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Sahai, Hardeo; Ojeda, Mario Miguel
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In the study of random and mixed effects models, our interest lies primarily in making inferences about the specific variance components. In this chapter, we consider some general methods for point estimation, confidence intervals, and hypothesis testing for linear models involving random effects. Most of the chapter is devoted to the study of various methods of point estimation of variance components. However, in the last two sections, we briefly address the problem of hypothesis testing and confidence intervals. There are now several methods available for estimation of variance components from unbalanced data. Henderson’s (1953) paper can probably be characterized as the first attempt to systematically describe different adaptations of the ANOVA methodology for estimating variance components from unbalanced data. Henderson outlined three methods for obtaining estimators of variance components.
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Sahai, Hardeo; Ojeda, Mario Miguel
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Consider two factors A and B with a and b levels, respectively, involving a factorial arrangement. Assume that n_{ij} (≥ 0) observations are taken corresponding to the (i, j)th cell. The model for this design is known as the unbalanced twoway crossed classification. This model is the same as the one considered in Chapter 4 except that now the number of observations per cell is not constant but varies from cell to cell. Models of this type frequently occur in many experiments and surveys since many studies cannot guarantee the same number of observations for each cell. This chapter is devoted to the study of a random effects model for unbalanced twoway crossed classification with interaction.
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Sahai, Hardeo; Ojeda, Mario Miguel
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The twoway crossed model considered in Chapter 3 uses the simple additive model, which makes an important assumption that the value of the difference between the mean responses at two levels of A is the same at each level B. However, in many situations, this simple additive model may not be appropriate. When the differences between the mean response at different levels of A tend to vary over the different levels of B, it is said that the two factors interact. If an experimenter makes more than one observation per cell, it permits him to investigate not only the main effects of both factors but also their interaction. In this chapter, we consider a twoway crossed model with more than one observation per cell, which allows the investigation of interaction terms between the two factors.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In the preceding chapter, we considered a random effects model involving a twoway nested classification. Examples of three and higherorder nested classifications occur frequently in many industrial experiments where raw material is first broken up into batches and then into subbatches, subsubbatches, and so forth. For example, in an experiment designed to identify various sources of variability in tensile strength measurements, one may randomly select a lots of raw material, b boxes are taken from each lot, c sample preparations are made from the material in each box, and finally n tensile strength tests are performed for each preparation. These factors often present themselves in a hierarchical manner and are appropriately specified as random effects. In this chapter, we consider a random effects model involving a threeway nested classification and indicate its generalization to higherorder nested classifications.
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Sahai, Hardeo; Ojeda, Mario Miguel
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Consider an experiment with two factors A and B where the levels of B are nested within the levels of A. Assume that there are a levels of A and within the ith level of A there are b_{i} levels of B and n_{ij} observations are taken at the jth level of B. The model for this design is known as the unbalanced twoway nested classification. This model is the same as the one considered in Chapter 6 except that now b_{i}s and n_{ij}s rather than being constants vary from one level to the other. Models of this type are frequently used in many experiments and surveys since the sampling plans cannot be balanced because of the availability of limited resources. In addition, unless the number of levels of factor A is very large, the estimate of its variance component may be very imprecise for a balanced design. In this chapter, we consider the random effects model for the unbalanced twoway nested classification.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In this chapter, we consider the random effects model involving two factors in a factorial arrangement where the numbers of observations in each cell are different. We further assume that the model does not involve any interaction terms. Consider two factors A and B and let there be n_{ij} (≥ 0) observations corresponding to the (i, j)th cell. The model for this design is known as the twoway crossed classification without interaction.
