David Bowers

Multiple Groups Analysis

The previous chapters have dealt with methods from which orthogonal factors emerge. Clusters of related variables are then obtained subsequently by orthogonal or oblique rotations to simple structure. This chapter is concerned with a method called multiple groups analysis from which an oblique factor matrix is obtained directly and all factors are extracted simultaneously. The number of such groups (and therefore factors) is assumed at the outset either on the basis of a priori knowledge of the field of study or by a systematic procedure which the interested reader can find in more advanced texts, such as Harman [1967]. It is, however, not serious if an incorrect assumption about the number of groups is made initially.

Principal Factor Analysis

In Chapter 2 we examined the method of factor analysis using the centroid technique. It was stated then that the aim of the analysis was to explain the correlations between the original observed variables in terms of their correlations with a smaller set of factors. In this chapter we will examine two other methods of analysis, principal factor analysis and principal component analysis. The approach of the two methods is similar and their aim, to aid interpretation of the underlying structure of the interrelationships between variables, is the same. But there is in fact, as we shall see later, a fundamental difference between the two methods.

Sampling Distribution of Sample Statistics

We stated in Chapter 9 that the process of statistical inference (whether estimation or hypothesis testing) depends on knowledge of what is referred to as the sampling distribution of sample statistics. To illustrate the idea of a sampling distribution, consider the example of the distribution of the variable X, which measures the expenditure by customers in a supermarket, so that X i measures the expenditure of the ith customer. Suppose we are interested in the mean expenditure, µ, of all customers, and intend to guess or estimate its value by taking a random sample of 100 customers, recording the expenditure of each customer in the sample, and calculating the mean expenditure of this sample, which we denote 1 (we have added the subscript 1 because this is the first of a number of samples we will consider). 1, is calculated using equation (4.4) i.e. 1 = ΣX i /n, where the X i are the sample values and n is the sample size. 1 will be a random variable because its value will not be known until an experiment is performed, the ‘experiment’ in this case consists of determining how much each customer spends.

The Simple Linear Regression Model

Up to now, we have been largely concerned with statistics in the context of only one variable. In the first five chapters of this book we discussed descriptive statistics of a single variable (except in Section 3.8 when we considered joint frequency distributions). In Chapters 6 to 8 we examined the ideas of probability and probability distributions, again in the context of a single variable (except in Section 8.5 when we considered briefly the idea of the covariance between two variables). In Chapters 9, 10, 11 and 12 we discussed statistical inference, and in particular estimation and hypothesis testing about a single variable. Finally in the previous chapter, we discussed inference about two variables.

An Introduction to Probability Theory

The first part of this book has been concerned with the efficient description of data (by means of suitable graphs and tables), and with the production of summary measures of location and dispersion. In descriptive statistics no attempt is made to generalise from any particular sample of data to draw wider implications about the population from which the sample was taken.

Data Presentation — Qualitative Data

In this chapter we will consider ways of presenting qualitative data, i.e. that which arises from the use of nominal measuring scales. In what follows we will deal with both graphical (i.e. diagrammatic) and tabular methods; we will not discuss the most suitable design of tables and diagrams, readers are strongly recommended to refer to Chapman (1986) for an excellent discussion of this.

Measures of Location

In the previous two chapters we have discussed ways in which data can be arranged and presented so that its principal features can be more easily observed. The descriptive procedures we examined were either graphical (bar charts, pie diagrams, histograms, ogives, etc.), or tabular (frequency distributions, relative frequency and cumulative frequency distributions, and so on).

Measures of Dispersion

In the previous chapter on measures of location we considered the first of two important summary descriptive measures. In this chapter we will discuss the second of these two measures, a measure of the spread or dispersion of the data. Recall that at the beginning of Chapter 4 we considered the frequency distributions of male and female absenteeism, measured in days per year and illustrated in Figure 4.2, and speculated about ways of measuring the dispersions of each of these distributions about their respective means.

Statistical Inference with Two Populations

In Chapters 11 and 12, we discussed estimation and hypothesis testing of the mean, proportion and variance of a single variable in a population. Very often we will be interested in comparing the parameters of two populations, for example: what is the difference between the mean salary of female teachers and male teachers, and is this difference statistically significant; is the difference in the proportion of households with video recorders in the UK and in France significant.

Discrete Probability Distributions

In Chapter 3 we saw that a frequency distribution is a listing of the values which a variable takes, together with a count (i.e. the frequency of occurrence), of the number of times each value occurs. In Chapter 6 we saw that an ‘experiment’ gives rise to a number of possible outcome values, and we discussed ways of determining what the probability of each of these different outcomes occurring is.