Why yet another book on “Design and Analysis of Experiments”?

In contrast to many books by good authors, this book takes a departure and focuses on two aspects like “Practical” and “Using R ” as depicted in its tile, “Practical Design and Analysis of Experiments in Statistics Using R”.

That is, the author has tried to focus more on the practical computation and graphic presentation of the layouts of the designs of experiments followed by practical computation of complex analyses of variances of various designs, involving fixed, random and mixed linear models, using R- codes.

There are so many books on R. But, mostly deal with the grammar of R language. The subject ‘Statistics’ as such still needs example based books containing development of the subject matter along with live running codes of R, which a normal user initially finds difficult to frame.

From the perspective of filling the same gap in this book “Practical Design and Analysis of Experiments in Statistics Using R” the author is presenting it to the students and professors of statistics and applied statistics, agricultural scientists and applied scientists, who use designs of experiments for analyzing their data.

Many examples have been taken from the fields of agriculture, biology, medical and social science.

There is no attempt of explaining the grammar of R language in this book, rather ready-made R-codes have been given here and there to illustrate the underlying concepts, as the author thinks that illustration automatically makes one understand the concepts gradually as a child does while growing.

Akhilesh Kumar Singh

The author is a Professor and Head, Department of Agricultural Statistics and Social Science at Indira Gandhi Agricultural Statistics

(IGKV), Raipur, Chhattisgarh, India.

He has been working on the computational aspects of his research works, using R, since 1998.

The author felt that the students and scientists, who are potential users of R, are not interested to learn the R codes, rather they want the methods of analyses for their data.

So, the he decided to develop the subject matter using R by taking examples, analyzing them and presenting the R codes used in the book itself for the benefit of the readers of the book.

Earlier, the author wrote a book, see Singh (2016), and conducted a 10-days training programme on “R-An Indispensable Open

Source Software Package for Statisticians, Agricultural Scientists and Students” for trainees.

The author has published more than 50 scientific research papers in national and international journals.

He has handled an award winning e-Governance project at Indira Gandhi Krishi Vishwavidyalaya (IGKV) as Nodal Officer, best awarded at two national levels of ICAR and Digital India of the Government of India, which is still functional at https://igkvmis.cg.nic.in.

1 Basics of Designs of Experiment

1.1 Introduction

1.2 Some Terminologies in Experimental Design

1.3 Basic Principles of Designs of Experiment

1.4 Analyses of Designs of Experiments

2 Completely Randomized Design (CRD)

2.1 Denition

2.2 Randomization and Layout

2.2.1 Examples of Randomized layout of CRD with given replications of treatments

2.2.1.1 Example: Randomization Using R library “agricolae”

2.2.1.2 Example: Same Example as in Example 2.2.1.1 but using standard method of randomization using R

2.2.1.3 Example: Same Example as in Example 2.2.1.1 but using “desplot” for randomized layout

2.3 Analysis of Variance: Single Factor Classication

2.3.1 Linear Statistical Model for CRD

2.3.2 Analysis of Variance Table (ANOVA Table) for CRD

2.3.3 Diagnostics of Residuals of the Linear Model for ANOVA

2.3.4 Analysis of Variance for Single Factor Classication Using R methods:

2.3.4.1 Example: One-Way-ANOVA with Post-Hoc Tukey’s Test using R methods for data set of Medley and Clements (1998)

2.3.4.2 Example: One factor One-Way-ANOVA with Planned Comparison using R

methods for data set of Keough and Raimondi (1995)

2.3.4.3 Example: One Factor One-Way-ANOVA with Planned Comparison of Orthogonal Polynomial Trends using R methods for data set of Keough and Raimondi (1995)

2.3.4.4 Example: One-Way-ANOVA with variance components using R methods for data set of Medley and Clements (1998), earlier Example 2.3.4.1

3 Randomized Block Design (RBD)

3.1 Denition

3.2 Randomization and Layout

3.2.1 Examples of Randomized layout of RBD with given replication of treatments

3.2.1.1 Example: Randomization Using R library agricolae

3.2.1.2 Example: Same Example as in Example 3.2.1.1 but using standard method of randomization using R

3.2.1.3 Example: Same Example as in Example 3.2.1.1 but using desplot for randomized layout

3.3 Analysis of Variance: One Way ANOVA with Randomized Blocking

3.3.1 Linear Statistical Model for RBD

3.3.2 Analysis of Variance Table (ANOVA Table) for RBD

3.3.3 Analysis of Variance for Single/Multi-Factor Classication with Blocking Using R

methods:

