*Hence it follows that the accuracy of the arithmetic mean is √n **times the accuracy of a single observation; a result of fundamental importance in the theory of errors.* Brunt, p10.

### Introduction

The first edition of *The Combination of Observations **(Brunt, 1917)* by David Brunt (1886-1965) appeared in 1917 and thus nearly sixty years before Eugene Glass’s influential paper (Glass 1976) of 1976. Glass was addressing a problem that Brunt did not consider, namely how does one produce a quantitative synthesis of results from a body of studies, when the type of measurement used varies from study to study. (What has since been referred to as a *Type C meta-analysis *(Senn 2000).) Nevertheless, the term *meta-analysis* was introduced by Glass and was even more influential than the technique and is now also used to describe the synthesis of results from studies using a common scale of measurement and, indeed, the majority of what are now called *meta-analyses *are of this sort.

Brunt explains the purpose of his book thus “The aim of this book is to give an account of the method of least squares, without entering into elaborate descriptions of instruments or experimental methods. ”(Page v) Least squares already had a venerable history by the time of Brunt’s book and is associated with the work of many leading mathematicians of the late 18th and early 19^{th} century (Stigler 1986), in particular Adrien-Marie Legendre (1752-1833) and Karl-Friederich Gauss (1777-1885).

The modern student of least squares will usually approach it as a technique in a subject confusingly referred to as *regression*, in which context it will be developed as a means of finding unique estimates of parameters in an over-determined system of linear equations. The first case that is usually treated is that of fitting a straight line with two unknowns, intercept and slope. However, an even simpler such problem is that of estimating a single constant and this is what meta-analysis does.

By the time of Brunt’s book, the theory of least squares was a well-developed subject, as was the particular aspect of it we would now call meta-analysis. For example, George Bidell Airy (1801-1892) had published a book (Airy 1861) with a title that included the phrase, *The Combination of Observations,* that Brunt was to use for his. Curiously, however, Airy’s book is not cited by Brunt, despite the fact that it was still well-known at the time of Brunt’s, as is attested to by reference to it in correspondence between Student (1876-1937) and Egon Pearson (1895-1980) (Pearson 1990). (See also Lehmann’s discussion (Lehmann 1999).) Furthermore, F.J.M. Stratton (1881-1960), whose help Brunt acknowledges and who was at Caius at the same time as Fisher, taught a course on Combination of Observations (Edwards 1997) which, given its title quite possibly made reference to Airy’s book. However, Brunt’s book was more detailed in its treatment, broader in its scope and generally more ambitious than Airy’s and Brunt may have felt there was no debt to acknowledge.

### David Brunt

…began the writing of his first book, *The combination of observations*, which was published in 1917 and once had a considerable vogue as a practical manual in statistics. (Sutton 1965, p42)

The quotation above is from Sutton’s biography for The Royal Society (Sutton 1965), upon which this section is based.

David Brunt was born into a Welsh-speaking family in the village of Staylittle, Montgomeryshire in 1886. He was schooled in Welsh until the age of 10, when the family moved to the English-speaking area of Llanhilleth. From that time on his education was in English, a language to which he rapidly adapted. In 1899 he entered the Abertillery Intermediate School where he proved to be a star pupil winning a scholarship to the University College of Wales, Aberystwyth, where he studied physics and mathematics, graduating with a distinguished first class honours in 1907. A scholarship then brought him to Trinity College, Cambridge where he gained firsts in part I and II of the mathematical tripos. He initially hoped to be an astronomer but in 1916 he enlisted with the Royal Engineers (Meteorological Section) and meteorology was the field in which he subsequently made his reputation.

In meteorology he made contribution to many topics, including atmospheric turbulence, periodicities in European weather, the dynamics of cyclones and anti-cyclones, eddy heat-transfer and predicting nighttime minimum temperatures, to name but a few. In 1934, he was appointed Professor at Imperial College London, where he was the head of the only department of meteorology at a British University. He was appointed FRS in 1939 and knighted in 1949.

He was an enthusiastic glider pilot and was chairman of the Council of the British Gliding Association from 1935-1946. He married Claudia Mary Elizabeth Roberts in 1915. They had one son. Both his wife and son predeceased him. He died in 1965. The Brunt Iceshelf in Antarctica is named after him.

