analysis in r.

Posted on January 15, 2022 by R in the Lab in R bloggers | 0 Comments

Basic design of experiments in R for one factor and two factors designs.

You can find all the code, data and results in the GitHub repository for this post: Basic design of experiments .

There is no signal without noise

It never hurts to go back to basics before tackling more complex things. The purpose of this post is to give a brief overview of the basics of design of experiments, their analysis and how to present results using R and packages like ggplot2 and agricolae . Included are one- and two-factor experiments.

What is the design of experiments?

The design of experiments (DOE) deals with the planning and performance of tests with the objective of generating data. Statistical analysis of these data will provide objective evidence that will allow the researcher to resolve questions about a given situation, process or phenomenon.

DOE can be applied to scientific research and in industry. Its goal is to generate knowledge and learning in an efficient manner. Ideally, this process can be considered as a cycle where initial hypotheses are refined as more information is obtained. Thus, we start with an initial hypothesis that we test through experimentation. If the data (our evidence) does not agree with the hypothesis, a new hypothesis is sought to explain the observed discrepancy.

Inductive-deductive loop – Pamela Toman

Hypothesis?

Hypotheses are tentative explanations of the research phenomenon that are stated as propositions or assertions. There are different types of hypotheses, such as:

• Research hypotheses. Tentative propositions about possible relationships between two or more variables.
• Null hypothesis. Propositions that deny or refute the relationship between variables.
• Alternative hypotheses. These are different or “alternate” possibilities to the research and null hypotheses.

Null hypothesis

Data and measurements

A datum is a number that is the product of a measurement. A measurement is a process that links abstract concepts with empirical indicators. Of course, measurements could not be made without a measurement instrument, which can be defined as the resource used by the researcher to record information or data.

Measuring absorbance

Other basic definitions in the design of experiments

It is a change in the operating conditions of a system or process, which is made with the objective of measuring the effect of the change on one or more properties of the product or result.

Schematic view of experiment design

Controlled experiment

Experimental unit

Piece(s) or sample(s) used to generate a value that is representative of the test result.

Response variable (Depedent variable)

Through these variable(s) we know the effect or results of each experimental test. They are the product of our measurements once we make changes in the system.

Controllable factors (Independent variable)

They are process variables that can be modified and fixed at a given level.

Response and independent variables

Non-controllable factors

These are variables that cannot be controlled during the experiment or the normal operation of the process.

Factors studied

These are the variables that are investigated in the experiment to observe how they affect or influence the response variables.

Levels and treatments

The different values assigned to each factor studied in an experimental design are called levels. A combination of levels of all the factors studied is called a treatment or design point. In the case of experimenting with a single factor, each level is a treatment.

Design matrix

It is the arrangement formed by the treatments that will be carried out, including the repetitions. In this same array are usually included the results of each response variable for each treatment and repetition. For me this is a concept of great importance, since from this matrix it is possible to perform directly the different statistical tests in R.

Random error

It is the observed variability that cannot be explained by the factors studied. It is the result of the effect of factors not studied and experimental error.

Experimental error

A component of random error that reflects the experimenter’s errors in the planning and execution of the experiment.

Basic principles of experimental design

• Randomization. Performing the experimental runs (treatments) in a randomized manner is essential to ensure, as far as possible, the assumption of independence of errors.
• Repetition. It is running a treatment or combination of factors more than once. This makes it possible to estimate random error and in turn calculate realistic statistics in data analysis.
• Blocking. This consists of nullifying the effect of factors in which we are not really interested and which may affect the response variables. Not taking into account their effect can lead us to erroneous conclusions about the factors in which we are interested.

Block example: Male and females

Some concepts of statistical inference

• Population. In a very general way, it can be defined as the totality of possible individuals, specimens, objects or measurements of interest on which a study is made. Populations can be finite or infinite.
• Parameters. Characteristics that, by means of their numerical value, describe the entire population.
• Representative sample. An appropriately selected part of a population that retains key aspects of a population.
• Statistic. A quantity obtained from the data of a sample that summarizes the characteristics of the sample.
• Statistical inference. These are statistical statements about the population or process based on the information contained in the sample.
• Probability distribution. It relates the set of values of a characteristic in a given population to the probability associated with each of these values.

Statistical inference

Probability distribution

Some hypothesis testing concepts

• Statistical hypothesis. An assertion about the values of the parameters of a population or process, which can be tested from the information in a sample.
• Test statistic. Formula with which a number is calculated from the data, the magnitude of which makes it possible to determine whether or not the null hypothesis is rejected.
• Acceptance region. These are the possible values of the test statistic where the null hypothesis is not rejected.
• Rejection region. It is the set of possible values of the test statistic that lead to rejecting the null hypothesis.
• Type I error. It is when a null hypothesis that is true is rejected. It can also be called false positive.
• Type II error. This is when a null hypothesis that is false is accepted. It is basically a false negative.
• Power of the test. It is the probability of rejecting the null hypothesis when it is false.
• Predefined significance. It is the maximum risk that the experimenter is willing to take with respect to the type I error. It is denoted by α.
• Observed significance. It is the area under the reference distribution beyond the value of the test statistic. It is called p-value.

Hypothesis testing

Experiments with a single factor

When the objective is to study the effect of a single factor on the mean of a response variable taking into account more than two values of the factor under consideration, it is advisable to perform a completely randomized experiment and analyze the results by analysis of variance (ANOVA). The reason for comparing means by ANOVA and not by Student’s t-tests is that the latter test increases false positives as the number of means to be compared increases, i.e., we will find differences between means where there are none.

The analysis of variance consists of separating the total variation observed in each of the sources that contribute to it.

Design and results

Obtaining a single factor design is very simple using R commands. Let’s say we want to compare the effect of three baking times (35, 45 and 60 min) in the average diameter of cookie batches. For each time we baked ten cookie batches so we have ten replicates. Note that each time and replicate have to be run in aleatory order:

Subsequently, the design can be exported in some other format such as CSV:

The results can be integrated into the design matrix directly, let’s simulate the average diameter with a small function:

Or, once the results have been recorded externally, they can be imported from a CSV file:

The aov function is used for the analysis of factorial designs. Previously, it is recommended to convert the factor values into the factor class and then perform the analysis. Here I going to analyze our first design ( one_fct_dgn ):

Note the aov requires a formula that explicitly specifies the relationship between the response and the factor(s) in the design. Once this has been done, it is possible to obtain an ANOVA table using the “anova” function:

And export it in the format of your choice:

Multiple range comparisons or tests

Once significant differences are established between at least two of the means, the next step is to perform a multiple comparison test, which will allow us to define which means are significantly different. There are several methods that can be used as the LSD method, Tukey’s method, or Duncan’s method. Nice functions for these methods can be found in agricolae package.

The results are consistent among the three methods. The section groups indicates which means are different, different letters mean significant differences. You can export the results of this kind of methods, as a TXT file, with the function capture.output . For example, to export the results of the Tukey method:

Verification of model assumptions

Something that is not very widespread in the scientific literature is the verification of the normality of the residuals, the equality between variances for the means of each treatment and the independence between each data recorded. Verification of these assumptions helps to validate the differences established as significant.

Shapiro-Wilks test to verify normality

This test is easy to perform:

Here, if p-value > 0.05, we can say that data come, approximately, from a normal distribution.

The results of this analysis can be exported with the capture.output function:

Bartlett’s test for equality of variances

To perform this test it is possible to use the bartlett.test function:

A p-value > 0.05 mean that treatment means have equal variances. We can export the results in the same way:

Independence

To verify independence between observations, residuals are plotted as a function of run order. If the residuals follow a defined pattern it will be a clear indication of lack of independence. For this plot I used the ggplot2 package:

Tests such as the Durbin-Watson test are also available to verify this assumption. Use the function durbinWatsonTest in the car package:

A p-value > 0.05 means that data observations are not auto correlated. You can also save the results in the usual way:

Display of results

To visualize the results of this type of experiments, bar graphs are usually used, whose height represents the magnitude of each mean, together with error bars representing the standard deviation and letters showing the significant differences established by the multiple comparisons test:

Alternative

Although bar graphs are widely used in the scientific literature of various types, their use hides the true dispersion of the data for each treatment. One thing I have also noticed is that people tend to judge differences between means by the overlap or separation of standard deviations, which seems to me to be more of a bias than an objective criterion.

