# What were the main statistical contributions of Ronald Fisher?

Richard Dawkins has described Ronald Fisher as "the father of modern statistics and experimental design", a line which is quoted in Fisher's Wikipedia biography. And also Anders Hald called him "a genius who almost single-handedly created the foundations for modern statistical science" in his book A History of Mathematical Statistics.

I just wonder what exactly he did so people give him such a high evaluation?

• This would be a great post for HSM. – Antoni Parellada Mar 12 '16 at 23:31
• @Antoni I think at some point in the future, as HSM continues to grow and thrive, HSM might become a better home for statistical history questions. But there's such a strong expertise base on CV, with many users who have a real interest in historical aspects, that CV is arguably the better place for now. (I think in the long run, CV will likely continue to be the better place for the more "conceptual" history questions.) – Silverfish Mar 13 '16 at 0:25
• I don't think the potentially 'opinion-based' nature of this question is the issue. I agree with @AntoniParellada: If this question doesn't belong on the History of Science and Mathematics SE site, it's not clear what would. We owe it to our SE colleagues to migrate it there. The original framing was perfectly fine. – gung - Reinstate Monica Mar 21 '16 at 10:07
• I agree, 'nice question', but maybe not here. Fisher's work's already been collected into a pretty volume Contributions to Mathematical Statistics that is easily obtained from any second-hand books shop. For a book-review see: jstor.org/stable/2332332 I am personally not capable to add better words and can only refer to Efron jstor.org/stable/2676745 What would indeed be interesting and adding information is a view from historians. (or philosophers since the different statistics views is a though question and I actually do not really get it, ie. I use all of them) – Sextus Empiricus Nov 7 '17 at 4:50
• @MatthewDrury We have a relatively popular [history] tag. History questions are on-topic on our site. If something is on topic here, we should IMHO not migrate it away even if it's on-topic somewhere else too. – amoeba Nov 9 '17 at 20:31

It is very difficult to write an answer to the question

What were the main statistical contributions of Ronald Fisher?

since there are already numerous excellent works on this topic, created by excellent writers, including great statisticians e.g.:

These works are very difficult to match in a few simple lines on an internet Q&A board. On top of that it is not quite easy to grasp the entirety of ideas from Fisher, as Efron wrote in his work on Fisher:

One difficulty in assessing the importance of Fisherian statistics is that it's hard to just say what it is. Fisher had an amazing number of important ideas and some of them, like randomization inference and conditionality, are contradictory. It's a little as if in economics Marx, Adam Smith and Keynes turned out to be the same person.

# Fisher was a pioneer

Already a simple, but very good, source to Fisher's contribution is Wikipedia. Just reading the article on the history of statistics (or you can use any other text) will give you some insight in the amount and importance of Fisher's contributions.

You will also see that it is partly time, location and luck that made Fisher a great contributor. Fisher was an important and influential statistician in the early 20th century when the basic foundations of applied statistics were created and the field was relatively small (comparable to the period of the 18th and 19th century in mathematics).

The first journal of statistics and the first statistics department at a university had just been started when Fisher entered the stage. Before the beginning of the 20th century, there were mostly methods to do regression and several ideas about distributions of residual terms and errors, used in such fields as astronomy.

Concepts of measurement errors and probability of results. This type of mathematics and logic (more close to pure mathematics, and... seen as more noble, and less condemned by serious mathematicians of that time), became applied more widely to fields of Fisher's choice: genetics, evolution, biology, agriculture. Since Fisher, an excellent mathematician, provided major contributions to these early developments (or may even be considered as the major driver for these developments), his work has been placed at an important position in the history of statistics.

# Basic concepts and tools

If you look at the topics in an introduction book on statistics (specifically the mathematical concepts, or inference) you might consider Fisher as the dominating contributor. It is also Fisher who wrote the first, and most influential, introduction to statistics books:

• Statistical Methods for Research Workers (1925)
• The Design of Experiments (1935) (using the tea cup experiment to explain among others, randomization, the use of latin squares, null hypothesis, significance, sensitivity/power, and basically everything;Yates provides a historic background to this work)

Note that online versions of these books exist SMRW and partially DE (see readings October 29 b).

From 1912 to 1925, Fisher:

• helped to improve the chi-square test (where Pearson and others were wrong about the number of degrees of freedom for many years),
• provided an exact test to calculate the p-value for goodness of fit with low number of observations (which was named after him as the Fisher's exact test),
• wrote a proof (as an undergraduate) for Gosset's 'student's distribution' (and developed it further during his work on small observation numbers, such as ideas of using $$N-1$$ degrees of freedom instead of the sample size $$N$$ when using sample statistics) (see historic description by Fisher's daughter Joan Fisher Box),
• developed analysis of variance and the F-distribution (also named after him), and
• (another "little" thing that he did as an undergraduate) was developing the basics and concepts for maximum likelihood (Aldrich's R. A. Fisher and the Making of Maximum Likelihood).

So roughly this covers most of the basic inferential tools that current introduction texts use. While doing this work on statistics Fisher tackled major problems in genetics that make people like Richard Dawkins admire him so much.

# Terminology

Fisher introduced many concepts and terms and improved statistical language. Two recent questions on this Q&A site relate to Fisher. The question why so many variables are squared in statistics and why we so often the $$L_2$$ norm instead of the $$L_1$$. It is Fisher who "proved" that the $$L_2$$ norm is a better (more efficient) estimator than the $$L_1$$ norm (assuming a perfect Gaussian distribution, which Fisher agreed later is not always true for 'real' errors), and introduced the terms deriving it as an 'efficient statistic' and a 'sufficient statistic' while doing so, as well as introducing the term 'variance' (in his 1920 paper A mathematical observation of the methods of determining the accuracy of observation by the mean error and the mean square error).