### The Combination of Observations

The material of Brunt’s book is arranged in 12 chapters. CHAPTER I, ERRORS OF OBSERVATION, makes some general remarks and CHAPTER II, THE LAW OF ERROR, provides various derivations of and proofs associated with what would now be called the Normal distribution. CHAPTER III, THE CASE OF ONE UNKOWN, is concerned with the estimation of a population mean when observations have equal weight. CHAPTER IV, OBSERVATIONS OF DIFFERENT WEIGHT, covers the more general case of unequal precision. Chapters III & IV thus enter into the matter of what we now call *meta-analysis*. The remaining eight chapters address matters that go beyond this and include estimation for more complex models including correlation, what would now be called multiple regression and Fourier analysis. CHAPTER VIII, THE REJECTION OF OBSERVATIONS and CHAPTER IX, ALTERNATIVES TO THE NORMAL LAW OF ERRORS cover matters that are still of interest to statisticians today, although the treatment will seem a little strange.

However, the reader who is interested in medical examples will be disappointed by the book. Of 22 datasets, twelve deal with astronomy, three with meteorology, five with other physical sciences and two with agriculture. Nevertheless, many of the techniques are the same as or similar to those used today on clinical trials. In the remainder of this paper Brunt’s approach to meta-analysis will be considered with the help of an example from astronomy given in CHAPTER IV.

### Brunt’s Recipe for the Combination of Observations

Figure 1 Example 1 from page 69 of The Combination of Observations.

Figure 1 is taken from the 1^{st} edition of Brunt’s book (but is unchanged in the 2^{nd} edition) as an illustration of Brunt’s method of weighted combination of observations. The method of calculation is clearly described by Brunt but some matters are worth clarifying further.

First, by *P.E.,* Brunt means *probable error (PE)*. This was at that time the usual way of expressing precision of observations but was eventually replaced by *standard error (SE)*. The probability that a random observation from a Normal distribution with mean μ lies within the range μ – *PE, μ + PE* is ½ . The *PE* is thus simply a multiple of the *SE* and the value of that multiple to four decimal places is 0.6745, and this is the value Brunt uses in his calculations. Note that this value is really only appropriate asymptotically and would be rather poor given that there are only six observations. This point will be picked up again in the discussion.

Second, the parallax measures being combined are circular but are treated as if they were linear. This is justifiable given the very small range of values given. Brunt’s book includes various types of fitting, including that appropriate to periodic data but simple linear combinations are clearly adequate here.

Third, Brunt does not give any explanation as to where the weights he specifies come from. In modern meta-analysis these would be inversely proportional to the squares of the standard errors for the estimates being combined. In earlier sections of the book he has warned the reader that, “The greatest disadvantage of this method of weighting lies in its arbitrary and personal nature” (p64).

Fourth, for ease of calculation, Brunt uses an arbitrary origin of 0”.4 and a scale factor of 1000. This method of calculation was commonly taught up to the 1970s when the author of this article was a student but subsequent generations will probably not have encountered it, as it is no longer needed, due to the availability of personal computers.

Fifth, Brunt estimates the PE using the observed differences between observations. In an earlier part of the book he discusses what we would now refer to as the fixed effects approach by which the PEs assigned to individual observations could be combined to estimate the PE of their weighted combination. He warns that, “If the differences between the individual observations that have to be combined are attributable to systematic errors entering into different determinations in different ways, it is clear that the P.E. of a determination can give no clear estimate of the reliability of that determination.” (p68) He then expresses the view that, “it is safer to… calculate the P.E. from the residuals “ (p69) unless there are few determinations, say 3 or 4 to be combined and there is no reason to suspect systematic variation. He does not, however, consider the possibility of adjusting the weights accordingly. The net result is that Brunt proposes what we should now call fixed effect weighting but with a random effects approach to estimating the uncertainty of the estimate.

### Discussion

The first (1917) edition of Brunt’s book is a scholarly and clear account of a theory of combining observations using least squares that had been developed over more than a century. The glaring omission is Student’s introduction of the t-distribution (Student 1908). Brunt quotes a multiplier to calculate the PE of 0.6745. The t-distribution would have to have in excess of 4000 degrees of freedom to give this value to four decimal places. In the example illustrated, he has six observations only and therefore five degrees of freedom. The appropriate multiplier from the t-distribution is then 0.7276. In other words, Brunt’s value is accurate to one decimal place, not the four quoted.