As an alternative to the bar chart, it is possible to make a graph showing each observation, the mean, and the letters obtained in the multiple comparisons test:

Two-factor designs

In this type of design the objective is to verify the individual and interaction effect of two factors. Before showing how to obtain and analyze them, it is necessary to recall some concepts:

• Qualitative factor. Its levels take discrete or nominal values.
• Quantitative factor. Its levels can take any value within a certain interval. The scale is continuous.
• Effect of a factor. It is the change observed in the response variable due to a change in the level of the factor.
• Interaction effect. Two factors interact significantly on the response variable when the effect of one depends on the level of the other.

To obtain this kind of designs we can use the function fac.design from DoE.base package. For this we need specify the number of factors and the number of levels of each factor:

For more details about this function, just type ?fac.design in your console. Also note that there is an extra column with the name “Blocks”, this is not big deal since there is just one and we won’t taking it into account further in the analysis. In future posts I will address block designs.

This design can also be exported in the desired format:

The results can also be integrated directly or imported once recorded in the design matrix. Here, I’m going to import and work with data previously recorded:

To analyze this type of design, the “aov” function is used, specifying each main effect and the interaction effect in the formula:

You can also use the short form Y ~ A*B , which will be displayed in full once you get the ANOVA table:

Note that previously I also converted the values of each factor in the factor class.

Multiple range comparisons tests

Once a significant interaction effect is established, the next step is to establish which means differ. For this, a Tukey test can be used, for which the interaction effect must first be explicitly specified in a separated aov model:

The results of Tukey’s test can be exported directly:

For the verification of the assumptions it is possible to proceed in a manner similar to that of the single-factor designs.

For Bartlett test I previously made an extra column specifying the combination between factors (treatments). This will indicate the groups to bartlett.test function:

The realization of this type of graphs requires keeping in mind the individual means of each treatment and the interaction effects, if there is some. To calculate them together with the standard deviations we used the “group_by” function and then “summarise” from the “dplyr” package:

To make the graph showing each observation, the mean, a confidence interval and the letters obtained in the multiple comparisons test, it is necessary to use the data.frame with the observations next to the one with the means:

A good way to visualize the interaction effect is through interaction.plot function:

That’s it! Thank you very much for visiting this site, I hope you find the content of this post useful. See you soon!

Juan Pablo Carreón Hidalgo 🤓

[email protected] https://twitter.com/JuanPa2601

The text and code on this tutorial is under Creative Commons Attribution 4.0 International License .

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An R package for Design and Analysis of Experiment

ehassler/MontgomeryDAE

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R package for montgomery's design and analysis of experiments 10th ed..

This repository is for Wiley's companion R package to Douglas C. Montgomery's "Design and Analysis of Experiments" 10th edition. The most current version of the PDF Guide will walk you through how to use R to work examples and exercises in the text.

The PDF guide also shows how to install the most current package . The package contains the content of the PDF in vignettes and also all of the data sets used in the book. It makes doing the exercises much easier.

One of the great features of github is that you can report bugs through the "Issues" tab. Please look there to see if any error you find has been reported, and if it has not then post a report of the error there.

Design and Analysis of Experiments and Observational Studies using R

A volume in the chapman & hall/crc texts in statistical science series.

Sample Course Documents

Sample course documents (i.e., slides, assignments, etc.) are available here .

scidesignR is an R package that contains the data used in the book.

1st Edition

Design and Analysis of Experiments with R

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Description

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.

Table of Contents

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

Critics' 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 "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

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Note that this "residual" for the within plot \(subplot\) part of the analysis is actually the sum of squares for the interaction of rows \(w\ hole plots\) with varieties \(subplot treatments\)---as in an RCBD.

- r_k\(i\) ~ N\(0, sigma^2_r\)

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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

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
• #748 in Statistics (Books)
• #2,159 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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• Open access
• Published: 17 September 2024

Improving rigor and reproducibility in western blot experiments with the blotRig analysis

• Cleopa Omondi 1 ,
• Austin Chou 1 ,
• Kenneth A. Fond 1 ,
• Kazuhito Morioka 1 ,
• Nadine R. Joseph 1 ,
• Jeffrey A. Sacramento 1 ,
• Emma Iorio 1 ,
• Abel Torres-Espin 1 , 3 , 4 ,
• Hannah L. Radabaugh 1 ,
• Jacob A. Davis 1 ,
• Jason H. Gumbel 1 ,
• J. Russell Huie 1 , 2 &
• Adam R. Ferguson 1 , 2  

Scientific Reports volume  14 , Article number:  21644 ( 2024 ) Cite this article

Metrics details

• Biochemistry
• Biological techniques
• Computational biology and bioinformatics
• Neuroscience

Western blot is a popular biomolecular analysis method for measuring the relative quantities of independent proteins in complex biological samples. However, variability in quantitative western blot data analysis poses a challenge in designing reproducible experiments. The lack of rigorous quantitative approaches in current western blot statistical methodology may result in irreproducible inferences. Here we describe best practices for the design and analysis of western blot experiments, with examples and demonstrations of how different analytical approaches can lead to widely varying outcomes. To facilitate best practices, we have developed the blotRig tool for designing and analyzing western blot experiments to improve their rigor and reproducibility. The blotRig application includes functions for counterbalancing experimental design by lane position, batch management across gels, and analytics with covariates and random effects.

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Introduction.

Proteomic technologies such as protein measurement with folin phenol reagent were first introduced by Lowry et al. in 1951 1 . The resulting qualitative data are typically confirmed by a second, independent method such as western blot (WB) 2 , 3 . The WB method, first described by Towbin et al. 4 and Burnette 5 in 1979 and 1981, respectively, uses specific antibody-antigen interactions to confirm the protein present in the sample mixture. Quantitative WB (qWB assay) is a technique to measure protein concentrations in biological samples with four main steps: (1) protein separation by size, (2) protein transfer to a solid support, (3) marking a target protein using proper primary and secondary antibodies for visualization, and (4) semi -quantitative analysis 6 . Importantly, qWB data is considered semi -quantitative because methods to control for experimental variability ultimately yield relative comparisons of protein levels rather than absolute protein concentrations 2 , 3 , 7 , 8 . Similarly, western blotting applying ECL (enhanced chemiluminescence) is considered a semi-quantitative method because it lacks cumulative luminescence linearity and offers limited quantitative reproducibility 9 . However, the emergence of highly sensitive fluorescent labeling techniques, which exhibit a wider quantifiable linear range, greater sensitivity, and improved stability when compared to the conventional ECL detection method, now permits the legitimate characterization of protein expression as linearly quantitative 10 . Current methodologies do not sufficiently account for diverse sources of variability, producing highly variable results between different laboratories and even within the same lab 11 , 12 , 13 . Indeed, qWB data exhibits more variability compared to other experimental techniques such as enzyme linked immunosorbent assay (ELISA) 14 . For example, results have shown that qWB can produce significant variability in detecting host cell proteins and lead to researchers missing or overestimating true biological effects 15 . This in turn results in publication of irreproducible qWB interpretations, which leads to loss of its credibility 13 . In the serious cases, qWB results may even provide clinical misdiagnosis 16 that could impact on a larger public health concern due to the prevalence of WB in biomedical research, such as diagnosis of SARS-CoV2 infection 17 .

The process of recognizing and accounting for variability in WB analyses will ultimately improve reproducibility between experiments. A growing body of studies has shown that this requires a fundamental shift in the experimental methodology across data acquisition, analysis, and interpretation to achieve precise and accurate results 2 , 3 , 11 , 12 , 13 .

