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The analysis of variance (ANOYA) models have become one of the most widely used tools of modern statistics for analyzing multifactor data. The ANOYA models provide versatile statistical tools for studying the relationship between a dependent variable and one or more independent variables. The ANOYA mod­ els are employed to determine whether different variables interact and which factors or factor combinations are most important. They are appealing because they provide a conceptually simple technique for investigating statistical rela­ tionships among different independent variables known as factors. Currently there are several texts and monographs available on the sub­ ject. However, some of them such as those of Scheffe (1959) and Fisher and McDonald (1978), are written for mathematically advanced readers, requiring a good background in calculus, matrix algebra, and statistical theory; whereas others such as Guenther (1964), Huitson (1971), and Dunn and Clark (1987), although they assume only a background in elementary algebra and statistics, treat the subject somewhat scantily and provide only a superficial discussion of the random and mixed effects analysis of variance.

Inhaltsverzeichnis

1. Introduction

Abstract
The variation among physical observations is a common characteristic of all scientific measurements. This property of observations, that is, their failure to reproduce themselves exactly, arises from the necessity of taking the observations under different conditions. Thus, in a given experiment, readings may have to be taken by different persons at different periods of time or under different operating or experimental conditions. For example, there may be a large number of external conditions over which the experimenter has no control. Many of these uncontrolled external conditions may not affect the results of the experiment to any significant degree. However, some of them may change the outcome of the experiment appreciably. Such external conditions are commonly known as the factors.
Hardeo Sahai, Mohammed I. Ageel

2. One-Way Classification

Abstract
In this chapter we consider the analysis of variance associated with experiments having only one factor or experimental variable. Such an experimental layout is commonly known as one-way classification in which sample observations are classified (grouped) by only a single criterion. It provides the simplest data structure containing one or more observations at every level of a single factor. One-way classification is a very useful model in statistics. Many complex experimental situations can often be considered as one-way classification. In the succeeding chapters, we discuss situations involving two or more experimental variables.
Hardeo Sahai, Mohammed I. Ageel

3. Two-Way Crossed Classification Without Interaction

Abstract
The major advantage of the one-way classification (one-factor design) discussed in the preceding chapter is its simplicity, which extends to the experimental layout, the model and assumptions underlying the analysis of variance, and the computations involved in the analysis. The major disadvantage of such a design is its relative inefficiency. The error variance will usually be large compared to that resulting from other designs. This is in part offset by the fact that no other design yields as many degrees of freedom for the error variance as does this design.
Hardeo Sahai, Mohammed I. Ageel

4. Two-Way Crossed Classification with Interaction

Abstract
Suppose that we relax the requirement of model (3.1.1) that there be exactly one observation in each of the a x b cells of the two-way layout. The model remains the same except that we could now use yijk to designate the k-th observation at the i-th level of A and the j-th level of B, that is, in the (i, j)-th cell. We now suppose that there are n (n ≥ 1) observations in each cell. With n = 1, the model (3.1.1) will be a special case of the model being considered here. With an arbitrary integer value of n, the analysis of variance will be a simple extension of that described in Chapter 3. However, an important and somewhat restrictive implication of the simple additive model discussed in Chapter III is that the value of the difference between the mean responses at two levels of A is the same at each level of B. However, in many cases, this simple additive model may not be appropriate. The failure of the differences between the mean responses at the different levels of A to remain constant over the different levels of B is attributed to interaction between the two factors. Having more than one observation per cell allows a researcher to investigate the main effects of both factors and their interaction. In this chapter, we study the model involving two factors with interaction terms.
Hardeo Sahai, Mohammed I. Ageel

5. Three-Way and Higher-Order Crossed Classifications

Abstract
Many experiments and surveys involve three or more factors. Multifactor layouts entail data collection under conditions determined by several factors simultaneously. Such layouts usually provide more information and often can be even more economical than separate one-way or two-way designs. The models and analysis of variance for the case of three or more factors are straightforward extensions of the two-way crossed model. The methods of analysis of variance for the two-way crossed classification discussed in the preceding two chapters can thus be readily generalized to three-way and higher-order classifications. In this chapter, we study the three-way crossed classification in some detail because it serves as an illustration as to how the analysis can be extended when four or more factors are involved. Generalizations to four-way and higher-order classifications are briefly outlined.
Hardeo Sahai, Mohammed I. Ageel

