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This chapter continues the discussion of the multi-set problem, presenting several forms of analysis of variance which have only a rudimentary form for the two sets Procrustes problem but which give more detailed information with K sets. The terms in this analysis of variance help with interpretation and throw more light on the possible choices of criteria suitable for fitting Procrustes models by least squares.

This chapter presents a reprint of Hald (1981), containing a detailed discussion of Thiele's contributions to statistics and a brief summary of some of his contributions to other areas. Topics covered include skew distributions, cumulants, estimation methods and k statistics, the linear model with normally distributed errors, analysis of variance, and a time series model combining Brownian motion and the linear model with normally distributed errors. Thiele's work is placed in a historical perspective and explained in modern terms.

Although detailed statistical analyses are beyond the scope of this book, there are some important general principles about organizing and analyzing data. Hypotheses have to be phrased in such a way that they can be tested and a level of ‘significance’ set for the outcome. This chapter presents a brief review of the most frequently used statistical tests that are available, including a discussion of parametric and non-parametric tests. The presentation of results and writing the project report or paper is an important part of any project.

This chapter outlines some basic statistical methods and shows their application in crystallography, particularly in analysing the results. Concepts include: random and systematic errors, precision and accuracy, and distributions and their properties. Important properties include the mean and standard deviation of a distribution. The normal (Gaussian) distribution is of particular importance in view of the Central Limit Theorem. The mean of a set of values may be weighted or unweighted, and the place of weights in crystallography is discussed, especially in structure refinement. Some statistical tests and tools are described and used, including normal probability plots and analyses of variance. Correlation and covariance among parameters and derived results are considered. Possible sources of systematic and other errors in crystal structures are listed and their impact assessed. A simple checklist is provided for assessing results.

Chapter 6 provides a description of one-way and two-way Analysis of Variance (ANOVA) and Analysis of Covariance (ANCOVA) statistical procedures. The chapter includes a presentation on the basic statistics that inform one-way ANOVA (i.e., the mean and the variance), components of the ANOVA summary table, estimation procedures, required assumptions, planned and post-hoc comparisons, and examples of the use of one-way ANOVA in social work research. Building on this, the next section provides an explanation of two-way ANOVA, including main and interaction effects, estimation procedures, required assumptions, and multiple comparisons as well as a description of examples of the use of two-way ANOVA in social work research. ANCOVA is covered in the third section, including basic statistics (i.e., adjusted mean and partitioned variance), the important role of the covariate, required assumptions as a consequence of including a covariate in the model, and ANCOVA examples from social work research. The final section provides step-by-step instructions for running one-way ANOVA in SPSS, using data from the National Education Longitudinal Study of 1988.

Analysis of variance is introduced using a worked analysis of some data collected by Charles Darwin on inbreeding depression in experimental maize plants. ANOVA is used to compare the height of a group of cross-pollinated plants with a group of self-fertilized seedlings. The least squares process is explained including the calculation of sums of squares and variances and the calculation of F-ratios. The organization of the results of a linear model into an ANOVA table is explained. The R code necessary to perform the analysis is worked through and explained in terms of the underlying statistical concepts and the interpretation of the R output demonstrated.

This chapter begins with an introduction to multivariate procedures, which allow social workers and other human services researchers to analyze multidimensional social problems and interventions in ways that minimize oversimplification. Examples of multivariate statistical procedures to predict and describe relationships include multivariate multiple regression (MMR), multivariate analysis of variance (MANOVA), and multivariate analysis of covariance (MANCOVA). Structural equation modeling (SEM) may be used for data simplification and reduction, description, and prediction. The discussion then turns to the rationale for multivariate analysis followed by a description of the organization and contents of this book.

This chapter builds on the previous chapter by elaborating from theories of causal knowledge presented earlier to practical considerations for the design, execution, and analysis of randomized experiments and randomized controlled trials in music education research. The straightforward statistical analysis of the two-group experimental designs is explained through the t test. The analysis of variance technique is explained for the analysis of experimental and quasi-experimental data involving more than two groups. The chapter closes with a discussion of the analysis of data arising from experiments where additional data, beyond group membership and the score on an outcome measure, is known about the participants (i.e., analysis of covariance).