Here we highlight experimental design practices that enable a statistics-driven approach to improve the reproducibility of qWBs. Specifically, we discuss major sources of variability in qWB including the non-linearity in antibody signal 2 , 3 ; imbalanced experimental design 13 ; lack of standardization in the treatment of technical replicates 3 , 18 ; and variability between protein loading, lanes, and blots 2 , 7 , 19 . To address these issues, we provide new comprehensive suggestions for quantitative evaluation of protein expression by combining linear range characterization for antibodies, appropriate counterbalancing during gel loading, running technical replicates across multiple gels, and by taking careful consideration of the analysis method. By applying these experimental practices, we can then account for more sources of variability by running analysis of covariance (ANCOVA) or generalized linear mixed models (LMM). Such approaches have been shown to successfully improve reproducibility compared to other methods 13 .

Good options for qWB protein bands analysis using free, downloadable tools are available for researchers. Amongst others, LI-COR Image Studio Lite can be used to measure the intensity of protein bands in western blots and calculate their relative abundance . Likewise, ThermoFisher ImageQuant Lite offers features such as the ability to perform background subtraction and normalization. However, to date, no specific tools are freely available to provide a map to counterbalance samples, which overcome imperfect uniform protein electrophoresis/transfer and perform statistical analysis. Here, we present blotRig, a tool for researchers with functionalities to counterbalance samples and perform statistical analysis.

To help improve WB rigor we developed the blotRig protocol and application harnessing a database of 6000 + western blots from N = 281 subjects (rats and mice) collected by multiple UCSF labs on core equipment. To demonstrate blotRig best practices in a real-world experiment, we carried out prospective multiplexed WB analysis of protein lysate from lumbar cord in rodent models of spinal cord injury (SCI) (N = 29 rats) in 2 groups (experimental group & control group). In order to show that these experimental suggestions could improve qWB reproducibility, we compared different statistical approaches to handling loading controls and technical replicates. Specifically, we applied two strategies to integrate loading controls: (i) normalizing the target protein levels by dividing by the loading control or (ii) treating the loading control as a covariate in a LMM. Additionally, we analyzed technical replicates in four ways: (1) assume each sample was only run once without replication, (2) treat each technical replicate as an independent sample, (3) use the mean of the three technical replicate values, and 4) treat the replicate as a random effect in a LMM. Altogether, we found that the statistical power of the experiment was significantly increased when we used loading control as a covariate with technical replicates as a random effect during analysis. In addition, the effect size was increased, and the p-value of our analysis decreased when using this LMM, suggesting the potential for greater sensitivity in our WB experiment when using this approach 20 . Through rigorous experimental design and statistical analysis we show that we can account for greater variability in the data and more clearly identify underlying biological effects.

Materials and methods

All experiments protocol were approved by the University Laboratory Animal Care Committee at University of California, San Francisco (UCSF, CA, USA) and followed the animal guidelines of the National Institutes of Health Guide for the Care and Use of Laboratory animals (National Research Council (US) Committee for the Update of the Guide for the Care and Use of Laboratory Animals, 2011). We followed The ARRIVE guidelines (Animal Research: Reporting In Vivo Experiments) to describe our in vivo experiments.

Male Simonsen Long Evans rats (188–385 g; Gilroy (Santa Clara, CA, USA), (N = 29) aged 3 weeks were housed under standard conditions with a 12-h light–dark cycle (6:30 am to 6:30 pm) and were given food and water ad libitum. The animals were housed mostly in pairs in 30 × 30 × 19-cm isolator cages with solid floors covered with a 3 cm layer of wood chip bedding. The experimenters were blind to the identity of treatments and experimental conditions, and all experiments were designed to minimize suffering and limit the number of animals required.

Anesthesia and surgery

We performed non-survival spinal cord injury and spared nerve injury surgeries on animals. Specifically, 3 week old female rats were anesthetized with continuous inhalation of isoflurane (1–5% mg/kg) while on oxygen (0.6–1 mg/kg) in accordance with the IACUC surgical and anesthesia guidelines. Preoperative 0.5% lidocaine local infiltration was applied once at surgical site, avoiding injection into muscle. Fur over the T7–T9 thoracic level was shaved. The dorsal skin was aseptically prepared with surgical iodine or chlorhexidine and 70% ethanol. A small longitudinal incision was made along the spine through the skin, fascia, and muscle to expose the T7-T9 vertebrae. Animals undergoing sham procedure did not undergo laminectomy and immediately proceeded to wound closure. Overlying muscle and subcutaneous tissue was sutured closed using an absorbable suture in a layered fashion. External skin was reinforced using monofilament suture or tissue glue as needed. Animals were euthanized after 30 min to extract spinal cord tissue through fluid expulsion.

Experimental methodology

In accordance with established quality standards for preclinical neurological research 21 , experimenters were kept blind to experimental group conditions throughout the entire study. Western blot loading order was determined a priori by a third-party coder, who ensured that a representative sample from each condition was included on each gel in a randomized block design. The number of subjects per condition was kept consistent across groups for each experiment to ensure that proper counterbalancing could be achieved across independent western runs. All representative western images presented in the figures represent lanes from the same gel. Sometimes, the analytical comparisons of interest were not available on adjacent lanes even though they come from the same gel because of our randomized counterbalancing procedure.

• Western blot

The example western blot data used in this paper are taken from a model of spared nerve injury in animals with spinal cord injury. The nerve injury model used is based on models from pain literature 22 , where two of the three branches of the sciatic nerve are transected, sparing the sural nerve (SNI) 23 . Two surgeons perform the procedure simultaneously, with injuries occurring 5 min apart. The spinal cord of animals was obtained based on fluid expulsion model 24 and a 1 cm section of the lumbar region was excised at the lumbar enlargement section. The tissue was then preserved in a -80 degree freezer until it was needed for an experiment, at which point it was thawed and used to run a Western blot. We conducted a Western blot analysis on 29 samples from animals using standard biochemical methods. We measured the protein levels of the AMPA receptor subunit GluA2 and used beta-actin as a loading control. The data from these experiments was then aggregated and used for statistical analysis.

Protein assay

We assayed sample protein concentration using a bicinchoninic acid (BCA assay (Pierce) for reliable quantification of total protein using a plate reader (Tecan; GeNios) with triplicate samples (technical replicates) detected against a Bradford Assay (BSA) standard curve. Technical replicates are multiple measurements that are performed under the same conditions in order to quantify and correct for technical variability and improve the accuracy and precision of the results (48). We ran the same WB loading scheme three times (technical replicates of the entire gel) and measured the protein levels of AMPA receptors.

Polyacrylamide gel electrophoresis and multiplexed near-infrared immunoblotting

The approach involved performing serial 1:2 dilutions with cold Laemmli sample buffer in room temperature; 15 μg of total protein per sample was loaded into separate lanes on a precast 10–20% electrophoresis gel (Tris–HCl polyacrylamide, BioRad) to establish linear range (Fig.  1 ). The blotRig software helps counterbalance sample positions across the gel by treatment condition. (Fig.  2 ). A kaleidoscope ladder was loaded on the first lane of each gel to confirm molecular weight (Fig.  2 ). The gel was electrophoresed for 30 min at 200 V in SDS buffer (25 mm Tris, 192 mm glycine, 0.1% SDS, pH 8.3; BioRad). Protein was transferred to a nitrocellulose membrane in cold transfer buffer (25 mm Tris, 192 mm glycine, 20% ethanol, pH 8.3). Membrane transfer was confirmed using Ponceau S stain (67) followed by a quick rinse and blocking in Odyssey blocking buffer (Li-Cor) containing Tween-20.

Determining linear range of antibodies to optimize parametric analysis of Western blot data. When small or large protein concentrations are loaded, there is often a possibility that their representation on western blot band density may become non-linear. If there is a disconnect between the observed and expected protein concentrations, results may be inaccurate. Thus determining the linear range wherein, a one-unit increase in protein is reflected in a linear increase in band density for each western blot antibody is a crucial initial step to ensure confidence in reproducibility of the linear models commonly applied to western blot data analysis.

