Student and the Lanarkshire milk experiment

Abstract

A detailed examination of the 1930 Lanarkshire Milk Experiment (LME) by the famous statistician William Sealy Gossett ("Student"), which appeared in Biometrika in 1931, is re-examined from a more modern perspective. The LME had a complicated design whereby 67 schools in Lanarkshire were allocated to receive either raw or pasteurised milk but pupils within the schools were allocated to either receive milk or to act as controls. Student's criticisms are considered in detail and examined in terms of subsequent developments on the design and analysis of experiments, in particular as regards appropriate estimation of standard errors of treatment estimates when an incomplete blocks structure has been used. An analogy with a more modern trial in osteoarthritis is made. Suggestions are made as to how analysis might proceed if the original data were available. Some lessons for observational studies in epidemiology are drawn and it is speculated that hidden clustering structures might be an explanation as to why results may vary from observational study to observational study by more than conventionally calculated standard errors might suggest.

Keywords: Cluster design; Incomplete blocks; Nutrition; Random effects; Randomisation; Standard errors; Student.

Fig. 1

Reproduction of Diagram 1 from Student’s paper. Height by age cohort for boys enrolled on the LME. Note that a baseline height is indicated o and that at 4 months by x. Pairs of o followed by x are measured in the same age cohort. However, although all points are joined, in moving from x to o a different age cohort is introduced

Fig. 2

Reproduction of Diagram 3 from Student’s paper. Weight by age cohort for boys enrolled on the LME. Note that a baseline weight is indicated o and that at 4 months by x. Pairs o followed by x are measured in the same age cohort. However, although all points are joined, in moving from x to o a different age cohort is introduced

Fig. 3

Variances and estimated variances for two types of contrasts and three different approaches to analysis. Top row: milk type versus control. Bottom row: comparison of milk types. Left hand column: analysis treating school as fixed. Right hand column: analysis treating school as random. Middle column: incorrect analysis ignoring school effects. Lines show theoretical values as given in the “Appendix”, points show simulated variances and diamonds show simulated estimated variances. A group represents one of the 28 age by sex by milk received combinations. It is assumed that will be 10 pupils per school for such a group and, for simplicity 33 school of each type. The within group variance is assumed to be 2.25 inches squared. The ratio of the between to within school variance is given by $\psi$. The results of 400 simulations are shown