This chapter presents a new analysis of experimental data that examines the relative effects of three factors—predation, food supply, and interspecific competition—on the population growth of voles. It aims to test the proposition that predation is an important factor which reduces population growth of small mammals during population increases as well as declines, an idea called Pearson's hypothesis. The chapter quantifies the relative impact on growth of prairie vole populations during the growing season of manipulating access by predators, food supply, and presence of a competing species (the meadow vole, M. ochrogaster). It finds that all three factors had substantial effects on population growth, but that predation had two–three times greater impact than did food supply or competition, using analysis of variance and linear modeling.

This chapter introduces the reader to more possibilities for thinking about causal questions and for laying the foundational concepts necessary for conducting data analyses that correspond to more complex experimental designs. The discussion of experimental design types presented in chapter 8 is expanded to include within-subjects designs, factorial designs, mixed designs, and designs for multivariate outcomes. Prototypical examples of each design type are presented along with the typical analysis tools used for testing the associated experimental hypotheses. Hypothetical examples of research designs that are suitable for illustrating analyses with repeated-measures ANOVA, factorial or multiway ANOVA, and MANOVA (multivariate analysis of variance).

Analysis of variance is introduced using a worked analysis of some data collected by Charles Darwin on inbreeding depression in experimental maize plants. ANOVA is used to compare the heights of a group of cross-pollinated plants with a group of self-fertilized seedlings. The least squares process is explained, including the calculation of sums of squares and variances and the calculation of F-ratios. The organization of the results of a linear model into an ANOVA table is explained. The R code necessary to perform the analysis is worked through and explained in terms of the underlying statistical concepts, and the interpretation of the R output is demonstrated.

The varying demands of justice are often thought to depend on a distinction between natural and social inequalities, but making this distinction has been little discussed, and it has been dismissed by philosophers of biology. It cannot be established by a simple causal criterion, nor by the use of the analysis of variance, nor by distinguishing the innate from the acquired. Whether an inequality can be socially controlled provides the most plausible criterion, so ‘natural’ and ‘social’ are misleading labels for types of inequality. The analysis of these depends, besides, on how fine-grained the descriptions are; so it implausible to think that the natural/social distinction, when drawn in terms of social control, is relevant to theories of justice.

The chapter sets out the aims of the book, the approach, what is covered in the book and what is not. The book starts by introducing several different variations of the basic linear model analysis (analysis of variance, linear regression, analysis of covariance, etc). Then two extensions are introduced: generalized linear models (for data with non-normal distributions) and mixed-effects models (for data with multiple levels and a hierarchical structure). The book ends by combining these two extensions into generalized linear mixed-effects models. To allow a learning-by-doing approach the R code necessary to perform the basic analysis is embedded in the text along with the key output from R.

This chapter summarizes similarities and differences between multivariate analysis of variance (MANOVA), multivariate analysis of covariance (MANCOVA), multivariate multiple regression (MMR), and structural equation modeling (SEM). It offers suggestions to guide their differential use. It also compares and contrasts MANOVA and MANCOVA versus MMR, MANOVA and MANCOVA versus SEM, and MMR versus SEM.

What are normal data? What are non-normal data? How do you tell them apart and how do you know what sort of data you have? Chapter 5 begins teaching statistical analysis with a focus on understanding the shape of one’s data and how to measure it. The RxP dataset is explored and the principal focus in this chapter is on normal data and on conducting an Analysis of Variance (ANOVA), although classic non-parametric statistics such as a Kruskal-Wallis test and Mann-Whitney U test are also introduced. Readers are taught how to test for data normality, assess model fit, interpret the summary output from the model, calculate summary statistics and p-values, and how to conduct post-hoc tests (Tukey’s HSD) to compare different levels of a single predictor.

Chapter 6 continues exploring the world of statistics that are covered within the linear model, namely two-way and three-way ANOVA, linear regression and analysis of covariance (ANCOVA). In each type of model, a detailed description of how to interpret the summary output is undertaken, including understanding how to interpret and plot interactions. Conducting post-hoc analyses and using the predict() function are also covered. The chapter ends by reinforcing earlier plotting skills in ggplot2 by walking through an example of making a professional looking figure with multiple non-linear regression curves and confidence intervals.