The Lanarkshire report is twenty pages long and carries twelve tables. Everything written about the experiment in the ninety years since has been worked from those tables, with one exception, Ethel Elderton’s paper of 1933, which was worked from the cards. This piece goes through the tables in order and then through the three re-readings, with the figures as each writer gave them. The report’s own habit of writing a decimal with a raised point is modernised; its probable errors, the figure after the plus-or-minus sign, are kept, because every argument about the experiment turns on them.

The counts

The report says that 17,159 children’s records were used after the exclusions: children absent at the final weighing, which was most of the loss; children without a date of birth; a few whose names did not settle their sex; obvious errors; and, “later in the inquiry,” children over twelve, nearly all of them controls, for whom there were too few feeders of the same age and sex to compare. The usable children were sorted into three groups, control, raw-milk and pasteurised-milk, and each group by sex and by year of age from five to eleven, which gives forty-two cells.

Table 1 gives the average initial weight of each cell, in pounds, with the number of children in brackets. The brackets are the only place the report says how many children of each kind there were, and they add up as follows. Boys: 4,320 controls, 2,236 on raw milk, 2,088 on pasteurised milk. Girls: 4,298 controls, 2,139 on raw milk, 2,133 on pasteurised milk. That is 8,618 controls and 8,596 feeders, 17,214 children in all, fifty-five more than the 17,159 in the text, a difference the report does not explain. Stephen Senn’s re-reading of 2022 prints the girls on pasteurised milk as 3,133 and the total as 18,214, and then puzzles over a gap of 978 between feeders and non-feeders which “seems rather large”; the gap is a slip of one digit in his table, and the real gap is twenty-two. The report never says how many of the sixty-seven schools took raw milk and how many pasteurised; the feeders divide 4,375 raw to 4,221 pasteurised, which is as near to even as the schools can have been.

The cells are not of equal size. The five-year-olds are few, forty-two control boys and twenty-six raw-milk boys, fifty-one control girls and sixteen raw-milk girls, because, as Student guessed, few children of five were in school or were thought up to it. The six- to ten-year-old cells hold between three and eight hundred each.

Tables 1 to 4: the start

Tables 1 and 2 give the average weight and height of each cell in February. Tables 3 and 4 give the differences between the groups, with their probable errors, in three columns for each sex: control minus raw, control minus pasteurised, raw minus pasteurised.

In weight the controls are heavier than the raw-milk children in every cell but the five-year-old boys, and heavier than the pasteurised-milk children in every cell. The differences run from half a pound to two pounds and more: control boys of six weighed 1.27 pounds more than raw-milk boys of six and 1.63 more than pasteurised-milk boys, with probable errors of about 0.23; control girls of ten weighed 1.36 pounds more than raw-milk girls and 2.24 more than pasteurised-milk girls, with probable errors of about 0.35. The report’s rule, given in a footnote to Table 5, is that a difference of three times its probable error “is generally regarded as significant.” Most of the differences in Table 3 pass that line, and the report says so: the control groups “were, at the beginning of the experiment, slightly, though in many instances, significantly heavier and taller.”

In height the picture is the same. Control boys of six were 0.58 inches taller than raw-milk boys and 0.86 taller than pasteurised-milk boys, with probable errors of a tenth of an inch; control girls of six, 0.46 and 0.85. The only cell in which a milk group was taller than its controls is the seven-year-old boys on pasteurised milk, by 0.08 of an inch, within its error. Between raw and pasteurised the differences in both tables are smaller and change sign from row to row, which is what two groups drawn alike would show.

Student turned the whole of Tables 3 and 4 into one sentence: the controls were “definitely superior both in weight and height to the ‘feeders’ by an amount equivalent to about 3 months’ growth in weight and 4 months’ growth in height.”

