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Stephen Senn
Consultant Statistician
Edinburgh, Scotland

A Diet of Terms

A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex difference in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and their weight the following June are recorded.(P304)

This is how Frederic Lord (1912-2000) introduced the paradox (1) that now bears his name. It is justly famous (or notorious). However, the addition of sex as a factor adds nothing to the essence of the paradox and (in my opinion) merely confuses the issue. Furthermore, studying the effect of diet needs some sort of control. Therefore, I shall consider the paradox in the purer form proposed by Wainer and Brown (2), which was subtly modified by Pearl and Mackenzie in The Book of Why (3) (See pp212-217). Continue reading →

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Stephen Senn
Consultant Statistician
Edinburgh, Scotland

The Many Halls Problem
It’s not that paradox but another

Generalisation is passing…from the consideration of a restricted set to that of a more comprehensive set containing the restricted one…Generalization may be useful in the solution of problems. George Pólya [1] (P108)

Introduction

In a previous blog  https://www.linkedin.com/pulse/cause-concern-stephen-senn/ I considered Lord’s Paradox[2], applying John Nelder’s calculus of experiments[3, 4]. Lord’s paradox involves two different analyses of the effect of two different diets, one for each of two different student halls, on weight of students. One statistician compares the so-called change scores or gain scores (final weight minus initial weight) and the other compares final weights, adjusting for initial weights using analysis of covariance. Since the mean initial weights vary between halls, the two analyses will come to different conclusions unless the slope of final on initial weights just happens to be one (in practice, it would usually be less). The fact that two apparently reasonable analyses would lead to different conclusions constitutes the paradox. I chose the version of the paradox outlined by Wainer and Brown [5] and also discussed in The Book of Why[6].  I illustrated this by considering two different experiments: one in which, as in the original example, the diet varies between halls and a further example in which it varies within halls. I simulated some data which are available in the appendix to that blog but which can also be downloaded from here http://www.senns.uk/Lords_Paradox_Simulated.xls so that any reader who wishes to try their hand at analysis can have a go. Continue reading →

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Stephen Senn
Consultant Statistician
Edinburgh

What is this n you boast about?

Failure to understand components of variation is the source of much mischief. It can lead researchers to overlook that they can be rich in data-points but poor in information. The important thing is always to understand what varies in the data you have, and to what extent your design, and the purpose you have in mind, master it. The result of failing to understand this can be that you mistakenly calculate standard errors of your estimates that are too small because you divide the variance by an n that is too big. In fact, the problems can go further than this, since you may even pick up the wrong covariance and hence use inappropriate regression coefficients to adjust your estimates.

I shall illustrate this point using clinical trials in asthma. Continue reading →