Counterbalancing to reduce bias. ( A) Experimental design. A simple hypothetical experimental design for illustrating counterbalancing. Two experimental groups (Wild Type vs Transgenic), with two treatments (Drug vs Vehicle) analyzed within each individual. This 2 (Experimental Condition) by 2 (Tissue Area) design yields four groups. ( B) Counter-balanced Gel Loading. The goal of appropriate counterbalancing is to optimize the sequence in which samples are loaded such that groups are represented equally across the gel. Those with red X have with the experimental groups and treatment condition grouped in the same area of the gel, and thus variability across the gel may be conflated with group differences. In contrast, those with the green check are organized so that experimental condition and treatment condition are better placed to reduce the possibility of any single group being over-represented in a particular area of the gel.

The membrane was blocked for 1 h in Odyssey Blocking Buffer (Li-Cor) containing 0.1% Tween-20, followed by an overnight incubation in primary antibody solution at 4 °C. Membrane incubation was done in a primary antibody solution containing Odyssey blocking buffer, Tween-20, appropriate primary antibody receptor targeting1:2000 mouse PSD-95 (cat # MA1-046,Thermofisher), 1:200 rabbit GluA1 (cat # AB1504, Millipore), 1:200 rabbit GluA2 (cat # AB1766, Millipore), 1:200 rabbit pS831(cat # 04–823, Millipore), 1:200 p880 (cat#07–294, Millipore) or 1:1,500 mouse actin loading control (cat # 612,857, BD Transduction)]. Following incubation, the membrane was washed 4 × 5 min with Tris-buffered saline containing 0.1% Tween 20 (TTBS) and incubated in fluorescent-labeled secondary antibody (1:30 K LiCor IRdye appropriate goat anti-rabbit in Odyssey blocking buffer plus 0.2% Tween 20) for 1 h in the dark. Subsequent to 4 × 5 min washes in TTBS, followed by a 5 min wash in TBS.

Membrane incubation was used to detect the presence of a specific protein or antigen on a membrane. In this case, the membrane was incubated with a fluorescently labeled secondary antibody solution that was specifically tuned to the emission spectra of the laser lines used by the Li-Cor Odyssey quantitative near-infrared molecular imaging system instrument. This allows for specific detection of the protein of interest on the membrane. The sample is then imaged using an infrared imaging system that is optimized for detecting the specific wavelengths of light emitted by the fluorescent label. Additional rounds of incubation and imaging are performed to detect additional proteins using the multiplexing functionality of the Li-Cor instrument, with each round adding new bands at different molecular weight ranges. This allows for the detection of multiple proteins in the same sample, maximizing the proteomic detection.

Quantitative near-IR densitometric analysis

Using techniques optimized in the our lab 25 , 26 , we established near-infrared labeling and detection techniques (Odyssey Infrared Imaging System, Li-Cor) to quantify linear intensity detection of fluorescently labeled protein bands. The biochemistry is performed in a blinded, counterbalanced fashion, and three independent replications of the assay are run on different days 27 . Fluorescent Western blotting utilizes fluorescent-labeled secondary antibodies to detect the target protein, which allows for more sensitive and specific detection compared to chemiluminescence 11 , 28 , 29 . Additionally, fluorescence imaging allows multiple detection of a target protein and internal loading control in the same blot, which enables more accurate correction of sample-to-sample and lane-to-lane variation 11 , 30 , 31 . This provides a more accurate and reliable quantification of the target protein, making it a popular choice for quantitative analysis of WB data.

It is good practice for the pipetting experimenter to remain blind to experimental conditions during gel loading, transfer, and densitometric quantification. We achieved this using de-identified tube codes and a priori gel loading sequences that were developed by an outside experimenter using the method implemented in the blotRig software.

Statistical analyses

Statistical analyses were performed using the R statistical software. Our WB data was analyzed using parametric statistics. The WB was run using three independent replications and covariance corrected by beta-actin loading control, with replication statistically controlled as a random factor. Significance was assessed at p < 0.05 25 , 26 , 32 , 33 , 34 . We report estimated statistical power and standardized regression coefficient effect sizes in the results section.

All ANOVAs were run using the stats R package; standardized effect size was calculated using the parameters R package 35 . Linear mixed models were run using the lme4 R package. Observed power was calculated by Monte Carlo simulation (1000x) run on the fitted model (either ANOVA or LMM) using the simR package 36 . For the development of the blotRig interface, the R packages used included: shiny, tidyverse, DT, shinythemes, shinyjs, and sortable ) 37 , 38 , 39 , 40 , 41 , 42 . You can access the blotRig analysis software, which includes code for inputting experimental parameters for all Western blot analysis, through the following link: https://atpspin.shinyapps.io/BlotRig/.

Designing reproducible western blot experiments

Determining linear range for each primary antibody.

Most WB analyses assume semi -quantitatively that the relationship between qWB assay optical density data (i.e. western band signal) and protein abundance is linear 2 , 3 , 11 , 18 . Accordingly, most qWB analyses use statistical tests (t-test; ANOVA) that assume a linear effect. However, recent studies have shown that the relationship can potentially be highly non-linear 19 As Fig.  1 illustrates, the WB band signal can become non-linearly correlated with protein concentrations at low and high values. This may result in inaccurate quantification of relative target protein amount in the experiment and violates the assumptions for linear model which can lead to false inferences. To address the assumption of linearity, it is important to first determine the optimal linear range for each protein of interest so that one can be confident that a unit change in band density reflects a linear change in protein concentration. This enables an experimenter to accurately quantify the protein of interest and apply linear statistical methods appropriately for hypothesis testing.

Counterbalancing during experimental design

Counterbalancing is the practice of having each experimental condition represented on each gel and evenly distributing them to prevent overrepresentation of the same experimental groups in consecutive lanes. For example, imagine an experimental design in which we are studying two experimental groups (wild type and transgenic animals) and are also looking at two treatment conditions (Drug and Vehicle). The best way to determine the effects and interactions between our experimental and treatment groups would be to create a balanced factorial design. A factorial design is one in which all combinations of levels across factors are represented. For the current example, a balanced factorial design would produce four groups, covering each possible combination (Drug-treated Wild Type, Vehicle-treated Wild Type, Drug-Treated Transgenic and Vehicle-treated Transgenic) (Fig.  2 A). During WB gel loading, experimenters often distribute their samples unevenly such that certain experimental conditions may be missing on some gels or samples from the same experimental condition are loaded adjacently on a gel. This is problematic because we know that polyacrylamide gel electrophoresis (PAGE) gels are not perfectly uniform, reflecting a source of technical variability 43 ; in the worst case, if we have only loaded a single experimental group on a gel and found a significant effect of the group, we cannot conclude if the effect is due to the experimental condition or a technical problem of the gel. At minimum, experimenters should ensure that every group in a factorial design is represented on each gel to avoid confounding technical gel effects with experimental differences. If the number of combinations is too large to represent on a single gel because of the number of factors or the number of levels of the factors, then a smaller "fractional factorial" design will provide maximal counterbalancing to ensure unbiased estimates of all factor effects and the most important interactions.

In addition, experimenters can further counter technical variability by arranging experimental groups on each gel to ensure adequately counterbalanced design assuming the uniformed protein concentration and fluid volume of all samples. This importantly addresses the variability due to physical effects within an individual gel. In our example, this means alternating the tissue areas and experimental conditions as much as possible to minimize similar samples from being loaded next to one another (Fig.  2 B). By spreading the possibility of technical variability across all samples by counterbalancing across and within gels, we can mitigate potential technical effects that can bias our results. Proper counterbalancing also enables us to implement more rigorous statistical analysis to account for and remove more technical variability 25 , 26 , 32 , 33 . Overall, this will help to ensure that experimenters can find the same result in the future and improve reproducibility.