3.3.3.1 Example: One Factor Two-Way-ANOVA, including Blocking Factor, Post-Hoc

Tukey’s Test using R methods for data set of Walter and O’Dowd (1992)

3.3.3.2 Example: Two-Way-Factorial-ANOVA in RBD with Post-Hoc Tukey’s Test using

R methods for data set of Doncaster and Davey (2007)

4 Latin Square, Crossover, and Graeco-Latin Square Designs

4.1 Latin Square Design (LSD):

4.2 Randomization and Layout

4.2.1 Examples of Randomized layout of LSD with given replication of treatments

4.2.1.1 Example: Randomization Using R library agricolae

4.2.1.2 Example: Same Example as in Example 4.2.1.1 but using standard method of randomization using R

4.2.1.3 Example: Same Example as in Example 4.2.1.1 but using desplot for randomized layout

4.3 Analysis of Variance: One Way Classication with Two Way Blocking

4.3.1 Linear Statistical Model for LSD

4.3.2 Analysis of Variance Table (ANOVA Table) for LSD

4.3.3 Analysis of Variance for Single/Multi-Factor Classication with Blocking in two

Directions Using R methods:

4.3.3.1 Example: One Factor Three-Way-ANOVA, including Two Blocking Factors, Post-Hoc

Tukey’s Test using R methods for Montgomery (2013) data set

4.4 Replicated Latin Square Design (RLSD):

4.4.1 Replicated LSD: Case-I: Same Row Levels and Same Column Levels in Different

Replicates of LSD

4.4.1.1 Example: Clinical Trial of Infant Baby in LSD for the dataset of Fleiss (2011): Case-I:

4.4.2 Replicated LSD: Case-II: Different Row Levels and Same Column Levels in Different Replicates of LSD, or vice-versa Different Column Levels with Same Row Levels in Different Replicates of LSD