Table 5: the report’s defence

Table 5 gives, for the controls, the correlation between a child’s initial weight and its gain in weight, and between its initial height and its gain in height, by sex and age. The coefficients are small, between about minus 0.23 and plus 0.17, and of no constant sign. The report takes this to mean that heavier or taller children did not gain more or less than lighter or shorter ones, and therefore that the controls’ head start did not matter: “the change of weight and height, within the period of experiment is, therefore, substantially independent of initial measurements.” Tables 10 and 11 repeat the calculation for the two milk groups and find much the same, with some negative coefficients for height in the younger raw-milk girls.

Student’s answer was that the argument “might have been a perfectly good argument had the height and weight been selected directly,” but that if the teachers had chosen by need, the feeders were not merely smaller children but needier ones, who would gain more from milk than the controls would have done; and no correlation of size with growth could speak to that.

Tables 6 to 9: the gains

Table 6 gives the average gain in weight, in ounces, over the four months, and Table 7 the gain in height, in inches. For boys, the controls gained 11.64 ounces at five, 13.75 at six, 11.17 at seven, 11.38 at eight, 9.53 at nine, 7.10 at ten and 6.14 at eleven; the raw-milk boys 14.88, 13.51, 14.85, 14.21, 13.43, 13.53 and 12.74; the pasteurised-milk boys 15.65, 9.96, 15.55, 15.21, 11.83, 10.39 and 11.05. For girls the controls gained 7.00, 11.21, 8.90, 9.77, 7.87, 9.51 and 12.62; the raw-milk girls 14.50, 10.61, 11.22, 13.40, 13.81, 15.08 and 24.92; the pasteurised-milk girls 6.62, 10.05, 12.94, 13.37, 12.52, 18.96 and 17.08. The report notices that the controls’ gains fall away with age in boys, the youngest gaining nearly twice what the oldest did, and that this is not so in the milk groups; Student would make that observation the centre of his argument about clothes.

In height the controls gained between 0.66 and 0.86 of an inch in every cell, and the feeders between 0.64 and 0.95. The milk groups are ahead in every cell except the five-year-old raw-milk girls.

Tables 8 and 9 give the differences between each milk group and the controls, with probable errors. In weight, the feeders are ahead everywhere above the age of seven, by between 2.3 and 12.3 ounces, and the errors are about an ounce, so nearly all of those pass the report’s line. Among the six-year-olds the sign reverses: the pasteurised-milk boys of six gained 3.79 ounces less than their controls, with a probable error of 0.79, which the report counts as one of only two significant differences against the milk; the other is the height of the five-year-old raw-milk girls. In height the feeders are ahead by between 0.04 and 0.12 of an inch, with errors of about 0.016, so most of those pass the line as well.

Table 12: the two milks

Table 12 gives raw minus pasteurised, cell by cell. In weight the differences are 3.55 ounces for raw among boys of six, 3.14 among boys of ten, 7.84 among girls of eleven, and 3.88 for pasteurised among girls of ten, each about three times its error; the rest are within an ounce or two of nothing and change sign. In height every difference among boys favours raw milk, by between nothing and 0.08 of an inch, and two of them, at nine and ten, are three or four times their error; among girls the differences go both ways. The report concludes that “there is no evidence of greater effect on growth in weight in raw compared with pasteurised milk or vice versa,” and that in height “one is not justified in believing that there is a substantial advantage of one over the other type of milk.”

Bartlett’s regrouping

Stephen Bartlett’s paper in the Journal of the Ministry of Agriculture for April 1931 reprints Tables 6, 7 and 12 and then recalculates each with the ages combined into three bands, five to seven, eight and nine, and ten and eleven, and into a single figure for all ages, for boys, for girls, and for both together. His reason was that fourteen small cells gave fourteen large probable errors, while “children between the ages of five and twelve years, in a single district and in one type of school, form a fairly uniform group.”