He can however, be forgiven for this oversight. The development of adjustment for uncertainty in the measurement of uncertainty took place over a long period. We now know that Student was anticipated in 1876 by Jakob Luroth (Pfanzagl and Sheynin 1996). Furthermore, Student’s paper received little attention when first published. It was only when his approach was adopted, modified and further developed by RA Fisher in 1925 in *Statistical Methods for Research Workers *(Fisher 1925) that the technique entered mainstream statistics. Its omission in Brunt’s first edition is therefore not surprising. However, it is also omitted in the 1931 edition and this means that whereas the book was an excellent account of the art of combining observations when it first appeared, it was seriously out of date by its second edition.

Meta-analysis has a much longer history than one might suppose. In his popular account of the subject (Hunt 1997), Morton Hunt attributes its origins to Karl Pearson’s paper of 1904 on the efficacy of inoculation against typhoid fever (Pearson 1904). However, it was nearly a century old by then, Adrien Legendre having proposed least squares (Legendre 1806) in 1806 as a means of determining the orbits of comets. Its continued development was then in the subject of astronomy and related physical sciences. Brunt’s book is an excellent summary of what had been achieved by the start of the 20^{th} century in that tradition.

Meta-analysis was then further developed in agricultural science (see, for example work by Yates and Cochran (Yates and Cochran 1938)) and then subsequently in education and medicine. At each stage, it seems, something was lost, both by those in the new field of application failing to appreciate what had already been done, and those in the old field being unaware of further developments elsewhere, as Brunt’s book illustrates. Nevertheless, even though soon outdated, the book remains an important historical account and can be read with profit, even today.

**Acknowledgements**

I am grateful to Anthony Edwards for drawing Brunt’s book to my attention and for helpful discussions on the relevant history and to Iain Chalmers for having encouraged me to write on the subject. I am also grateful to George Davey Smith and Iain Chalmers for helpful comments and to Patricia Atkinson for editorial help.

This James Lind Library article has been republished in the *Journal of the Royal Society of Medicine *2024;117:182-185. Print PDF

**References**

Airy GB (1861). On the algebraical and numerical theory of errors of observations and the combination of observations. Cambridge: MacMillan and Co.

Brunt D (1917). The Combination of Observations. University Press.

Edwards A (1997). Three early papers on efficient parametric estimation. Statistical Science 12:35-47.

Fisher RA (1925). Statistical Methods for Research Workers. Edinburgh, Oliver and Boyd.

Glass GV (1976). Primary, secondary and meta-analysis of research. Educational Researcher 10: 3-8.

Hunt M (1997). How Science Takes Stock: The Story of Meta-analyis. New York, Russel Sage Foundation.

Legendre AM (1806). Nouvelles méthodes pour la détermination des orbites des comètes: avec un supplément contenant divers perfectionnemens de ces méthodes et leur application aux deux comètes de 1805, Courcier.

Lehmann EL (1999). “Student” and Small-Sample Theory. Statistical Science 14: 418-426.

Pearson ES (1990). Student, A Statistical Biography of William Sealy Gosset (Edited and augmented by RL Plackett with the assistance of GA Barnard). Oxford, Clarendon Press.

Pearson K (1904). Report on certain enteric fever inoculation statistics. BMJ 3:1243-1246.

Pfanzagl J, Sheynin O (1996). Studies in the history of probability and statistics .44. A forerunner of the t-distribution. Biometrika 83: 891-898.

Senn SJ (2000). The many modes of meta. Drug Information Journal 34: 535-549.

Stigler SM (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Cambridge, Massachusetts. Belknap Press.

Student (1908). The probable error of a mean. Biometrika 6: 1-25.

Sutton OG (1965). David Brunt, 1886-1965. The Royal Society London.

Yates F, Cochran WG (1938). The analysis of groups of experiments. Journal of Agricultural Science 28; 556-580.

