Analysis of Variance in Experimental Design

Frontmatter

1. Review of Statistical Concepts

Abstract

This text is written for those who have already had an intermediate level, noncalculus course in statistics. In this chapter we will review certain basic concepts and cover some fine points that may have been overlooked in earlier study. This chapter will also introduce the special notation used in the book, and my own statistical biases.

Harold R. Lindman

2. Analysis of Variance, One-Way, Fixed Effects

Abstract

In the simplest version of the t test, the means of two independent groups of scores are compared. The simplest form of the F test, or analysis of variance, is an extension of the t test to the comparison of more than two groups. For example, consider the data in Table 2.1.

Harold R. Lindman

3. Comparing Groups

Abstract

In Chapter 2 we learned how to test for general differences among the groups in a one-way design, and how to estimate the overall size of those differences. However, none of those techniques enabled us to tell where the differences were, that is, exactly which means were significantly different from which others. In this chapter and the next we will discuss methods of finding and testing for specific differences. In this chapter we will concentrate on testing for differences that we planned to test before obtaining the data. In Chapter 4 we will discuss ways of testing for differences that were not planned, but were suggested by the data. Both kinds of techniques are useful. In a well-planned experiment there are often specific differences in which we are interested; however, we should also be aware of unexpected differences in the data.

Harold R. Lindman

4. Other Multiple Comparison Methods

Abstract

A complete data analysis often requires more than a simple overall F test or a limited number of planned comparisons. Many important discoveries are “after the fact”—unanticipated relationships found in the data. Such relationships cannot be rested by planned comparisons; the choice of a comparison on the basis of apparent differences among the obtained means would introduce a strong bias in favor of rejecting the null hypothesis. Other techniques for making multiple comparisons exist, but they have very low power; in many cases, they will not find a significant difference unless it is large enough to be obvious without a test.

Harold R. Lindman

5. Two-Way Analysis of Variance

Abstract

Often, a simple one-way analysis of variance, with or without planned comparisons, is the best way to analyze data. Sometimes, however, simple one-way models are not appropriate, and even when they are appropriate, more complicated methods may be better suited to our needs.

Harold R. Lindman

6. Random Effects

Abstract

In the previous chapters, we assumed that scores within a group were randomly selected. However, the groups from which scores were to be taken were assumed to have been the deliberate choice of the experimenter. For the data in Table 4.1, for example, the two drugs were assumed to have been chosen because we were particularly interested in those two drugs. They were not assumed to have been selected randomly from a large population of potential drugs. In some cases, however, the groups or “treatments” themselves may have been selected randomly from a large number of potential treatments. In this chapter we will consider methods for analyzing such data.

Harold R. Lindman

7. Higher-Way Designs

Abstract

The principles discussed so far with respect to one-way and two-way designs can be extended to designs having any arbitrary number of factors. We need only extend the basic model to include the additional factors; however, the extension involves additional interaction effects. For example, consider a three-way analysis of variance (2 × 2 × 3), such as the one in Table 7.1.

Harold R. Lindman

8. Nested Designs

Abstract

In each of the designs discussed so far, every cell contained scores. However, sometimes it is impossible, impractical, or otherwise undesirable to obtain scores in every cell. This chapter and the next discuss some designs in which not all cells contain scores. These incomplete designs are convenient for some purposes, but they also have disadvantages. When choosing a design, the advantages and disadvantages must be weighed against each other.

Harold R. Lindman

9. Other Incomplete Designs

Abstract

If Factor B is nested in Factor A, it is impossible to separate the B main effect from the AB interaction. When effects cannot be separated, they are confounded. Whenever the factors in a design are not completely crossed, some effects will be confounded. However, in some designs the factors are neither completely crossed nor completed nested. The 4 × 6 design in Table 9.1 is an example; in this design, the factors are not completely crossed, yet neither is nested within the other. The analysis of most such partially nested designs is very difficult, but for certain designs it is relatively simple.

Harold R. Lindman

10. One-Way Designs with Quantitative Factors

Abstract

In some designs, meaningful numerical values can be assigned to the factor levels. An example of this might be a study of extinction (i.e., “unlearning”) of a learned response after 10, 20, 30, 40, 50, and 60 learning trials. The six numbers of learning trials are the six levels of the factor being studied; the data are the numbers of trials to extinction. The labels on the factor levels in this experiment are meaningful numerical values: Thirty trials are 20 more than 10, 60 are 40 more than 20, and so on. If the cell means from such an experiment were plotted in a graph, they might look like those in Figure 10.1 (taken from the data in Table 10.1); in this graph the numerical values of the factor levels dictate both their order and their spacing along the X axis. By contrast, for the data plotted in Figure 3.1, both the ordering and the spacing of the factor levels were arbitrary.

Harold R. Lindman

11. Trend Analyses in Multifactor Designs

Abstract

Frequently, a multifactor design has one or more numerically valued factors. When this occurs, trend analyses can be performed on main effects and interactions involving these factors. Just as in the one-way design, the theory of trend analysis differs for fixed and random numerical factors. For multifactor designs with random factors, however, the problem can be complicated; random factors with numerical values exert important influences on tests involving only fixed effects. In this chapter we will discuss designs in which all of the numerical factors are fixed; in Chapter 15 we will discuss designs that can be regarded as having numerically valued random factors.

Harold R. Lindman

12. Basic Matrix Algebra

Abstract

The remaining material requires an elementary knowledge of matrix algebra. This chapter contains the basic concepts necessary to understand it. If you know no matrix algebra, it should teach you enough to understand the remaining chapters. If you are not sure of your knowledge of matrix algebra, you should probably at least scan this material. If you already have a basic knowledge of matrix algebra, you may skip most of this chapter, although you should read at least the last section.

Harold R. Lindman

13. Multivariate Analysis of Variance

Abstract

This chapter and the next deal specifically with multivariate extensions of the analysis of variance. They are concerned with measures on more than one dependent variable. The two examples in Chapter 12 are illustrative. In Table 12.2 the experiment of Table 2.1 is extended to include measures of IQ and chronological age as well as of intellectual maturity. There are thus three dependent variables. In Table 12.3 the data in Table 5.1 are extended to include severity of illness as a second dependent variable.

Harold R. Lindman

14. Analysis of Covariance

Abstract

In multivariate analysis of variance we test all of the dependent variables simultaneously. In analysis of covariance we use some dependent variables as “controls” when testing for others. If we cannot control for certain variables experimentally by making them the levels of a factor, we may be able to control for them statistically by analysis of covariance. In this chapter, we will first give a simple example. We will then describe the model for analysis of covariance, the problems of interpretation, and the assumptions that must be made. Next, we will discuss the advantages and disadvantages of analysis of covariance, as compared with other possible ways of solving the same problem. Finally, we will describe the general analysis of covariance, with examples. The reader who wants only a general understanding of analysis of covariance can skip the final section.

Harold R. Lindman

15. General Linear Model

Abstract

Every analysis we have studied in this text has been based on a model equation containing terms that are added to obtain the observed scores. All of these are instances of the general linear model. The univariate general linear model is most easily represented in matrix terms as

$$X = AP + E $$

(15.1)

where X is a vector of N observed scores, P is a vector of m unknown parameters, A is an N × m matrix of coefficients, and E is a vector of N random errors. (In this chapter N is a scalar representing the total number of scores; n represents the number of scores in each group.)

Harold R. Lindman

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