# Foundations

In the 1922 paper On the Mathematical foundations of theoretical statistics Fisher provides a short and simple overview of the main concepts, just to name the list of definitions: 'centre of location', 'consistency', 'distribution', 'efficiency', 'estimation', 'intrinsic accuracy', 'isostatistical regions', 'likelihood', 'location', 'optimum', 'scaling', 'specification', 'sufficiency', 'validity'. It requires a historian to see what Fisher contributed here in the sense of being the originator of concepts, and this also relates to Efron's statement. It is difficult to grasp what exactly is contributed by whom. But certainly Fisher helped to improve the statistical language and thinking.

In that article Fisher starts mentioning the problem of applying terms like 'mean' and 'variance' to both the true distribution value as well as the estimated value.

(I will try to avoid to put Fisher somewhere in a 'school' such as frequentist or Bayesian. I'd say he was just 'sufficiently' practical to whatever question was at hand).

In his further work Fisher developed early concepts of linear discriminant analysis:

what linear function of the four measurements $$X=\lambda_1 x_1 + \lambda_2 x_2 + \lambda_3 x_3 + \lambda_4 x_4$$ will maximize the ratio of the difference between the specific means to the standard deviations within the species?

and the concept of estimation by likelihood that Fisher explored further, and has two concepts named after him, Fisher information and Fisher score . See Theory of statistical estimation, 1925, Two new properties of mathematical likelihood, 1934, and The logic of inductive inference, 1935.

• R.A. Fisher Guide, by John Aldrich. An enormous source, if not the largest, with information on Fisher, with many further references.
• Michael Hardy's answer on Mathoverflow on a question about the greatest Mathematicians: https://mathoverflow.net/a/173374

Written by StackExchangeStrike

• Thank you @Martijn! I went through your answer and did small edits here and there, mostly to make the formatting clearer, and to fix some typos. I hope you will not mind. I am happy to award my bounty to this answer; very good contribution. It's especially great to have all these references. – amoeba Nov 13 '17 at 20:43
• Not at all, great edits, that's why I made it community wiki. This is a big question to answer. Although I am a big fan of Fisher and have went trough several of his articles, I was feeling like I should not have answered this (loaded) question. – Sextus Empiricus Nov 13 '17 at 21:46
• I advertised your answer in our chat and now I see that it became the most upvoted one in this thread. I think it's well deserved. – amoeba Nov 14 '17 at 15:40

Some concepts he invented: Sufficiency, efficiency, ANOVA, ancillarity, p-value and probably a host of others (most importantly design of experiments).

The likelihood function and mle's had precursors, but was popularized by him.

• +1 While Fisher certainly should get credit in relation to it, the concept of the p-value appears to have existed, at least informally, before FIsher's work. Pearson is clearly calculating p-values in his 1900 paper on the chi square goodness of fit test, and treats what he calculates (if only described in passing), as if it were on obvious, accepted thing to do. One gets the impression that it wasn't seen as a new concept introduced in that paper. Of course similar things might be said of many concepts ... they're often "around" for a while before someone formalizes it. – Glen_b Mar 12 '16 at 23:49

Sir Ronald Aylmer Fisher is credited for numerous aspects of experimental design and modern statistical theory and practice. Some of his most important contributions include significance testing (Bandyopadhyay and Cherry 2011), maximum likelihood estimation (MLE), permutation (re-sampling) distributions, sufficiency, asymptotic optimality theory (Efron 1998), and experimental design components including randomization, replication, blocking, confounding, and the analysis of variance (ANOVA). Also of note is his contention of Mendel's Pea Plant experiment. He claimed it was "too good to be true."

Consider reading that Efron (1998) paper, "RA Fisher in the 21st century". Let me quote the abstract:

Fisher is the single most important figure in 20th century statistics. This talk examines his influence on modern statistical thinking, trying to predict how Fisherian we can expect the 21st century to be. Fisher’s philosophy is characterized as a series of shrewd compromises between the Bayesian and frequentist viewpoints, augmented by some unique characteristics that are particularly useful in applied problems. Several current research topics are examined with an eye toward Fisherian influence, or the lack of it, and what this portends for future statistical developments. Based on the 1996 Fisher lecture, the article closely follows the text of that talk.

References

• Bandyopadhyay, Prasanta S., and Steve Cherry. "Elementary probability and statistics: A primer." Philosophy of Statistics 7 (2011): 53.

• Efron, Bradley. "R. A. Fisher in the 21st century." Statistical Science (1998): 95-114.

• His version of significance testing was fiducial inference which has been controversial and not accepted the way the Neyman-Pearson theory has been. Other contributions were monumental and part of the foundation of statistics. – Michael R. Chernick Nov 9 '17 at 19:20
• Well, fiducial inference seems to be finding some fans now. I note that nobody has mentioned Fisher Information, yet, nor useful guidelines like "You analyze as your randomize." – Björn Nov 9 '17 at 19:40
• Bjorn - yes, definitely missed the Fisher Information part. Probably because I copied this text from a paper I am writing up re: fisher information. Ha! – Jessica Burnett Nov 10 '17 at 14:42