Technical replication

Technical replicates are used to measure the precision of an assay or method by repeating the measurement of the same sample multiple times. The results of these replicates can then be used to calculate the variability and error of the assay or method 13 . This is important to establish the reliability and accuracy of the results. Most experimenters acknowledge the importance of running technical replicates to avoid false positives and negatives due to technical error 13 . Even beyond extreme results, technical replicates can account for the differences in gel makeup, human variability in gel loading, and potential procedural discrepancies. In fact, most studies run at least duplicates; however, the experimental implementation of replicates (e.g., running replicates on the same gel or separate gels) as well as the statistical analysis of replicates (e.g., dropping “odd-man-out” or taking the mean or standard deviation) can differ greatly 44 , 45 . This experimental variability ultimately impedes our ability to meaningfully compare results. For experimenters to establish accuracy and advance reproducibility in WB experiments, it is important to implement standardized and rigorous protocols to handle technical replicates 11 , 13 . In doing so, we can further reduce the technical variability with statistical methods during analysis.

As underscored previously, we recommend that technical replicates are counterbalanced on separate gels to mitigate any possible gel effect. Additionally, by running triplicates, we can treat replicates as a random effect in a LMM during statistical analysis. Importantly, triplicates provide more values to measure the distribution of technical variance to ensure the robustness of the LMM than only running duplicates. This approach isolates and removes technical variance from biological variation which ultimately improves our sensitivity for true experimental effects 46 .

In the following demonstration of statistical methods, we replicated all WB analyses in triplicate with a randomized counterbalanced design. We then explore how the way in which technical replicates and loading controls are incorporated into analysis can have a significant impact on both the sensitivity of our results and the interpretation of the findings. An example mockup of a dataset illustrating the various ways in which western blot data are typically prepared for analysis can be found in Fig.  3 .

Western Blot Gel and Replication Strategies. ( A) Illustration of Western Blot Gel. This depiction of a typical multiplexed western blot gel highlights the antibody-labeled target protein bands of interest (green/yellow) and housekeeping protein loading control that is always run and quantified in the same sample and lane as the target of interest. Total protein stain (fluorescent ponceau stain) is shown in red can can be used as an alternative loading control. Specific, quantification is typically executed on a single antibody-labeled channel for the target protein and housekeeping protein loading control (gray scale image). ( B) Balanced Factorial Technical Replicate Strategy. Here we show the western blot data for the first 3 subjects from an example dataset. In a balanced factorial design, an equal number of samples from all possible experimental groups are represented on each gel. This table shows the subject number, the technical replicate, experimental group, and the band quantifications for both the target protein and the loading control. A ratio of target protein and loading control is also calculated. ( C) Other Common Technical Replicate Strategies. In this example table are two of the other ways western blot data are typically formatted. Some experimenters choose to not include technical replicates, with only one sample from each subject quantified. In another replication strategy, technical replicates are averaged. Averaging may bias or skew the data. We recommend running technical replicates on separate gels or batches, and using gel/batch as a random factor when analyzing western blot data.

Statistical methodology to improve western blot analysis

Loading control as a covariate.

Most qWB assay studies use loading controls (either a housekeeping protein or total protein within lane) to ensure that there are no biases in total protein loaded in a particular lane 2 , 11 , 27 . The most common way that loading controls are used to account for variability between lanes is by normalizing the target protein expression values by dividing it by the loading control values (Fig.  3 ) , resulting in a ratio between target protein to loading control 2 , 47 , 48 . However, ratios may violate assumptions of common statistical test used to analyze qWB (e.g., t-test, ANOVA, etc.) 49 This ultimately hinders the ability to statistically account for the variance in qWB outcomes and have a reliable estimate of the statistics. An alternative approach to improve the parametric properties would be to include loading control values as a covariate—a variable that is not our experimental factors but that may affect the outcome of interest and presents a source of variance that we may account for 50 . For instance, we know the amount of protein loaded is a source of variability in WB quantification, so we can use the loading control as a covariate to adjust for that variance. In doing so, we extend the method of ANOVA into that of ANCOVA 51 . This approach accounts for the technical variability present between lanes while meeting the necessary assumptions for parametric statistics which helps curb bias and averts false discoveries.

Replication and subject as a random effect

Most WB studies use ANOVA, a test that allows comparison of the means of three or more independent samples, for quantitative analysis of WB data 49 . One of the assumptions in ANOVA is the independence of observations 49 . This is problematic because we often collect multiple observations from the same analytical unit, for example different tissue samples from a single subject, or technical replicates. As a result, those observations don’t qualify as independent and should be analyzed using models controlling for variability within units of observations (e.g., the animal) to mitigate inferential errors (false positives and negatives) 52 caused by what is known as pseudoreplication. This arises when the quantity of measured values or data points surpasses the number of actual replicates, and the statistical analysis treats all data points as independent, resulting in their full contribution to the final result 53 .

In addition, when conducting experiments, it is important to consider the randomness of the conditions being observed. Treating both subjects and conditions as fixed effects can lead to inaccurate p-values. Instead, subjects/ animals should be treated as random effects and the conditions should be considered as a sample from a larger population 54 . This is especially important when collecting data from different replicates or gels, as the separate technical replicate runs should be considered as random.

In Fig.  4 we use a simple experimental design comparing the difference in a target protein between two experimental groups to demonstrate four of the most common ways researchers tend to analyze western blot data: (1) running each sample once without replication, (2) treating each technical replicate as an independent sample, (3) taking the mean of technical replicate values, and (4) treating subject and replication as a random effect (Fig.  4 ). We then tested how effect size, power, and p value are affected by each of these strategies to get a sense of how much these estimates vary between analyses. For each of these strategies, we also tested the difference between using the ratio of target protein to loading controls versus using loading control as a statistical covariate. For further exploration of the way these data are prepared and analyzed, see the data workup in Supplementary Figs.  1 and 2 .

Effect of different replication and loading control strategies on statistical outcomes. Eight possible strategies are shown, representing the most common ways in which replication and loading controls are treated in a typical Western blot analysis. Four replication strategies: either no replication at all, 3 technical replicate gels treated as independent, mean of three replicates, or replicate treated as a random effect in a linear mixed model. These are crossed with two loading control strategies: either target protein is divided by loading control, or loading control is treated as a covariate in a linear mixed model. ( A) Effect Size: Standardized effect size coefficient is generally improved when loading control is treated as a covariate, compared to using a ratio of the target protein and loading control values. ( B) Power: By treating each replication as independent the statistical power is increased (due to the inaccurate assumption that technical replicates are not related, thus artificially tripling the n). Conversely, including the variability inherent in technical replicates as a part of the statistical model, we work to identify and account for a major source of variability, thus improving power in a more appropriate way. ( C) P value: As expected, when each replication is inaccurately treated as independent the p value is low (due to artificially inflated n). We found that using the mean of replications and loading controls as covariates also resulted in a p value below 0.05. The smallest p value was found when including replication as a random factor. Across each of these statistical measures, only when replication is included as a random factor and loading control as a covariate do we see a strong effect size, high power, and low p value.

In the first scenario, we imagined that no technical replication was run at all (by using only the first replication). With this strategy, we found that standardized effect size is weak, power is low, and the p value was high (Fig.  4 ). Second, we demonstrate how analytical output would be different if we did run three technical replicates, but treated each as independent. As discussed above, this strategy does not take into account the fact that each sample is being run three times, and consequently the overall n of your experiment is artificially tripled! As one might expect, observed power is quite high, and our p value is low (< 0.05). Power is increased by an increase in sample size, so it is not surprising that the power is much higher if we erroneously report that we have a 3X larger sample size (i.e., pseudoreplication) 53 . In this case, the observed power is inflated and an artifact of inappropriate statistics, and the probability of a false positive is considerably increased with respect to the expected 5%.

So, what would be a more appropriate way to handle technical replicates? One method that researchers often use is to take the mean of their technical replicates. This does ensure that we are not artificially inflating our sample size, which is certainly an improvement over the previous strategy. With this strategy, we do find that our p value is less than 0.05 (when loading control is treated as a covariate). But we also see that our power is still low. We have effectively taken our replicates into account by collapsing across them within each sample, but this can be dangerous. If there is wide variation across replicates of a particular sample, then taking the mean of three replicates could produce an inaccurate estimate of the ‘true’ sample value. Ideally, we want to find a solution where instead of collapsing this variation, we add it to our statistical model so that we can better understand what amount of variation is randomly coming from within technical replicates, and in turn what amount of variation is actually due to potential differences in our experimental groups.