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Sahai, Hardeo; Ojeda, Mario Miguel
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The oneway classification discussed in Chapter 2 involved the levels of only a single factor. It is the simplest model in terms of experimental layout, assumptions, computations, and analyses. However, in many investigations, it is desirable to measure response at combinations of levels of two or more factors considered simultaneously. Two factors are said to be crossed if the data contain observations at each combination of a level of one factor with a level of the other factor. Consider two factors A and B, where a levels are sampled from a large population of levels of A and b levels are sampled from a large population of levels of B, and one observation is made on each of the ab cells. This type of layout is commonly known as the balanced twoway crossed random model with one observation per cell. It can also be viewed as a randomized complete block design where both blocks and treatments are regarded as random.
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Sahai, Hardeo; Ojeda, Mario Miguel
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The completely nested or hierarchical classification involving several stages arises in many areas of scientific research and applications. For example, in a large scale sample survey, experiments may be laid down on very many blocks, and the blocks are then naturally classified by cities, the cities by states in which they occur; and the states by the regions, and so forth. In a genetic investigation of dairy production, the units could be cattle classified by sires, sires classified by their dams, and so on. Frequently, the designs employed in these investigations are unbalanced, sometimes inadvertently. In this chapter, we shall briefly outline the analysis of variance for an unbalanced rway nested classification.
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Sahai, Hardeo; Ojeda, Mario Miguel
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Volume I of the text was devoted to a study of various models with the common feature that the same numbers of observations were taken from each treatment group or in each submost subcell. When these numbers are the same, the data are referred to as balanced data; in contrast, when the numbers of observations in the cells are not all equal, the data are known as unbalanced data. In general, it is desirable to have equal numbers of observations in each subclass since the experiments with unbalanced data are much more complex and difficult to analyze and interpret than the ones with balanced data. However, in many practical situations, it is not always possible to have equal numbers of observations for the treatments or groups. Even if an experiment is wellthoughtout and planned to be balanced, it may run into problems during execution due to circumstances beyond the control of the experimenter; for example, missing values or deletion of faulty observations may result in different sample sizes in different groups or cells. In many cases, the data may arise through a sample survey where the numbers of observations per group cannot be predetermined, or through an experiment designed to yield balanced data but which actually may result in unbalanced data because some plants or animals may die, patients may drop out or be taken out of the study. For example, in many clinical investigations involving a followup, patients may decide to discontinue their participation, they may withdraw due to side effects, they may die, or they are simply lost to followup.
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Sahai, Hardeo; Ojeda, Mario Miguel
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Crossed classifications involving several factors are common in experiments and surveys in many substantive fields of research. Consider three factors A, B, and C with a, b, and c levels, respectively, involving a factorial arrangement. Assume that n_{ijk} (≥ 0) observations are taken corresponding to the (i, j, k)th cell. The model for this design is known as the unbalanced threeway crossedclassification model. This model is the same as the one considered in Chapter 5 except that now the number of observations per cell is not constant but varies from cell to cell including some cells with no data. Models of this type frequently occur in many experiments and surveys since many investigations cannot guarantee the same number of observations for each cell. In this chapter, we briefly outline the analysis of random effects model for the unbalanced threeway crossedclassification with interaction and indicate its extension to higherorder classifications.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In the preceding two chapters, we considered random models involving two factors. In many fields of research, an investigator often works with experiments or surveys involving more than two factors; which entails simultaneous data collection under conditions determined by several factors. This type of design is usually more economical and can provide more information than separate oneway or twoway layouts. The analysis of variance of the twoway crossed model can be readily extended to situations involving three or more factors. In this chapter, we study random effects models involving three factors in somewhat greater detail. The extension of the model to experiments involving four or more factors is also indicated briefly.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In previous chapters, we have considered random effects models for various crossed and nested designs with equal numbers in all cells and subclasses, and which are called balanced complete models. In this chapter, we present a unified treatment of balanced random effects models in terms of the socalled general linear model.