## Senn S (2024). An Early 20th Century Handbook on ‘Meta-analysis’: David Brunt’s The Combination of Observations.

## © Stephen Senn, School of Health and Related Research, University of Sheffield, Sheffield, UK. Email: stephen@senns.uk

Cite as:Senn S (2024). An Early 20th Century Handbook on ‘Meta-analysis’: David Brunt’s The Combination of Observations. JLL Bulletin: Commentaries on the history of treatment evaluation (https://www.jameslindlibrary.org/articles/an-early-20th-century-handbook-on-meta-analysis-david-brunts-the-combination-of-observations/)## Introduction

The first edition of

The Combination of Observations(Brunt, 1917)by David Brunt (1886-1965) appeared in 1917 and thus nearly sixty years before Eugene Glass’s influential paper (Glass 1976) of 1976. Glass was addressing a problem that Brunt did not consider, namely how does one produce a quantitative synthesis of results from a body of studies, when the type of measurement used varies from study to study. (What has since been referred to as aType C meta-analysis(Senn 2000).) Nevertheless, the termmeta-analysiswas introduced by Glass and was even more influential than the technique and is now also used to describe the synthesis of results from studies using a common scale of measurement and, indeed, the majority of what are now calledmeta-analysesare of this sort.Brunt explains the purpose of his book thus “The aim of this book is to give an account of the method of least squares, without entering into elaborate descriptions of instruments or experimental methods. ”(Page v) Least squares already had a venerable history by the time of Brunt’s book and is associated with the work of many leading mathematicians of the late 18th and early 19

^{th}century (Stigler 1986), in particular Adrien-Marie Legendre (1752-1833) and Karl-Friederich Gauss (1777-1885).The modern student of least squares will usually approach it as a technique in a subject confusingly referred to as

regression, in which context it will be developed as a means of finding unique estimates of parameters in an over-determined system of linear equations. The first case that is usually treated is that of fitting a straight line with two unknowns, intercept and slope. However, an even simpler such problem is that of estimating a single constant and this is what meta-analysis does.By the time of Brunt’s book, the theory of least squares was a well-developed subject, as was the particular aspect of it we would now call meta-analysis. For example, George Bidell Airy (1801-1892) had published a book (Airy 1861) with a title that included the phrase,

The Combination of Observations,that Brunt was to use for his. Curiously, however, Airy’s book is not cited by Brunt, despite the fact that it was still well-known at the time of Brunt’s, as is attested to by reference to it in correspondence between Student (1876-1937) and Egon Pearson (1895-1980) (Pearson 1990). (See also Lehmann’s discussion (Lehmann 1999).) Furthermore, F.J.M. Stratton (1881-1960), whose help Brunt acknowledges and who was at Caius at the same time as Fisher, taught a course on Combination of Observations (Edwards 1997) which, given its title quite possibly made reference to Airy’s book. However, Brunt’s book was more detailed in its treatment, broader in its scope and generally more ambitious than Airy’s and Brunt may have felt there was no debt to acknowledge.## David Brunt

The quotation above is from Sutton’s biography for The Royal Society (Sutton 1965), upon which this section is based.

David Brunt was born into a Welsh-speaking family in the village of Staylittle, Montgomeryshire in 1886. He was schooled in Welsh until the age of 10, when the family moved to the English-speaking area of Llanhilleth. From that time on his education was in English, a language to which he rapidly adapted. In 1899 he entered the Abertillery Intermediate School where he proved to be a star pupil winning a scholarship to the University College of Wales, Aberystwyth, where he studied physics and mathematics, graduating with a distinguished first class honours in 1907. A scholarship then brought him to Trinity College, Cambridge where he gained firsts in part I and II of the mathematical tripos. He initially hoped to be an astronomer but in 1916 he enlisted with the Royal Engineers (Meteorological Section) and meteorology was the field in which he subsequently made his reputation.

In meteorology he made contribution to many topics, including atmospheric turbulence, periodicities in European weather, the dynamics of cyclones and anti-cyclones, eddy heat-transfer and predicting nighttime minimum temperatures, to name but a few. In 1934, he was appointed Professor at Imperial College London, where he was the head of the only department of meteorology at a British University. He was appointed FRS in 1939 and knighted in 1949.