To achieve this, we need to model both the fixed effect of all groups in a full factorial design, and the random effect of replication across western blot gels. When we use both fixed and random effects, this is referred to as a linear mixed model (LMM). When using this strategy, we find that our effect size remains strong, and our p value is low. But importantly, we now have strong observed power (Fig.  4 ). This suggests that we can achieve greater sensitivity in our WB experiment when using this approach . Specifically, if we implement careful counterbalancing while designing our experiments, then we can use the variability between gels to our advantage during analysis using linear mixed effects model 55 .

LMM is recommended because it takes into account both the multiple observations within a single subject/animal in a given condition and differences across subjects observed in multiple conditions. This reduces chances of inaccurate p-values and improves reliability 56 . Further, treating both subjects and replication as random effects generalizes the results to the population of subjects and also to the population of conditions 57 .

Real world application of blotRig software for western blot experimental design, technical replication, and statistical analysis

We have designed a user interface that is designed to facilitate appropriate counterbalancing and technical replication for western blot experimental design. The ‘blotRig’ application is run through RStudio, and can be found here: https://atpspin.shinyapps.io/BlotRig/ Upon starting the blotRig application, the user is prompted to upload a comma separated values (CSV) spreadsheet. This spreadsheet should include separate columns for subject ID and experimental group. The user is then prompted to enter the total number of lanes that are available on their particular western blot gel apparatus. The blotRig software will first run a quality check to confirm that each subject ID (unique sample or subject) is only found in one experimental group. If duplicates are found, a warning will be shown that specifies which subjects are repeated across groups. If no errors are found, a centered gel map will be generated that illustrates the western blot gel lanes into which each subject should be loaded (Fig.  5 A). The decision for each lane loading is based on two main principles outlined above: (1) each western blot gel should hold a representative sample of each experimental group (2) samples from the same experimental group are not loaded in adjacent lanes whenever possible. This ensures that proper counterbalancing is achieved so that we can limit the chances that the inherent variability within and across western blot gels is confounded with the experimental groups that we are interested in experimentally testing.

Example of the blotRig Gel Creator interface. ( A ) Illustration of the blotRig interface. User has entered their sample IDs, experimental groups, and the number of lanes per western blot gel. ( B) The blotRig system then creates a counterbalanced gel map that ensures each gel contains a representative from each experimental group. This illustration shows the exact lane for each gel in which each sample should be run.

Once the gel map has been generated, the user can then select to export this gel map to a CSV spreadsheet. This sheet is designed to clearly show which gel each sample is on, which lane on each gel a sample is found, what experimental group each sample belongs to, and importantly, a repetition of each of these values for three technical replicates (Fig.  5 B). User will also see columns for Target Protein and Loading Control. These are the cells where the user can then input their densitometry values upon completing their western blot runs. Once this spreadsheet is filled out, it is then ready to go for blotRig analysis.

To analyze western blot data, users can upload the completed template that was exported in the blotRig experimental design phase or their own CSV file under the ‘Analysis’ tab (Fig.  6 ). The blotRig software will first ask the user to identify which columns from the spreadsheet represent Subject/SampleID, Experimental Group, Protein Target, Loading Control, and Replication. The blotRig software will again run a quality check to confirm that there are no subject/sample IDs that are duplicated across experimental groups. If no errors are found, the data will then be ready to analyze. The blotRig analysis will then be run, using the principles discussed above. Specifically, a linear mixed-model runs using the lmer R package, with Experimental Group as a fixed effect, Loading Control as a covariate, and Replication (nested within Subject/Sample ID) as a random factor. Analytical output is then displayed, giving a variety of statistical results from the linear mixed model output table, including fixed and random effects and associated p values (Fig.  6 ). A bar graph of group means and 95% confidence interval error bars will also be generated, along with a summary of the group means, standard error of the mean, and upper/lower 95% confidence intervals. These outputs can be directly reported in the results sections of papers, improve the statistical rigor of published WB reports. In addition, since the entire pipeline is opensource, the blotRig code itself can be reported to support transparency and reproducibility.

Workflow for running statistical analysis of replicate western blot data using blotRig. First, fill out spreadsheet with subject ID, experimental group assignment, number of technical replication, the densitometry values for your target proteins and loading controls. After saving this spreadsheet as a.csv file, the file can be uploaded to blotRig. Tell blotRig the exact names of each of your variables, then click ‘Run Analysis’. This will produce a statistical output using linear mixed model testing for group differences using loading control as a covariate and replication as a random effect. Bar graph with error bars and summary statistics can then be exported.

Although the western blot technique has proven to be a workhorse for biological research, the need to enhance its reproducibility is critical 13 , 19 , 27 . Current qWB assay methods are still lacking for reproducibly identifying true biological effects 13 . We provide a systematic approach to generate quantitative data from western blot experiments that incorporates key technical and statistical recommendations which minimize sources of error and variability throughout the western blot process. First, our study shows that experimenters can improve the reproducibility of western blots by applying the experimental recommendations of determining the linear range for each primary antibody, counterbalancing during experimental design, and running technical triplicates 13 , 27 . Furthermore, these experimental implementations allow for application of the statistical recommendations of incorporating loading controls as covariates and analyzing gel and subject as random effects 58 , 59 . Altogether, these enable more rigorous statistical analysis that accounts for more technical variability which can improve the effect size, observed power, and p-value of our experiments and ultimately better identify true biological effects.

Biomedical research has continued to rely on p-values for determining and reporting differences between experimental groups, despite calls to retire the p-value 60 . Power (sensitivity) calculations have also become increasingly common. In brief, p-values and the related alpha value are associated with Type I error rate—the probability of rejecting the null hypothesis (i.e., claiming there is an effect) when there is no true effect 61 . On the other hand, power effectively measures the probability of rejecting the null hypothesis (i.e. stating there is not effect) when there is indeed a true underlying effect—a concept that is closely related to reducing the Type II error rate 59 , 62 . Critically, empirical evidence estimates that the median statistical power of studies in neuroscience is between ∼ 8% and ∼ 31%, yet best practices suggest that an experimenter should aim to achieve a power of 80% with an alpha of 0.05 20 . By being underpowered, experiments are at higher likelihood of producing a false inference. If an underpowered experiment is seeking to reproduce a previous observation, the resulting false negative may throw into question the original findings and directly exacerbate the reproducibility crisis 59 . Even more alarmingly, a low power also increases the likelihood that a statistically significant result is actually a false positive due to small sample size problems 61 . In our analyses, we show that our technical and statistical recommendations lower the p-value (indicating that the observed relationship between variables is less likely to be due to chance) as well as observed power of our experiments. This translates into the ability to better avoid false negatives when there is a true effect as well as reduce the likelihood of false positives when there is not a true experimental effect, both of which will ultimately improve the reproducibility of qWB assay experiments.

Another useful component of statistical analyses that is not as commonly reported but is critically related to p-value and power is effect size. Effect size is a statistical measure that describes the magnitude of the difference between two groups in an experiment 63 . It is used to quantify the strength of the relationship between the variables being studied 63 . The estimated effect size is important because it answers the most frequent question that researchers ask: how big is the difference between experimental groups, or how strong is the relationship or association? 63 . The combination of standardized effect size, p-value and power reflect crucial experimental results that can be broadly understood and compared with findings from other studies 62 , thus improving comparability of qWB experiments 49 , 63 . In particular, studies with large effect sizes have more power: we are more likely to detect a true positive experimental effect and avoid the false negative if the underlying difference between experimental groups is large 46 . In some cases, the calculated effect size is greatly influenced by how sources of variance are handled during analysis 13 . Our results demonstrate that by reducing the residual variance (by modeling the random effect of replication) the estimated effect size of our experiment increases. This could mean that the magnitude of the difference between the groups in our experiment is larger than it was originally thought to be. This could be due to a variety of factors such as improving the experimental design, sample size, or the measurement of the variables 13 . Likewise, conducting a power analysis is an essential step in experimental design that should be done before collecting data to ensure that the study is adequately powered to detect an effect of a certain size 64 .