6. Two-Way Nested (Hierarchical) Classification

Abstract
In Chapters 3 through 5 we considered analysis of variance for experiments commonly referred to as crossed classifications. In a crossed-classification, data cells are formed by combining of each level of one factor with each level of every other factor. We now consider experiments involving two factors such that the levels of one factor occur only within the levels of another factor. Here, the levels of a given factor are all different across the levels of the other factor. More specifically, given two factors A and B, the levels of B are said to be nested within the levels of A, or more briefly B is nested within A, if every level of B appears with only a single level of A in the observations. This means that if the factor A has a levels, then the levels of B fall into a sets of b 1, b 2,…,b a levels, respectively, such that the i-th set appears with the i-th level of A. These designs are commonly known as nested or hierarchical designs where the levels of factor B are nested within the levels of factor A.
Hardeo Sahai, Mohammed I. Ageel

7. Three-Way and Higher-Order Nested Classifications

Abstract
The results of the preceding chapter can be readily extended to the case of three-way and the general q-way nested or hierarchical classifications. As an example of a three-way nested classification, suppose a chemical company wishes to examine the strength of a certain liquid chemical. The chemical is made in large vats and then is barreled. To study the strength of the chemical, an analyst randomly selects three different vats of the product. Three barrels are selected at random from each vat and then three samples are taken from each barrel. Finally, two independent measurements are made on each sample. The physical layout can be depicted schematically as shown in Figure 7.1 In this experiment, barrels are nested within the levels of the factor vats and samples are nested within the levels of the factor barrels. This is the so-called three-way nested classification having two replicates or measurements. In this chapter, we consider the three-way nested classification and indicate its generalization to higher-order nested classifications.
Hardeo Sahai, Mohammed I. Ageel

8. Partially Nested Classifications

Abstract
In the preceding chapters, we discussed classification models involving several factors that are either all crossed or all nested. Occasionally, in a multifactor experiment, some factors will be crossed and others nested. Such designs are called partially nested (hierarchical), crossed-nested, nested-factorial, or mixed-classification designs. For example, suppose that in a study involving an industrial experiment it is desired to test three different methods of a production process. For each method, five operators are employed. The experiment is carried out over a period of four days and three observations are obtained for each combination of method, operator, and day. Because of the nature of the experiment, the five operators employed under Method I are really individuals different from the five operators under Method II or Method III and the five operators under Method II are different from those under Method III. The physical layout of such an experiment can be depicted schematically as shown in Figure 8.1 In this experiment, the days are crossed with the methods and operators, and operators are nested within methods.
Hardeo Sahai, Mohammed I. Ageel

9. Finite Population and Other Models

Abstract
As discussed earlier, so far in this volume we have been primarily concerned with random effects models or Model II based on the infinite population theory, that is, when the treatments included in the experiment are assumed to be a random sample from a population of treatments having infinite size or when the experimenter selects the levels at random from a large number of possible levels of a factor usually considered as infinite. However, as described in Section 1.4, there are situations when the treatments selected may be a sample from a finite population and then the assumptions of an infinite population may be inappropriate. For example, in a large laboratory, there could be a total of 10 analysts and the data obtained on just three of them could be used to make inferences concerning a new method for the determination of arginine content as used by the entire group of 10 analysts.
Hardeo Sahai, Mohammed I. Ageel

10. Some Simple Experimental Designs

Abstract
In the previous chapters we developed techniques suitable for analyzing experimental data. It is important at this point to consider the manner in which the experimental data were collected as this greatly influences the choice of the proper technique for data analysis. If an experiment has been properly designed or planned, the data will have been collected in the most efficient manner for the problem being considered. Experimental design is the sequence of steps initially taken to ensure that the data will be obtained in such a way that analysis will lead immediately to valid statistical inferences. The purpose of statistically designing an experiment is to collect the maximum amount of useful information with a minimum expenditure of time and resources. It is important to remember that the design of the experiment should be as simple as possible consistent with the objectives and requirements of the problem. The purpose of this chapter is to introduce some basic principles of experimental design and discuss some commonly employed experimental designs of general applications.
Hardeo Sahai, Mohammed I. Ageel

11. Analysis of Variance Using Statistical Computing Packages

Abstract
The widespread availability of modern high speed mainframes and microcomputers and myriad accompanying software have made it much simpler to perform a wide range of statistical analyses. The use of statistical computing packages or software can make it possible even for a relatively inexperienced person to utilize computers to perform a statistical analysis. Although there are numerous statistical packages that can perform the analysis of variance, we have chosen to include for this volume three statistical packages that are most widely used by scientists and researchers throughout the world and that have become standards in the field),1 The packages are the Statistical Analysis System (SAS), the Statistical Product and Service Solutions (SPSS),2 and the Biomedical Programs (BMDP). In the following we provide a brief introduction to these packages and their use for performing an analysis of variance, and related statistical tests of significance.3,4,5
Hardeo Sahai, Mohammed I. Ageel

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