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This chapter discusses one‐way random effects model, two‐way random effects model, and two‐way mixed effects model. The introduced random effects models can be classified also as mixed effects models, since they have a general mean as a fixed effect. They are formulated as a general Gaussian linear mixed effects model. The chapter also discusses problems of negative estimators of variance components. The negative best unbiased estimator is a difficult problem, often encountered in the estimation of variance components. The chapter then introduces the idea of Smith and Murray to justify the negative estimator by regarding a certain variance component as a covariance. Next, the variance components are estimated by the moment method. Some authors evaluated the variances of estimation by the methods G 1 and G 2 of Yates and G 3 of Henderson in various unbalanced two‐way designs.

Statistical analyses were applied at the Hanford Site, USA, to assess groundwater contamination problems that included (1) determining local backgrounds to ascertain whether a facility is affecting the groundwater quality and (2) determining a 'pre-Hanford' groundwater background to allow formulation of background-based cleanup standards. The primary purpose of this paper is to extend the random effects models for (1) assessing the spatial, temporal, and analytical variability of groundwater background measurements; (2) demonstrating that the usual variance estimate s2, which ignores the variance components, is a biased estimator; (3) providing formulas for calculating the amount of bias; and (4) recommending monitoring strategies to reduce the uncertainty in estimating the average background concentrations. A case study is provided. Results indicate that (1) without considering spatial and temporal variability, there is a high probability of false positives, resulting in unnecessary remediation and/or monitoring expenses; (2) the most effective way to reduce the uncertainty in estimating the average background, and enhance the power of the statistical tests in general, is to increase the number of background wells; and (3) background for a specific constituent should be considered as a statistical distribution, not as a single value or threshold. The methods and the related analysis of variance tables discussed in this paper can be used as diagnostic tools in documenting the extent of inherent spatial and/or temporal variation and to help select an appropriate statistical method for testing purposes.

Along with linguistic messages, prosody is an essential paralinguistic component of emotional speech. Prosodic parameters such as intensity, fundamental frequency (F0), and duration were studied worldwide to understand the relationship between emotions and corresponding prosody features for various languages. For evaluating prosodic aspects of emotional Marathi speech, the Marathi language has received less attention. This study aims to see how different emotions affect suprasegmental properties such as pitch, duration, and intensity in Marathi's emotional speech. This study investigates the changes in prosodic features based on emotions, gender, speakers, utterances, and other aspects using a database with 440 utterances in happiness, fear, anger, and neutral emotions recorded by eleven Marathi professional artists in a recording studio. The acoustic analysis of the prosodic features was employed using PRAAT, a speech analysis framework. A statistical study using a two-way Analysis of Variance (two-way ANOVA) explores emotion, gender, and their interaction for mean pitch, mean intensity, and sentence utterance time. In addition, three distinct linear mixed-effect models (LMM), one for each prosody characteristic designed comprising emotion and gender factors as fixed effect variables, whereas speakers and sentences as random effect variables. The relevance of the fixed effect and random effect on each prosodic variable was verified using likelihood ratio tests that assess the goodness of fit. Based on Marathi's emotional speech, the R programming language examined linear mixed modeling for mean pitch, mean intensity, and sentence duration.

This chapter discusses several of the commonly used experimental designs and their statistical analyses. The principles of experimental design and methods of statistical analysis of experimental results considered in the chapter are commonly referred to as analysis of variance methods. The chapter first discusses one‐way experimental designs and completely randomized designs. It then focuses on randomized complete block (RCB) designs in which people have one factor of prime interest, while the other factor is referred to blocks. In RCB designs, the blocks are used to eliminate the effects of a nuisance variable, and it may be that our only interest in studying block effects is to find out whether the creation of blocks was justified. The chapter also considers two‐way experimental layouts also called two‐way factorial experimental designs. It further explains how to analyze data coming out of experiments with fixed, random, or mixed effects.

Treatment effects may be moderated by individual or contextual characteristics or by other concurrent or consecutive treatments. This chapter reveals the conceptual confusion in past literature and aims to clarify the definitions of “moderated treatment effects” in terms of potential outcomes. The chapter then reviews experimental designs including randomized block designs, factorial designs, and multisite randomized trials and the corresponding analytic methods that are suitable for evaluating moderated treatment effects. It also introduces principal stratification, a relatively new approach for disclosing the heterogeneity of treatment effects across latent subpopulations.

A marginal model for binomial data where the experimental units involving one or two random factors is presented. Two variance-covariance models are derived based on the multiplicative error formulation. The parameters of mean and variance components are estimated using quasi-likelihood and method of moments, respectively. The model is applied to analyzing multivariate overdispersed binomial data from a developmental toxicity experiment.

SUMMARY A mixed effects model is assumed for comparing treatments based on repeated measurements. The covariance structure of the data is interpreted as some systematic inhomogeneity of individual profiles along the time axis, rather than as serial correlation. Then a method of comparing treatment effects is proposed as well as that of testing the homogeneity of individual profiles. A follow-up analysis of residuals for the resulting model is also mentioned.