Design and Analysis of Experiments

Frontmatter

Chapter 1. Principles and Techniques

Abstract

Basic principles and techniques behind the design and analysis of experiments are discussed briefly in this chapter. Replication and blocking are introduced as design tools for decreasing variability. Randomization is introduced as a tool for eliminating systematic and personal biases. A brief distinction between exploratory and confirmatory analysis is provided.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 2. Planning Experiments

Abstract

This chapter highlights the importance of thorough experimental planning and provides a step-by-step guide for this process. The purpose and significance of the guide is illustrated with real experiments. The chapter also briefly introduces some standard experimental designs, such as completely randomized designs, block designs, and split-plot designs, whose detailed descriptions are provided in later chapters.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 3. Designs with One Source of Variation

Abstract

The design of an experiment (including determination of the number of observations to be collected), the model, and the analysis of the experiment are all dependent upon one another. This chapter introduces the analysis of an experiment based on a completely randomized design. A one-way analysis of variance model together with its assumptions is described, and estimation of the model parameters using least squares is discussed in detail. The estimation of the error variance is described, together with the calculation of a confidence bound. A hypothesis test, based on the F distribution, for testing equality of treatment effects is developed, and the analysis of variance table (ANOVA) is introduced. The chapter provides a detailed discussion of the calculation of sample sizes in a one-way analysis of variance model setting using the power of a test. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 4. Inferences for Contrasts and Treatment Means

Abstract

There are many types of comparisons among treatments that can be undertaken in the analysis of an experiment. Contrasts for pairwise comparisons, treatment-versus-control comparisons, trends, and difference of averages are introduced and examined in detail in this chapter. Confidence intervals and hypothesis testing for these contrasts are developed for the one-way analysis of variance model. The necessity for a multiplicity adjustment when examining more than one contrast is explained, and the Bonferroni, Scheffé, Tukey, and Dunnett methods of multiple comparisons are described. The chapter provides a detailed discussion of the calculation of sample sizes using the width of confidence intervals. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 5. Checking Model Assumptions

Abstract

Every model contains underlying assumptions about its form and about the distribution of error variables. In this chapter discusses methods of checking such assumptions for the one-way analysis of variance model, including checking the normality, constant variance, and independence of the errors. In this chapter, and throughout the book, the model assumption checks are made by examining residual plots. In the case of unequal variances, a transformation of data is suggested as well as methods for data analysis which incorporate unequal variances. The normality assumption is checked through construction of half-normal probability plots. A real experiment illustrates the techniques, and the use of SAS and R software is illustrated.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 6. Experiments with Two Crossed Treatment Factors

Abstract

When there are two treatment factors to be examined, their interaction needs to be considered too. In this chapter, the meaning of interaction is explained in detail. For completely randomized designs, the cell-means model and two analysis of variance models are presented together with the checking of model assumptions. Least squares parameter estimates are derived and formulas for methods of multiple comparison of treatments are given. Tests for the presence of interactions and significance of main-effects are developed, and analysis of variance tables (ANOVA) are presented. The chapter also discusses calculation of sample sizes. The analysis of experiments with one observation per treatment combination is discussed based on orthogonal contrasts. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 7. Several Crossed Treatment Factors

Abstract

When an experiment involves three or more treatment factors, the presence of high order interactions must be considered. Completely randomized designs for such factorial experiments form the topic of this chapter. Several different models are examined and, in the case of equal sample sizes per treatment combination, formulas for confidence intervals and hypothesis tests are developed. Since the total number of treatment combinations can become large, methods are presented for analyzing experiments with a single observation per treatment combination and no degrees of freedom for error. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 8. Polynomial Regression

Abstract

This chapter presents polynomial regression models for modelling the response from a factor with quantitative levels. The parameters in the model are estimated via least squares and the fit of the model is assessed with a lack-of-fit test. The simple linear regression model is examined in detail and confidence intervals and hypothesis tests about the parameter values are developed. The investigation of linear and quadratic trends in the data via orthogonal polynomials is also discussed. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 9. Analysis of Covariance

Abstract

Experiments which involve variables (covariates) that affect the response but that are not of direct interest nor can be controlled during the design of the experiment can be analyzed by the technique of analysis of covariance. This technique adjusts the treatment parameter estimates for the estimated values of the covariates. This chapter describes standard analysis of covariance models. Treatment parameter estimates are obtained via least squares, and analysis of covariance tests and confidence interval methods for the comparison of treatment effects are also developed. The concepts introduced in this chapter are illustrated through examples and use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 10. Complete Block Designs

Abstract

In the presence of controllable nuisance factors, the device of blocking divides the experimental material into homogeneous blocks in such a way that treatments can be compared under similar conditions. This chapter describes complete block designs (including randomized block designs), together with block design models, model assumption checks, multiple comparisons, sample size calculations, and analysis of variance. The analyses of complete block designs are illustrated through two real experiments, one having factorial treatment combinations. The use of R and SAS software is described.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 11. Incomplete Block Designs

Abstract

When a block design is needed for controlling the effects of nuisance factors, but the blocks cannot be made sufficiently large to accommodate all the treatments, incomplete block designs can be used instead. Basic design issues of block size, connectedness, and randomization are discussed in this chapter. Balanced incomplete block designs, group divisible designs, and cyclic designs, which are three efficient types of incomplete block designs, are described in greater detail. Formulas for the analysis of incomplete block designs are given, including simplifications for balanced incomplete block designs and group divisible designs. A specific experiment designed as a cyclic group divisible design is described and analyzed. A discussion of sample size calculation and factorial experiments in incomplete block designs is also included. The concepts introduced in this chapter are illustrated through several real experiments. The use of SAS and R software is illustrated for both the design and analysis of incomplete block designs.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 12. Designs with Two Blocking Factors