He was an enthusiastic glider pilot and was chairman of the Council of the British Gliding Association from 1935-1946. He married Claudia Mary Elizabeth Roberts in 1915. They had one son. Both his wife and son predeceased him. He died in 1965. The Brunt Iceshelf in Antarctica is named after him.

## The Combination of Observations

The material of Brunt’s book is arranged in 12 chapters. CHAPTER I, ERRORS OF OBSERVATION, makes some general remarks and CHAPTER II, THE LAW OF ERROR, provides various derivations of and proofs associated with what would now be called the Normal distribution. CHAPTER III, THE CASE OF ONE UNKOWN, is concerned with the estimation of a population mean when observations have equal weight. CHAPTER IV, OBSERVATIONS OF DIFFERENT WEIGHT, covers the more general case of unequal precision. Chapters III & IV thus enter into the matter of what we now call

meta-analysis. The remaining eight chapters address matters that go beyond this and include estimation for more complex models including correlation, what would now be called multiple regression and Fourier analysis. CHAPTER VIII, THE REJECTION OF OBSERVATIONS and CHAPTER IX, ALTERNATIVES TO THE NORMAL LAW OF ERRORS cover matters that are still of interest to statisticians today, although the treatment will seem a little strange.However, the reader who is interested in medical examples will be disappointed by the book. Of 22 datasets, twelve deal with astronomy, three with meteorology, five with other physical sciences and two with agriculture. Nevertheless, many of the techniques are the same as or similar to those used today on clinical trials. In the remainder of this paper Brunt’s approach to meta-analysis will be considered with the help of an example from astronomy given in CHAPTER IV.

## Brunt’s Recipe for the Combination of Observations

Figure 1 Example 1 from page 69 of The Combination of Observations.

Figure 1 is taken from the 1

^{st}edition of Brunt’s book (but is unchanged in the 2^{nd}edition) as an illustration of Brunt’s method of weighted combination of observations. The method of calculation is clearly described by Brunt but some matters are worth clarifying further.First, by

P.E.,Brunt meansprobable error (PE). This was at that time the usual way of expressing precision of observations but was eventually replaced bystandard error (SE). The probability that a random observation from a Normal distribution with mean μ lies within the range μ –PE, μ + PEis ½ . ThePEis thus simply a multiple of theSEand the value of that multiple to four decimal places is 0.6745, and this is the value Brunt uses in his calculations. Note that this value is really only appropriate asymptotically and would be rather poor given that there are only six observations. This point will be picked up again in the discussion.Second, the parallax measures being combined are circular but are treated as if they were linear. This is justifiable given the very small range of values given. Brunt’s book includes various types of fitting, including that appropriate to periodic data but simple linear combinations are clearly adequate here.

Third, Brunt does not give any explanation as to where the weights he specifies come from. In modern meta-analysis these would be inversely proportional to the squares of the standard errors for the estimates being combined. In earlier sections of the book he has warned the reader that, “The greatest disadvantage of this method of weighting lies in its arbitrary and personal nature” (p64).

Fourth, for ease of calculation, Brunt uses an arbitrary origin of 0”.4 and a scale factor of 1000. This method of calculation was commonly taught up to the 1970s when the author of this article was a student but subsequent generations will probably not have encountered it, as it is no longer needed, due to the availability of personal computers.

Fifth, Brunt estimates the PE using the observed differences between observations. In an earlier part of the book he discusses what we would now refer to as the fixed effects approach by which the PEs assigned to individual observations could be combined to estimate the PE of their weighted combination. He warns that, “If the differences between the individual observations that have to be combined are attributable to systematic errors entering into different determinations in different ways, it is clear that the P.E. of a determination can give no clear estimate of the reliability of that determination.” (p68) He then expresses the view that, “it is safer to… calculate the P.E. from the residuals “ (p69) unless there are few determinations, say 3 or 4 to be combined and there is no reason to suspect systematic variation. He does not, however, consider the possibility of adjusting the weights accordingly. The net result is that Brunt proposes what we should now call fixed effect weighting but with a random effects approach to estimating the uncertainty of the estimate.

