Mohammed I. Ageel

One-Way Classification

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 one-way classification or the completely randomized design. The one-way classification is the simplest and most useful model in statistics. In a one-way 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 one-way random model can be adapted to provide solutions to more complex models. Models involving two or more factors will be considered in succeeding chapters.

Analysis of Variance Using Statistical Computing Packages

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

Partially Nested Classifications

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.

Three-Way and Higher-Order Crossed Classifications

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.

Some Simple Experimental Designs

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.

Two-Way Nested (Hierarchical) Classification

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 b1, b2,…,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.

Finite Population and Other Models

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.