Abstract

This chapter examines row-column designs which involve two blocking factors that do not interact. Latin square designs and Youden designs are examined as simple examples of row-column designs. The model and an overview of analysis of variance, confidence intervals, and multiple comparisons for general row-column design is provided. The analysis simplifications that occur in Latin square and Youden designs are then described. The chapter provides a brief description of model assumption checking as well as an extension of the model to cover factorial experiments in row-column designs. The concepts introduced in this chapter are illustrated through examples and through the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 13. Confounded Two-Level Factorial Experiments

Abstract

Factorial experiments that involve several treatment factors tend to be large, and many such experiments can only support one observation per treatment combination. When such experiments need to be arranged in blocks, some of the treatment contrasts are confounded (muddled with) some of the block contrasts. For factors at two levels, methods of designing such experiments so that information is available on as many important treatment contrasts as possible are developed and illustrated in this chapter 13. Partial confounding in multi-replicate factorial experiments in blocks is explored. The chapter also compares traditional incomplete block designs with the multiple use of single-replicate confounded designs. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 14. Confounding in General Factorial Experiments

Abstract

In this chapter discusses confounding in single replicate experiments in which at least one factor has more than two levels. First, the case of three-levelled factors is considered and the techniques are then adapted to handle m-levelled factors, where m is a prime number. Next, pseudofactors are introduced to facilitate confounding for factors with non-prime numbers of levels. Asymmetrical experiments involving factors or pseudofactors at both two and three levels are also considered, as well as more complicated situations where the treatment factors have a mixture of 2, 3, 4, and 6 levels. Analysis of an experiment with partial confounding is illustrated using the SAS and R software packages

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 15. Fractional Factorial Experiments

Abstract

When there are numerous treatment factors to be examined but a limited budget, it may only be possible to observe a small proportion of the treatment combinations. Some main-effect and interaction contrasts cannot then be distinguished and are said to be aliased. Such fractional factorial experiments form the topic of in this chapter 15, where two methods of design construction are discussed to enable minimum aliasing between important contrasts. The first method is to select one block from a single-replicate block design. The second method uses the concept of an orthogonal array. Saturated designs, supersaturated designs, and definitive screening designs are introduced for searching for influential factors among a large number of potentially important factors. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 16. Response Surface Methodology

Abstract

Experiments for fitting a predictive model involving several continuous variables are known as response surface experiments. The objectives of response surface methodology include the determination of variable settings for which the mean response is optimized and the estimation of the response surface in the vicinity of this good location. The first part this chapter discusses first-order designs and first-order models, including lack of fit and the path of steepest ascent to locate the optimum. The second part of the chapter introduces second-order designs and models for exploring the vicinity of the optimum location. The application of response surface methodology is demonstrated through a real experiment. The concepts introduced in this chapter are illustrated through the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 17. Random Effects and Variance Components

Abstract

If the levels of a treatment factor used in an experiment are selected at random from a large population of possible levels, the effects of that factor are known as random effects. Random-effects models are discussed in this chapter. Of main concern is the variability of the effects of all the levels in the population rather than comparison of the effects of the few levels observed. The sample size calculation and modeling are introduced via experiments containing one random effect and are then extended to experiments with two or more random effects. The analysis of mixed-effects models is also discussed in which both fixed effects and random effects are present. The concepts introduced in this chapter are illustrated through examples and the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 18. Nested Models

Abstract

A factor is said to be nested within a second factor if each of its levels is observed in conjunction with a single level of the second factor. This chapter discusses how to recognize nesting when it occurs, how to formulate the associated models, and how to analyze the effects in these models. For fixed-effects nested models, the estimable contrasts are identified and the corresponding confidence intervals and hypothesis tests are developed. Estimation of variance components in random-effects nested models is described. The chapter concludes with the analysis of nested models using the SAS and R computer packages.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 19. Split-Plot Designs

Abstract

When the levels of some treatment factors are more difficult to change during the experiment than those of others, split-plot designs are necessary. In a split-plot design, the experimental units are called split plots, and are nested within whole plots, which themselves may or may not be nested within blocks. The split plots within each whole plot are assigned at random to the levels of one or more of the treatment factors. The levels of other treatment factors are assigned to whole plots and remain constant for all split plots within a whole plot. A typical model for the split-plot type of designs and the associated analysis are introduced in this chapter through examples. Designs with an extra level of nesting (split-split-plot designs) are briefly described as well. The concepts introduced in this chapter are illustrated through the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Chapter 20. Computer Experiments

Abstract

In contrast to experiments discussed in previous chapters which are carried out in the physical world, the experiments introduced in this chapter are carried out on a computer. Computer experiments can be run if the physical process of interest can be described by a mathematical model and if computer code can be written to compute the response from the mathematical model. A computer experiment is usually deterministic and will return the same response if observed more than once at the same input variable settings. Maximin Latin hypercube designs are introduced as commonly used designs for computer experiments. The chapter describes Gaussian stochastic process models for modeling the responses. The model parameters are estimated using the maximum likelihood technique. Prediction of output values and construction of prediction intervals are also discussed. The concepts introduced in this chapter are illustrated through a real experiment and the use of SAS and R software.

Angela Dean, Daniel Voss, Danel Draguljić

Backmatter