Increasingly, power analysis is becoming a requirement for publications and grant proposals 65 . This is because a study with low statistical power is more likely to produce false negative results, which means that the study may fail to detect a real effect that actually exists. This can lead to the rejection of true hypotheses, wasted resources, and potentially harmful conclusions. In brief, given an experimental effect size and variance, we can calculate the sample size needed to achieve an alpha of 0.05 and power of 0.8; an increased sample size reduces the standard error of mean (SEM), which is the measured spread of sample means and consequently increases the power of the experiment 66 . We have demonstrated that our experimental and statistical recommendations lead to a lower p value (Fig.  3 C) and effect size (Fig.  3 B) without changing the sample size. This may be of greatest interest to researchers: more rigorous analytics ultimately improves experimental sensitivity without relying solely on increasing the sample size.

Reducing the sample size of an experiment can be beneficial for several reasons, one of which is cost-effectiveness. A smaller sample size can lead to a reduction in the number of animals or other resources that are needed for the study, which can result in lower costs. Additionally, it can also save time and reduce the duration of the experiment, as fewer subjects need to be recruited, and the data collection process can be completed more quickly. However, it is important to note that reducing the sample size can also lead to decreased statistical power. As a result, reducing sample size too much can increase the risk of a type II error, failing to detect significance when there is a true effect 62 .Therefore, it is important to consider the trade-off between sample size and power when designing an experiment, and to use statistical techniques like power analysis to ensure that the sample size is sufficient to detect an effect of a certain size. Moreover, when using animals in research, it's always important to consider the ethical aspect and the 3Rs principles of reduction, refinement, and replacement 55 .

Despite our best efforts in creating a balanced, full factorial experimental design, there will always be random variation in biological experiments. Fixed effects such as experimental group differences are expected to be generalizable if the experiment is replicated. Random effects (such as gel variation) on the other hand are unpredictable across experiments. Western blot analyses are particularly susceptible to this random gel variation, as different values may be observed for technical replicates run on different gels. By using a linear mixed model paired with rigorous full factorial design, we can ensure that we account for as much of that random variation as possible. When we acknowledge, identify, and model random effects we enhance the possibility of discovering our fixed effect of experimental treatment, if one exists.

The linear mixed model framework discussed above assumes that our western blot outcome measures are on a linear scale. As described above, parametric work to identify the linear range of a protein of interest is critical for ensuring that the results of a LMM (or ANOVA and t-test) are accurate and interpretable. While we recommend using loading control (or total protein control) as a covariate in a linear mixed model, many bench researchers may prefer to use the within-lane loading control (or total protein) to normalize target protein values. It is important to consider that in doing so, one creates a ratio value that is multiplicative instead of linear. This property has the side effect of artificially distorting the variance. To account for this non-linearity, we recommend that one uses semi-parametric mixed models such as generalized estimating equations with a gamma distribution link function that appropriately represents ratio data.

There has been recent recognition that an appropriate study design can be achieved by balancing sample size (n), effect size, and power 31 . The experimental and statistical approach presented in this study provide insight into how more rigorous planning for western blot experimental design and corresponding statistical analysis without depending on p-values only can acquire precise data resulting in true biological effects. Using blotRig as a standardized, integrated western blot methodology, quantitative western blot may become highly reproducible, reliable, and a less controversial protein measurement technique 18 , 28 , 67 .

Study reporting

This study is reported in accordance with ARRIVE guidelines.

Supporting information

This article contains supporting information. You can access the blotRig analysis software, which includes code for inputting experimental parameters for all Western blot analysis, through the following link: https://atpspin.shinyapps.io/BlotRig/

Data availability

The datasets and computer code generated or used in this study are accessible in a public, open-access repository at https://doi.org/10.34945/F51C7B and https://github.com/ucsf-ferguson-lab/blotRig/ respectively.

Abbreviations

American association for accreditation of laboratory animal care

Animal research reporting of in vivo experiments

American veterinary medical association

Institutional animal care and use committee

Quantitative western blot

Enzyme linked immunosorbent assay

Severe acute respiratory syndrome coronavirus 2

Analysis of covariance

Analysis of variance

Spinal cord injury

Spared nerve injury

α-Amino-3-hydroxy-5-methyl-4-isoxazoleproprionic acid

Glutamate receptor 1

Glutamate receptor 2

Linear mixed models

Tris-buffered saline containing 0.1% Tween 20

Polyacrylamide gel electrophoresis

Standard error of mean

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Acknowledgements

The authors would like to thank Alexys Maliga Davis for data librarian services.

This work was supported by a National Institutes of Health/National Institute of Neurological Disorders and Stroke grant (R01NS088475) to A. R. F. NIH NINDS: R01NS122888, UH3NS106899, U24NS122732, US Veterans Affairs (VA): I01RX002245, I01RX002787, I50BX005878, Wings for Life Foundation, Craig H. Neilsen Foundation. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Correspondence and requests for materials should be addressed to A.R.F.

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Cleopa Omondi, Austin Chou, Kenneth A. Fond, Kazuhito Morioka, Nadine R. Joseph, Jeffrey A. Sacramento, Emma Iorio, Abel Torres-Espin, Hannah L. Radabaugh, Jacob A. Davis, Jason H. Gumbel, J. Russell Huie & Adam R. Ferguson

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C.O: Writing-original draft preparation, Investigation, Validation, Data Curation, Visualization, Formal Analysis, Writing-Review & Editing; A. C: Formal analysis, Writing-Review & Editing; K. A. F: Software, Writing-Review & Editing; K. M: Methodology, Writing-Review & Editing; N. R. J: Writing—Review & Editing; J. A. S: Investigation, Project Administration, Writing—Review & Editing; E. I: Resources, Writing-Review & Editing; A.T.E: Software, Writing—Review & Editing; H. L. R: Software, Writing—Review & Editing; J. A. D: Investigation, Writing—Review & Editing; J. H. G: Investigation, Writing – Review & Editing; J. R. H: Conceptualization, Methodology, Validation, Formal Analysis, Investigation, Data Curation, Writing- Review & Editing, Visualization, Supervision; A. R. F:Conceptualization, Methodology, Validation, Formal Analysis, Resources, Investigation, Data Curation, Writing-Review & Editing, Visualization, Supervision, Project Administration, Funding Acquisition.

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Omondi, C., Chou, A., Fond, K.A. et al. Improving rigor and reproducibility in western blot experiments with the blotRig analysis. Sci Rep 14 , 21644 (2024). https://doi.org/10.1038/s41598-024-70096-0

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DOI : https://doi.org/10.1038/s41598-024-70096-0

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Experimental and computational analysis of the structure-activity relationship of ionic gel electrolytes based on bistrifluoromethanesulfonimide salts for supercapacitors

Peer review under responsibility of The Chinese Ceramic Society.

Graphical Abstract

Ionic gel (IG) electrolytes are emerging as promising components for the development of next-generation supercapacitors (SCs), offering benefits in terms of safety, cost-effectiveness, and flexibility. The ionic conductivity, stability, and mechanical properties of the gel electrolyte are relevant factors to be considered and the key to improving the performance of the SC. However, the structure–activity relationship between the internal structure of IGs and their SC properties is not fully understood. In the current study, the intuitive and regular structure–activity relationship between the structure and properties of IGs was revealed via combining computational simulation and experiment. In terms of conductivity, the ionic liquid (IL) ([EMIM][TFSI]) in the IG has a high self-diffusion coefficient calculated by molecular dynamics simulation (MDS), which is conductive to transfer and then improves the conductivity. The radial distribution function of the MDS shows that the larger the g ( r ) between the particles in the polymer network, the stronger the interaction. For stability, IGs based on [EMIM][TFSI] and [EOMIM][TFSI] ILs have higher density functional theory calculated binding energy, which is reflected in the excellent thermal stability and excellent capacitor cycle stability. Based on the internal pore size distribution and stress-strain characterization of the gel network ([ME3MePy][TFSI] and [BMIM][TFSI] as additives), the highly crosslinked aggregate network significantly reduces the internal mesoporous distribution and plays a leading role in improving the mechanical properties of the network. By using this strategy, it will be possible to design the ideal structure of the IG and achieve excellent performance.