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Sahai, Hardeo; Ojeda, Mario Miguel
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Consider three factors A, B, and C, where B is nested within A and C is nested within B. Suppose that each A level has b_{i}B levels, each B level has c_{ij}C levels, and n_{ijk} observations are taken from each C level. This is an example of a threeway unbalanced nested design and is frequently encountered in many areas of scientific applications. For example, suppose a clinical study involves monthly blood analysis of patients participating in the study. Two blood tests are made on each patient and three analyses are made from each test. Here, tests are nested within patients and analyses are made within tests. It may happen that on certain occasions some patients fail to appear for their blood tests and this makes the design unbalanced. In this chapter, we will study the random effects model for the threeway nested classification involving an unbalanced design.
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Sahai, Hardeo; Ojeda, Mario Miguel
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The nature and magnitude of variability of repeated observations plays a fundamental role in many fields of scientific investigation. For example, questions such as, the determination of sample size to estimate an effect with a given precision in a factorial experiment, estimation of standard errors of sample estimates in a complex survey, and selection of breeding programs to estimate genetic parameters, require the knowledge of the nature and magnitude of variability of measurement errors. The analysis of variance as understood and practiced today is concerned with the determination of sources and magnitude of variability introduced by one or more factors or stages of a process. The methodology was developed primarily by R. A. Fisher during the 1920s, who defined it as “separation of the variance ascribable to one group of causes from the variance ascribable to other groups.” Fisher is also credited with introducing the terms “variance” and “analysis of variance” into statistics. Since its introduction by Fisher (1925), the analysis of variance has been the most widely used statistical tool to obtain tests of significance of treatment effects. The technique has been developed largely in connection with the problems of agricultural experimentation. Scheffé (1959, p. 3) gives the following definition of the analysis of variance:
“The analysis of variance is a statistical technique for analyzing measurements depending on several kinds of effects operating simultaneously, to decide which kinds of effects are important and to estimate the effects. The measurements or observations may be in an experimental science like genetics or nonexperimental one like astronomy.”
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Sahai, Hardeo; Ojeda, Mario Miguel
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In the preceding two chapters, we have considered experimental situations where the levels of two factors are crossed. In this and the following chapter we onsider experiments where the levels of one of the factors are nested within the levels of the other factor. The data for a twoway nested classification are similar hat of a single factor classification except that now replications are grouped into different sets arising from the levels of the nested factor for a given level of the main factor. Suppose the main factor A has a levels and the nested factor B has ab levels which are grouped into a sets of b levels each, and n observations are made at each level of the factor B giving a total of abn observations. The nested or hierarchical designs of this type are very important in many industrial and genetic investigations. For example, suppose an experiment is designed to investigate the variability of a certain material by randomly selecting a batches, b samples are made from each batch, and finally n analyses are performed on each sample. The purpose of the investigation may be to make inferences about the relative contribution of each source of variation to the total variance or to make inferences about the variance components individually. For another example, suppose in a breeding experiment a random sample of a sires is taken, each sire is mated to a sample of b dams, and finally n offspring are produced from each siredam mating. Again, the purpose of the investigation may be to study the relative magnitude of the variance components or to make inferences about them individually.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In this chapter, we consider the random effect model involving only a single factor or variable in an experimental study involving a comparison of a set of treatments, where each of the treatments can be randomly assigned to experimental units. Such a layout is commonly known as the oneway classification or the completely randomized design. The oneway classification is the simplest and most useful model in statistics. In a oneway random effects model, treatments, groups, or levels of a factor are regarded to be a random sample from a large population. It is the simplest nontrivial and widely used variance component model. Moreover, the statistical concepts and tools developed to handle a oneway random model can be adapted to provide solutions to more complex models. Models involving two or more factors will be considered in succeeding chapters.
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Sahai, Hardeo; Ojeda, Mario Miguel
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In Chapter 2, we considered the socalled balanced oneway random effects model where n_{i}s are all equal. Equal numbers of observations for each treatment group or factor level are desirable because of the simplicity of organizing the experimental data and subsequent analysis. However, as indicated in Chapter 9, for a variety of reasons, more data may be available for some levels than for others. In this chapter, we consider a oneway random effects model involving unequal numbers of observations for different groups. This model is widely used in a number of applications in science and engineering.
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