## Discussion

The first (1917) edition of Brunt’s book is a scholarly and clear account of a theory of combining observations using least squares that had been developed over more than a century. The glaring omission is Student’s introduction of the t-distribution (Student 1908). Brunt quotes a multiplier to calculate the PE of 0.6745. The t-distribution would have to have in excess of 4000 degrees of freedom to give this value to four decimal places. In the example illustrated, he has six observations only and therefore five degrees of freedom. The appropriate multiplier from the t-distribution is then 0.7276. In other words, Brunt’s value is accurate to one decimal place, not the four quoted.

He can however, be forgiven for this oversight. The development of adjustment for uncertainty in the measurement of uncertainty took place over a long period. We now know that Student was anticipated in 1876 by Jakob Luroth (Pfanzagl and Sheynin 1996). Furthermore, Student’s paper received little attention when first published. It was only when his approach was adopted, modified and further developed by RA Fisher in 1925 in

Statistical Methods for Research Workers(Fisher 1925) that the technique entered mainstream statistics. Its omission in Brunt’s first edition is therefore not surprising. However, it is also omitted in the 1931 edition and this means that whereas the book was an excellent account of the art of combining observations when it first appeared, it was seriously out of date by its second edition.Meta-analysis has a much longer history than one might suppose. In his popular account of the subject (Hunt 1997), Morton Hunt attributes its origins to Karl Pearson’s paper of 1904 on the efficacy of inoculation against typhoid fever (Pearson 1904). However, it was nearly a century old by then, Adrien Legendre having proposed least squares (Legendre 1806) in 1806 as a means of determining the orbits of comets. Its continued development was then in the subject of astronomy and related physical sciences. Brunt’s book is an excellent summary of what had been achieved by the start of the 20

^{th}century in that tradition.Meta-analysis was then further developed in agricultural science (see, for example work by Yates and Cochran (Yates and Cochran 1938)) and then subsequently in education and medicine. At each stage, it seems, something was lost, both by those in the new field of application failing to appreciate what had already been done, and those in the old field being unaware of further developments elsewhere, as Brunt’s book illustrates. Nevertheless, even though soon outdated, the book remains an important historical account and can be read with profit, even today.

Acknowledgements

I am grateful to Anthony Edwards for drawing Brunt’s book to my attention and for helpful discussions on the relevant history and to Iain Chalmers for having encouraged me to write on the subject. I am also grateful to George Davey Smith and Iain Chalmers for helpful comments and to Patricia Atkinson for editorial help.This James Lind Library article has been republished in the

Journal of the Royal Society of Medicine2024;117:182-185. Print PDFReferencesAiry GB (1861). On the algebraical and numerical theory of errors of observations and the combination of observations. Cambridge: MacMillan and Co.

Brunt D (1917). The Combination of Observations. University Press.

Edwards A (1997). Three early papers on efficient parametric estimation. Statistical Science 12:35-47.

Fisher RA (1925). Statistical Methods for Research Workers. Edinburgh, Oliver and Boyd.

Glass GV (1976). Primary, secondary and meta-analysis of research. Educational Researcher 10: 3-8.

Hunt M (1997). How Science Takes Stock: The Story of Meta-analyis. New York, Russel Sage Foundation.

Legendre AM (1806). Nouvelles méthodes pour la détermination des orbites des comètes: avec un supplément contenant divers perfectionnemens de ces méthodes et leur application aux deux comètes de 1805, Courcier.

Lehmann EL (1999). “Student” and Small-Sample Theory. Statistical Science 14: 418-426.

Pearson ES (1990). Student, A Statistical Biography of William Sealy Gosset (Edited and augmented by RL Plackett with the assistance of GA Barnard). Oxford, Clarendon Press.

Pearson K (1904). Report on certain enteric fever inoculation statistics. BMJ 3:1243-1246.

Pfanzagl J, Sheynin O (1996). Studies in the history of probability and statistics .44. A forerunner of the t-distribution. Biometrika 83: 891-898.

Senn SJ (2000). The many modes of meta. Drug Information Journal 34: 535-549.

Stigler SM (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Cambridge, Massachusetts. Belknap Press.

Student (1908). The probable error of a mean. Biometrika 6: 1-25.

Sutton OG (1965). David Brunt, 1886-1965. The Royal Society London.

Yates F, Cochran WG (1938). The analysis of groups of experiments. Journal of Agricultural Science 28; 556-580.