4.4.2.1 Example: Clinical Trial of Infant Baby in LSD Fleiss (2011): Case-II:

4.4.3 Replicated LSD: Case-III: Different Row Levels and Different Column Levels in

Different Replicates of LSD

4.4.3.1 Example: Clinical Trial of Infant Baby in LSD Fleiss (2011): Case-III:

4.5 Crossover or Changeover or Designs Balanced for Residual Effects:

4.5.0.1 Example: Clinical Trial of Crossover 2×2 design Schutz (2012): Case-II:

4.6 Graeco-Latin Square Design (GLSD):

4.6.0.1 Example: One Factor Four-Way-ANOVA, including Three Blocking Factors, for

Graeco-Latin Square design using R methods for Montgomery (2013) data set

5 Incomplete Block Design and Balanced Incomplete Block Design

5.1 Incomplete Block Design:

5.1.1 Kinds of Incomplete Block Designs:

5.2 Balanced Incomplete Block Design (BIBD)

5.2.1 Some Properties of BIBD:

5.2.2 Randomization and Layout:

5.2.3 Examples of Randomized layout of IBD/BIBD with given replication of treatments

5.2.3.1 Example: Randomization of BIBD Using R library agricolae

5.2.3.2 Example: Same as Example 5.2.3.1 Randomization and Anatomy of BIBD Using R

libraries dae and desplot

5.2.3.3 Example: Same as Example 5.2.3.1. ANOVA using R library dae based on

Anatomy of BIBD

5.2.3.4 Example: BIBD ANOVA of Chick Embryo see Cox and Reid (2000), Section 4.2.6

Examples

5.2.3.5 Example: BIBD ANOVA of Cropping Systems Research at Jorhat Project Directorate of Cropping Systems Research

5.2.3.6 Example: BIBD ANOVA of Wear testing Example of Box et al. (1978), Appendix 8D

6 Factorial Experiments

6.1 Introduction:

6.2 Single factor experiment versus factorial experiments:

6.3 Factorial experiments: Simple, Main and Interaction Effects:

6.3.1 Some Remarks:

6.4 Regression Model Representation of Main Effect, Interaction Effect: Response curves:

6.5 Types of Factorial Experiments:

6.6 Analyses of Factorial Experiments:

6.6.1 Example: 22 Factorial Experiment without interaction:

6.6.2 Example: 22 Factorial Experiment with interaction:

6.6.3 Example: 3 x 2 General Factorial Experiment with interaction:

6.7 Analyses of Response Curves / Surface in Factorial Experiments:

6.7.1 Example: Response Curves in 32 Factorial Experiment with interaction:

6.7.2 Response Curves for mean.Yield.Variety.Spacing by tting Orthogonal polynomial:

6.7.2.1 Linear/Quadratic Orthogonal Polynomial Fit on Spacing of Varieties:

6.7.2.2 Removing Quadratic effects and then Fitting Linear Orthogonal Polynomial

Fit and its ANOVA:

6.7.2.3 Plotting the Orthogonal Linear Model:

6.7.3 Example: Response Surface in 3×4 Factorial Experiment with interaction:

6.7.3.1 Response Surface for Ascor.acid on Orthogonal Polynomials of (Temp- Weeks) Treatment combinations:

6.7.3.2 Fitting Response Surface plots against signicant combination of treatments

Temp:Weeks:

6.8 Blocking and Confounding in Factorial Experiments:

6.8.1 R-codes for implementing confounding with blocks in symmetric factorial designs

6.8.2 Example: Complete confounding with Randomization: 23 factorial experiment, with highest order interaction confounded with block effect in each replicate with randomization:

6.8.3 Example: Partial confounding: 23 factorial experiment, with at least one interaction is confounded with block effect of each replicate :

7 Split Plot Design and its Variants

7.1 Introduction:

7.2 Split plot design in CRD with Two factors Experiment

7.2.1 Randomization and Layout

7.2.2 Examples of Randomized layout of Split plot in CRD with given replication of treatments

7.2.2.1 Example: Randomization Using R library agricolae

7.2.2.2 Example: Same Example as in Example 7.2.2.1 but using standard method of randomization using R

7.2.2.3 Example: Same Example as in Example 7.2.2.1 but using desplot for randomized layout

7.3 Split plot design in RBD: Two factors experiment

7.3.1 Randomization and Layout

7.3.2 Examples of Randomized layout of Split plot in RBD with given replication of treatments

7.3.2.1 Example: Randomization Using R library agricolae

7.3.2.2 Example: Same Example as in Example 7.3.2.1 but using standard method of randomization using R

7.3.2.3 Example: Same Example as in Example 7.3.2.1 and Example 7.3.2.2 but using desplot for randomized layout

7.4 Split plot design in LSD: Two factors experiment

7.4.1 Randomization and Layout

7.4.2 Examples of Randomized layout of Split plot in LSD with as many replication as treatments

7.4.2.1 Example: Randomization Using R library agricolae

7.4.2.2 Example: Same Example as in Example 7.4.2.1 but using standard method of randomization using R

7.4.2.3 Example: Same Example as in Example 7.4.2.1 and Example 7.4.2.2 but using desplot for randomized layout

7.5 Analysis of Variance: Two Factor Split plot in CRD

7.5.1 Linear Statistical Model for split plot design in CRD

7.5.2 Analysis of Variance (ANOVA) Table for split plot in CRD

7.5.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table 7.1: Two Factor Split plot in CRD (Fixed effects: 1-main plot, 2-sub plot; Random effect: Replication nested into main plot):

7.5.3.1 Example: Two factor Split plot ANOVA in CRD with Post-Hoc analysis using emmeans package of R for data set of Potcner and Kowalski (2004)

7.5.3.2 Example: Two factor Split plot ANOVA in CRD with Post-Hoc analysis using emmeans package of R for data set of Box et al. (2005)

7.5.4 Analysis of Variance Using Linear Mixed Effect Model R methods Involving Variance

Components in Contrast to ANOVA of Table 7.1: Two Factor Split plot in CRD (Fixed effects:

1-main plot, 2-sub plot; Random effect: Replication nested into main plot):

7.5.4.1 Example: R methods of Linear Mixed Effect Model: Two Factor Split plot in CRD (Fixed effects: 1-main plot, 2-sub plot; Random effect: Replication nested into main plot) with lme4 and emmeans packages of R for data set of Potcner and Kowalski (2004) and Box et al. (2005)