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Development and analysis of Hastelloy-X alloy butt joint made by laser beam welding

• Published: 18 September 2024
• Volume 49 , article number  262 , ( 2024 )

Cite this article

• G Sathishkumar 1 ,
• S Senthil Murugan 2 &
• P Sathiya 1  

The investigation describes the processing and analysis of laser welded Hastelloy-X (HX) alloy joints through this paper. HX alloys are employed in aerospace and nuclear industries, especially for high-temperature applications. In this research, a series of laser beam welding (LBW) experiments denoted as E1 to E9 were conducted on the HX base metal (BM), adhering to the L9-orthogonal array (L9-OA) design matrix. CO 2 laser technology was employed to fabricate HX butt joints. The laser power, focal length and welding speed were the variables. Then, the character of each joint was analyzed by sophisticated testing methods as per ASTM standards. The results showed that the character of each sample was varied depending on the selection of parameters. The E5 sample had a maximum tensile strength (TS) and ductility with 93 % joint efficiency. The grain elongation and refinement in the weld zone (WZ) were confirmed through microstructures and electron back scattered diffraction (EBSD) studies. The corrosion character of each joint (E1 to E9) was analyzed using the potentiostatic polarization method. The E1 sample had the highest corrosion resistance. The corrosion rate was in the range of 7.4E−03 to 8.6E−05 mm/yr. The dry sliding wear test (A1 to A9) was carried out as per L9-OA on the E1 sample, since this weld parameter had good corrosion resistance, by varying applied load, sliding distances and sliding velocities. Wear test results showed an increase in wear rate with the increase in load and sliding distance. A wear map was also drawn using the results to find out the association between wear rates and wear parameters. Weld speed had influenced the strength of the joint and laser power had shown an impact on the corrosion and wear of the HX alloy joints. The weld zone, corroded sample and the worn-out surfaces of weld joints were further analyzed using an optical microscope (OM) and Field Emission Scanning Electron Microscopy (FESEM).

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Sathishkumar, G., Senthil Murugan, S. & Sathiya, P. Development and analysis of Hastelloy-X alloy butt joint made by laser beam welding. Sādhanā 49 , 262 (2024). https://doi.org/10.1007/s12046-024-02603-y

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DOI : https://doi.org/10.1007/s12046-024-02603-y

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Bioelectrical impedance vector analysis (biva) for assessment of hydration status: a comparison between endurance and strength university athletes.

1. Introduction

2. materials and methods, 2.1. study design and sampling, 2.1.1. athletic population, 2.1.2. reference population, 2.1.3. selection criteria, 2.2. recruitment, 2.3. data collection, 2.3.1. pre-hydration assessment questionnaire, 2.3.2. anthropometric measurements and body composition, 2.3.3. urine collection, 2.3.4. sweat rate, 2.4. ethical considerations, 2.5. statistical analysis, 3.1. urine color and biva parameters, 3.2. agreement between biva and (usg and sr), 3.2.1. bland–altman plots.

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3.2.2. Canonical Correlation Analysis

3.2.3. examination of data characteristics and comparative analysis, 3.3. pre–post measurements, 4. discussion, 4.1. strengths, 4.2. limitations, 4.3. future studies, 5. conclusions, supplementary materials, author contributions, institutional review board statement, informed consent statement, data availability statement, acknowledgments, conflicts of interest, appendix a. urine color chart, appendix b. urine specific gravity equivalent hydration status, appendix c. eligibility questionnaire.

Appendix D. Pre-Hydration Assessment Questionnaire

• Other, please specify: _____________
• How much time do you usually spend doing vigorous physical activities? _____ hours per day       _____ minutes per day _____Don’t know/Not sure
• Medical/Biochemical/Clinical Assessment:
• How often do you have a bowel movement? _____________________________
• If you take laxatives, what type/brand and how often? _____________________
• Heartburn, Bloating, Gas, Constipation, Diarrhea, Nausea, Vomiting:
• When was the last time you had Blood Test? ____________
• Recent Blood test results:
• Water Retention: Yes No
• Dietary Assessment:
• If yes, what is the reason?
• Are you taking any drugs or medications? If yes, specify name, quantity and duration:______________________________________________________ ▢ Yes, please specify: _______________ ▢ No ▢ Don’t know
• Do you consume any of the following foods? ▢ Beets ▢ Blackberries ▢ Carrots ▢ Fava beans ▢ Rhubarb
• Which meals do you eat regularly, check all that apply: ▢ Breakfast ▢ AM Snack ▢ Lunch ▢ Afternoon Snack ▢ Dinner ▢ Late Night Snack
• Are you vegetarian? ▢ Yes ▢ No
• Have you ever followed a dietary regime? ▢ Yes, now ▢ Yes, previously ▢ No ▢ If yes, please give details: ____________________________________________________________________________________________________________________

• How many fruits and vegetables in all do you consume daily on average? ____________ Fruits _________Vegetables
• How often do you consume fast foods? ▢ Never ▢ Two to three times monthly ▢ One or two times weekly ▢ Daily
• How many liters of water do you consume daily? ▢ Less than one liter ▢ One liter to two liters ▢ Two to three liters ▢ More than three liters

• Do you consume alcohol? If yes, specify ________________ ▢ Never or rarely ▢ Only in weekends ▢ Once or twice weekly ▢ Once or twice daily ▢ More than twice daily
• Do you smoke? ▢ Yes ▢ No If yes, how much on average? __________________________
• On average how many hours do you sleep per night? ▢ More than 10 h ▢ Between 8–10 h ▢ Between 6–8 h ▢ Between 4–6 h ▢ Less than 4 h
• How important do you think good nutrition is to sports performance? ▢ Very important ▢ Important ▢ Moderately important ▢ Of little importance ▢ Unimportant
• How important do you think hydration status is to sports performance? ▢ Very important ▢ Important ▢ Moderately important ▢ Of little importance ▢ Unimportant
• “The more supplements I take, the better I will perform”: ▢ Agree ▢ Disagree ▢ Strongly disagree ▢ Neither agree nor disagree
• Have you received any previous nutritional advice? ▢ Yes ▢ No
• Do you have access to a sports nutritionist/dietitian? ▢ Yes, through CHDC ▢ Yes, outside AUB ▢ No Anthropometric Assessment: Usual body weight: _________
• During these past 6 months, did your weight change? ▢ Yes, I gained weight ▢ Yes, I lost weight ▢ No ▢ Not sure How many kg(s)_____________________?
• Would you like to receive the hydration tests and body composition results by phone/mail? ▢ Yes, by phone ▢ Yes, by mail ▢ No, I don’t want to receive the results Instructions to prepare you for the hydration tests: - Avoid caffeine and alcohol consumption prior to the experiment. - Do not exercise intensely at least 12 h prior to the experiment. - Avoid food in the 4 h prior to the experiment.

Appendix E. Data Collection Sheet

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Abdelnour, M.; Berkachy, R.; Nasreddine, L.; Fares, E.-J. Bioelectrical Impedance Vector Analysis (BIVA) for Assessment of Hydration Status: A Comparison between Endurance and Strength University Athletes. Sensors 2024 , 24 , 6024. https://doi.org/10.3390/s24186024

Abdelnour M, Berkachy R, Nasreddine L, Fares E-J. Bioelectrical Impedance Vector Analysis (BIVA) for Assessment of Hydration Status: A Comparison between Endurance and Strength University Athletes. Sensors . 2024; 24(18):6024. https://doi.org/10.3390/s24186024

Abdelnour, Maria, Rédina Berkachy, Lara Nasreddine, and Elie-Jacques Fares. 2024. "Bioelectrical Impedance Vector Analysis (BIVA) for Assessment of Hydration Status: A Comparison between Endurance and Strength University Athletes" Sensors 24, no. 18: 6024. https://doi.org/10.3390/s24186024

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