Combined, the gains in weight were: for boys, 10.06 ounces on no milk, 13.78 on raw and 12.42 on pasteurised; for girls, 9.74, 14.41 and 13.82; for all children, 9.90, 14.09 and 13.13. The gains in height for all children were 0.729 inches on no milk, 0.811 on raw and 0.781 on pasteurised. Raw minus pasteurised, for all children, came to 0.86 of an ounce with a probable error of 0.42, which he described with a sentence that has been quoted ever since: “The cautious statistician might say that this is not a significant result, but he would admit that the chances are about 5 to 1 that raw milk is better.” In height his table gives 0.044 of an inch for boys and 0.026 for girls, each with an error of 0.009, and 0.036 for all, with an error of 0.008; his text calls the combined figure “nearly six times the probable error which is undoubtedly significant.” He warned in a note that his errors were computed from assumed standard deviations, forty ounces for weight and 0.6 of an inch for height, chosen “to ensure that the final figures are too high rather than too low.”

Student accepted Bartlett’s regrouped figures for what they were and built his own table on them.

Student’s percentages

Student’s one table gives the gain of feeders over controls by Bartlett’s three age bands, with both milk groups combined, and then expresses each as a percentage of the controls’ own gain. In weight the feeders gained 1.13 ounces more than the controls among boys of five to seven and 1.24 among girls, which is 9 and 13 per cent of what the controls gained; 3.15 and 4.47 ounces at eight and nine, 30 and 51 per cent; and 5.21 and 7.88 ounces at ten and eleven, 78 and 73 per cent. In height the extra gains were 0.083 and 0.059 of an inch in the youngest band, 0.071 and 0.098 in the middle and 0.037 and 0.055 in the oldest: 11 and 8 per cent, 10 and 14, 5 and 8.

The pattern is the argument. The milk’s advantage in height is about a tenth of the controls’ growth at every age. Its advantage in weight climbs from a tenth in the five- to seven-year-olds to three-quarters in the ten- and eleven-year-olds. Either milk did seven times as much for an eleven-year-old’s weight as for a six-year-old’s while doing the same for their height, or the June weights were being lowered by something that grew with age and did not touch height. Student’s candidate was the clothes: “older children wear about 1 lb. more clothes in February than they do in June, while in the case of younger children a more limited wardrobe permits of fewer discards,” and the better-off controls had more to take off than the feeders. His conclusion was that “the figures for weights were wholly unreliable” as a measure of the milk, and that the heights were the only numbers worth reading.

Elderton’s pairs

Elderton did not use the tables. The Department of Health for Scotland lent the cards to Karl Pearson, and she and Margaret Beer sorted them by sex and year of birth, then by height, and matched each milk-fed child to a control “of the same initial height, the same weight within 4 ounces and the same age within a month.” There were too few such pairs, so the weight was loosened to 8 ounces, which she justified from Percy Stocks’s study of twins, in which children taken to be identical differed by as much as ten ounces; later the age was loosened to two months, which nearly doubled the pairs. Three comparisons had to be made separately, since a child seldom had a match in both other groups: raw against control, pasteurised against control, raw against pasteurised.

Her first finding was about who could be paired. The matched children were less variable than Glasgow school children of the same ages by about forty per cent in height and thirty in weight: “the shortest and lightest and tallest and heaviest will not appear in this selected data,” and the conclusions “apply only to very average children.”

With all ages combined (her Table II A) the raw-milk boys gained 0.111 of an inch more in height than their paired controls, with a probable error of 0.019, over 337 pairs, and the raw-milk girls 0.108, error 0.018, over 336; the pasteurised-milk boys 0.056, error 0.017, over 342 pairs, and the pasteurised-milk girls 0.127, error 0.017, over 325. In weight the raw-milk boys gained 0.59 of an ounce more than their controls, error 1.08, which is nothing; the raw-milk girls 6.32, error 1.13; the pasteurised-milk boys 2.84, error 1.16; the pasteurised-milk girls 4.26, error 1.11. Raw against pasteurised, over 188 pairs of boys and 149 of girls, raw milk gave 0.039 and 0.091 of an inch more height, errors 0.023 and 0.028, and 2.58 and 0.45 of an ounce less weight, errors 1.43 and 1.77.

