In the late nineteenth and earlier twentieth centuries, Galton, Pearson and Yule, addressing in part social issues as well as those arising in their prime concern with evolutionary biology, created ‘the new English statistics’, which provided an essential methodological foundational for sociological science. Galton moved on from Quetelet in seeing the Gaussian error curve as the basis for determining not different social types but rather the extent of individual variation within human populations. Through struggling with the problems of a naturalistic sociology that he sought to develop, founded on the inheritance of individual characteristics, notably ability, he introduced the techniques of regression and correlation into quantitative data analysis. Pearson and Yule then significantly refined and advanced these techniques and also pioneered – if in the course of a fierce controversy – the analysis of contingency tables.

We concluded the previous chapter by introducing two methods of inference concerning the parameter vector. Since the Bayesian approach was one of them, we focus here on the competing frequentist or classical approach in its attempt to draw conclusions about the value of this vector. We introduce hypothesis testing, test statistics and their critical regions, size, and power. We then introduce desirable properties (lack of bias, uniformly most powerful test, consistency, invariance with respect to some class of transformations, similarity, admissibility) that help us find optimal tests. The Neyman–Pearson lemma and extensions are introduced. Likelihood ratio (LR), Wald (W), score and Lagrange multiplier (LM) tests are introduced for general hypotheses, including inequality hypotheses for the parameter vector. Monotone LR and the Karlin–Rubin theorem are studied, as is Neyman's structure and its role in finding optimal tests. The exponential family features prominently in the applications. Finally, distribution-free (nonparametric) tests are studied and linked to results in earlier chapters.