The Chasm between Scientific and Statistical Inference Demonstrated by Lord’s Paradox

Lord (1967) published a two page paper with simple data presented in a graph. He wished to show the absurdity of using ANCOVA, without good reason, to reach a scientific conclusion. The use of ‘paradox’ in the title misled people to think the use of ANCOVA might have been valid, so Lord (1969) published another two page paper to clarify. That did not end the confusion. Statisticians and causal theorists have been publishing long articles every few years since 1969 arguing that Lord was wrong (as would be the very many scientists who would agree with Lord), and arguing that ANCOVA could be justified for the data Lord presented. This history illustrates the divide between scientific inference and statistical inference, closely related to the difference between deduction (statistics) and induction/abduction (science). It is telling that not one of the many publications since 1969 have shown a model capable of generating the data shown in Lord’s original paper and also justifying the ANCOVA conclusions. Rather theoretical arguments have been given that there ought to be one. Scientists of course build theories; their theories are approximations to reality but attempt to explain in the simplest way consistent with present and past data the primary causal mechanisms that are operating to produce the data. The many statisticians and causal theorists analyzing Lord’s paradox since 1969 seem to have missed this point.