The weights were the trouble. Over the four months the paired control boys gained on average 11.6 ounces and the girls 8.5, against the pound or more that a year’s growth divided by three would lead one to expect, and the spread was enormous: the standard deviation of the gain was twenty ounces in every group, and 19 per cent of the control boys and 25 per cent of the control girls weighed less in June than in February. Her tables of the distribution show children recorded as losing more than two pounds. “One wishes more than ever that the children had been weighed without clothes.”

Her conclusions were that milk-fed children “generally gain in height over those who do not,” that they generally gain in weight too, girls more than boys and older girls more than younger, and that “there is no evidence that raw milk has an advantage over pasteurised or pasteurised over raw in increasing growth when the two are directly compared on this selected material.” She also tried to find whether the smaller children gained more from milk than the larger, and could not, because the pairing had thrown the small and the large away.

Pearson’s probabilities

Karl Pearson added an appendix that applies a method of his own to Elderton’s twelve comparisons, five age groups at a time. Each age group’s difference of means, divided by its standard deviation, is turned into a probability, the five probabilities are multiplied, and the product is turned, through the incomplete gamma function, into the probability that five random samples would differ from one another as much as these. For the height of boys on pasteurised milk against their controls he works the example in full and arrives at 0.0026: “only 26 times in 10,000 trials would on the average a pair of samples occur differing so much from one another as these two do.”

His table of results gives, for stature, less than one in twenty million for raw milk against controls in both sexes, 0.0026 for pasteurised milk in boys and less than one in twenty million in girls; for weight, 0.0588 for raw milk in boys and 0.00011 in girls, 0.1130 for pasteurised milk in boys and 0.00022 in girls; and for raw against pasteurised, 0.7381 and 0.6461 for boys’ stature and weight, 0.0088 for girls’ stature and 0.6032 for girls’ weight. “Raw milk undoubtedly accelerated the growth of stature,” he wrote; pasteurised milk very probably had; for weight the boys showed nothing certain and the girls a great deal, which he tried to explain by the boys spending the milk in play and the girls storing it, “that katabolism of the male and anabolism of the female on which some writers, perhaps too emphatically, have insisted.” And between the two milks there was “no evidence whatever” of a difference in boys or in girls’ weight, “but we have the remarkable result that in girls the two types of milk are not indifferent with regard to the acceleration of growth in stature, randomness here is highly improbable, and raw milk seems more advantageous than pasteurised.” Why raw milk should build bone in girls and not in boys he could not say, and he called for “fuller physiological investigation.”

None of this touches Student’s objection. Every one of Elderton’s pairs was a control chosen by a teacher against a feeder chosen by a teacher, matched afterwards on two numbers that the teacher had not been looking at.

Sources

• Gerald Leighton and Peter L. McKinlay, Milk Consumption and the Growth of School Children: Report on an Investigation in Lanarkshire Schools, Department of Health for Scotland, HMSO, 1930, Tables 1 to 12 and the footnote to Table 5. gwern.net
• Stephen Bartlett, “Nutritional Value of Raw and Pasteurized Milk,” Journal of the Ministry of Agriculture 38 (1), April 1931, pp. 60–64, with the recalculated Tables 6, 7 and 12 and the note on probable errors. archive.org
• “Student,” “The Lanarkshire Milk Experiment,” Biometrika 23, December 1931, pp. 398–406, the table on p. 403 and Diagrams 1 to 4. gwern.net
• Ethel M. Elderton, “The Lanarkshire Milk Experiment,” with Karl Pearson’s appendix, Annals of Eugenics 5, October 1933, pp. 326–338, Tables I to VIII. gwern.net
• Stephen Senn, “Student and the Lanarkshire milk experiment,” European Journal of Epidemiology 38, 2023, pp. 1–10, Tables 1 and 2. PMC9867657