This chapter builds on our previous work and takes a succession of giant strides towards realistic quantitative analysis of relationships between two or more variables. After completing this material it will be possible to examine a large range of potential historical problems, and to read critically many of the studies in history and the social sciences that make use of regression and related quantitative techniques.

We will begin in §9.1 with a brief discussion of the concept of a ‘model’ and of the associated methodology of quantitative research into the relationships between two or more variables. The basic technique of linear regression (discussed in chapters 4 and 8) is then extended in §9.2 to examine the reasons why the observed values of the dependent variable deviate from the regression line, and to consider the implications of these deviations. In §9.3 a new test statistic, the F-test, is introduced and used to test the significance of the multiple regression as a whole. Finally, §9.4 is devoted to a further useful summary statistic, the standard error of the estimate.

Historical research and models of relationships between variables

One of the principal features of quantitative research in the historical and social sciences is the attempt to analyse and explain the behaviour of some particular variable.