7.6 Analysis of Variance: Two Factor Split plot design in RBD

7.6.1 Linear Statistical Model for split plot design in RBD

7.6.2 Analysis of Variance Table (ANOVA Table) for split plot in RBD

7.6.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table 7.5: Two

Factor Split plot in RBD (Fixed effects: 1-main plot, 2-sub plot; Random Block effect):

7.6.3.1 Example: Two factor Split plot ANOVA in RBD with Post-Hoc analysis using emmeans package of R for data set of Gomez and Gomez (1984), p 102

7.7 Analysis of Variance: Two Factor Split plot in LSD

7.7.1 Linear Statistical Model for split plot design in LSD

7.7.2 Analysis of Variance Table (ANOVA Table) for split plot in LSD

7.7.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table 7.7: Two Factor Split plot in LSD (Fixed effects: 1-main plot, 2-sub plot; Random Row/Column effects):

7.7.3.1 Example: Two factor Split plot ANOVA in LSD with Post-Hoc analysis using emmeans package of R for data set of Smith (1951)

7.8 Strip plot design or Strip block design: Two factors experiment

7.9 Strip plot design in RBD: Two factors experiment

7.9.1 Randomization and Layout

7.9.2 Examples of Randomized layout of Strip plot in RBD with given replication of treatments

7.9.2.1 Example: Randomization Using R library agricolae

7.9.2.2 Example: Same Example as in Example 7.9.2.1 but using standard method of randomization using R

7.9.2.3 Example: Same Example as in Example 7.9.2.1 and Example 7.9.2.2 but using desplot for randomized layout

7.10 Analysis of Variance: Two Factor Strip plot in RBD

7.10.1 Linear Statistical Model for strip plot design in RBD

7.10.2 Analysis of Variance Table (ANOVA Table) for strip plot in RBD

7.10.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table 7.9: Two

Factor Strip plot in RBD (Fixed effects: 1-main plot, 2-strip plot; Random Block effect):

7.10.3.1 Example: Two factor Strip plot ANOVA in RBD with Post-Hoc analysis using emmeans package of R for data set of Gomez and Gomez (1984), p 102

7.10.3.2 Example: Two factor Strip plot ANOVA in LSD with Post-Hoc analysis using emmeans package of R for data set of Little and Hills (1978)

7.11 Split split plot design (SSP Design):

7.11.1 SSP Design: Three factors experiment (1-main plot, 2-split plot, 3-split- split plot):

7.12 Split split plot design in RBD (SSP design in RBD):

7.12.1 SSP design in RBD: Three factors experiment: (1-main plot, 2-sub plot, 3- sub sub plot)

7.12.1.1 Randomization and Layout

7.12.1.2 Examples of Randomized layout of Split split plot in RBD with given replication of treatments .

7.13 Analysis of Variance (SSP Design in RBD):

7.13.1 Linear Statistical Model for Split split plot design in RBD

7.13.2 Analysis of Variance Table (ANOVA Table) for Split split plot design in RBD .

7.13.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table

7.12: Three Factor Split split plot in RBD (Fixed effects: 1-main plot, 2-sub plot,

3-sub sub plot; Random Block effect):

7.13.3.1 Example: Three factor Split split plot ANOVA in RBD with Post-Hoc analysis using emmeans package of R for data set of Gomez and Gomez (1984), p 143

7.14 Strip split plot design (StSP Design):

7.14.1 StSP Design: Three factors experiment (1-main plot, 2-strip plot, 3-strip- split plot):

7.15 Strip split plot design in RBD (StSP design in RBD):

7.15.1 StSP design in RBD: Three factors experiment: (1-main plot, 2-strip plot,

3-strip split plot)

7.15.1.1 Randomization and Layout

7.15.1.2 Examples of Randomized layout of Strip split plot in RBD with given replication of treatments

7.16 Analysis of Variance (StSP Design in RBD):

7.16.1 Linear Statistical Model for Strip split plot design in RBD

7.16.2 Analysis of Variance Table (ANOVA Table) for Strip split plot design in RBD

7.16.3 Analysis of Variance Using Standard R methods Based on ANOVA of Table

7.14: Three Factor Strip split plot in RBD (Fixed effects: 1-main plot, 2-strip plot,

3-strip split plot; Random Block effect):

7.16.3.1 Example: Three factor Strip split plot ANOVA in RBD with Post-Hoc analysis using emmeans package of R for data set of see Cox and Rotti (1978)

7.17 Split Plot or Variant Design with Factorial Experiment:

7.17.1 Examples of factorial arrangement in main plot or sub plot of split plot design or sub sub plot etc of split plot variant designs:

7.17.1.1 Example: Three factor Pooled Split plot ANOVA in RBD (Two factors in main plot and one in split plot) with Post-Hoc analysis using emmeans package of R for data set of see Gomez and Gomez (1984), p.339

7.17.1.2 Example: Rancidness of Animal Meat Pieces: Three factor Split plot ANOVA in CRD (One factor in main plot and two factors in split plot) with Post- Hoc analysis using emmeans package of R for data set of see Stryhn et al. (2019), eNote-7, p.14

7.17.1.3 Example: Split Strip plot experiment on soybeans: Three factor Split strip plot ANOVA in RBD (One factor in main plot and two factors in sub plot of split plot design laid out as strip factors) with Post-Hoc analysis using emmeans package of R for data set of see Schabenberger and Francis (2002)

8 Analysis of Covariance and Miscellaneous Topics

8.1 Analysis of Covariance (ANCOVA)

8.1.1 Introduction:

8.1.2 Some Examples and Uses of ANCOVA:

8.1.3 Analysis of Covariance (ANCOVA) with RBD

8.1.3.1 Randomization and Layout

8.1.3.2 Analysis of Covariance: Single Factor Classication with Randomized Blocking

8.1.3.3 Linear Statistical Model for ANCOVA in RBD

8.1.3.4 Analysis of Variance Table (ANCOVA Table) for RBD

8.1.4 Analysis of Covariance for Single/Multi-Factor Classication with Blocking Using R

methods:

8.1.4.1 Example: One Factor Two-Way-ANCOVA with One Covariate, including Blocking

Factor, using R methods for data set of Snedecor (1946), Table 12.13, page 332

8.1.4.2 Example: One Factor Two-Way-ANCOVA with One Covariate, including Blocking Factor, Post-Hoc Tukey’s Test using R methods for data set of Steel and Torrie (1980), page 411-417

8.1.5 Analysis of Covariance (ANCOVA) with CRD, LSD, factorial experiments or split- plot variant

8.1.5.1 Example: One Factor ANCOVA with One Covariate, in Split plot design in RBD, Post-Hoc Tukey’s Test using R methods for data set of Gomez and Gomez (1984), page

442

8.1.6 Estimation of Two Missing Observations using Analysis of Covariance (ANCOVA)

8.1.6.1 Example: One Factor ANCOVA for Estimating Two Missing Observations with One Covariate, in Split plot design in RBD, using R methods for data set of Gomez and Gomez (1984), page 442

8.2 Transformations in ANOVA/ANCOVA

8.2.1 Introduction:

8.2.2 Some Variance Stabilizing Transformations:

8.2.3 Box-Cox Transformation:

8.2.3.1 Evolution of Box-Cox Transformation:

8.2.3.2 Estimation of Transformation parameter _ of Box-Cox Transformation:

8.2.3.3 Unit of Means Post Transformation:

8.2.4 Examples of Application of Transformations in ANOVA/ANCOVA Using R:

8.2.4.1 Example: Application of Box-Cox Transformations in ANOVA/ANCOVA Using R for data set of Montgomery (2013), pages 85-86 used in Montgomery (2013),

8.2.4.2 Example: Application of Arcsine Transformations in ANOVA/ANCOVA Using R for data set of Gomez and Gomez (1984), page 307:

8.3 Repeated Measures Design and ANOVA/ANCOVA

8.3.1 Introduction:

8.3.2 Statistical Formulation and Linear Statistical Model of Repeated Measures

Design:

8.3.2.1 Formulation of Intra-Class Correlation for the Repeated Measures Design:

8.3.3 ANOVA/ANCOVA for Repeated Measures Design:

8.3.3.1 Example: Sixty calves – repeated measure design along with polynomial analysis for data set of Kenward (1987), Table-1, pages 296-308

8.3.3.2 Example: Yields of Asparagus for year wise repeated measure cutting treatments for data set of Snedecor and Cochran (1989)