All models are wrong

All models are wrong is a common aphorism and anapodoton in statistics. It is often expanded as "All models are wrong, but some are useful". The aphorism acknowledges that statistical models always fall short of the complexities of reality but can still be useful nonetheless. The aphorism originally referred just to statistical models, but it is now sometimes used for scientific models in general.^[1]

The aphorism is generally attributed to George E. P. Box, a British statistician, although the underlying concept predates Box's writings.

The phrase "all models are wrong" was first attributed to George Box in a 1976 paper published in the Journal of the American Statistical Association. In the paper, Box uses the phrase to refer to the limitations of models, arguing that while no model is ever completely accurate, simpler models can still provide valuable insights if applied judiciously.^[2] In their 1983 book on generalized linear models, Peter McCullagh and John Nelder stated that while modeling in science is a creative process, some models are better than others, even though none can claim eternal truth.^[3][4] In 1996, an Applied Statistician's Creed was proposed by M.R. Nester, which incorporated the aphorism as a central tenet.^[5]

Although the aphorism is most commonly associated with George Box, the underlying idea has been historically expressed by various thinkers in the past. Alfred Korzybski noted in 1933, "A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness."^[6] In 1939, Walter Shewhart discussed the impossibility of constructing a model that fully characterizes a state of statistical control, noting that no model can exactly represent any specific characteristic of such a state.^[7] John von Neumann, in 1947, remarked that "truth is much too complicated to allow anything but approximations.^[2] In 1960, Georg Rasch noted that no models are ever true, not even Newton's laws, emphasizing that models should not be evaluated based on truth but on their applicability for a given purpose.^[1]

Box used the aphorism again in 1979, where he expanded on the idea by discussing how models serve as useful approximations, despite failing to perfectly describe empirical phenomena.^[8] He later reiterated this sentiment in his later works, where he discussed how models should be judged based on their utility rather than their absolute correctness.^[9][7]

David Cox, in a 1995 commentary, argued that stating all models are wrong is unhelpful, as models by their nature simplify reality. He emphasized that statistical models, like other scientific models, aim to capture important aspects of systems through idealized representations.^[10]

In their 2002 book on statistical model selection, Burnham and Anderson reiterated Box’s statement, noting that while models are simplifications of reality, they vary in usefulness, from highly useful to essentially useless.^[11]

J. Michael Steele used the analogy of city maps to explain that models, like maps, serve practical purposes despite their limitations, emphasizing that certain models, though simplified, are not necessarily wrong.^[12] In response, Andrew Gelman acknowledged Steele’s point but defended the usefulness of the aphorism, particularly in drawing attention to the inherent imperfections of models.^[13]

Philosopher Peter Truran, in a 2013 essay, discussed how seemingly incompatible models can make accurate predictions by representing different aspects of the same phenomenon, illustrating the point with an example of two observers viewing a cylindrical object from different angles.^[14]

In 2014, David Hand reiterated that models are meant to aid in understanding or decision-making about the real world, a point emphasized by Box’s famous remark.^[15]

• Anscombe's quartet – Four data sets with the same descriptive statistics, yet very different distributions
• Bonini's paradox – As a model of a complex system becomes more complete, it becomes less understandable
• Lie-to-children – Teaching a complex subject via simpler models
• Map–territory relation – Relationship between an object and a representation of that object
• Pragmatism – Philosophical tradition
• Reification (fallacy) – Fallacy of treating an abstraction as if it were a real thing
• Scientific modelling – Scientific activity that produces models
• Statistical model – Type of mathematical model
• Statistical model validation – Evaluating whether a chosen statistical model is appropriate or not
• Verisimilitude – Resemblance to reality

• Anderson, C. (23 June 2008), "The end of theory", Wired
• Box, G. E. P. (1999), "Statistics as a catalyst to learning by scientific method Part II—A discussion", Journal of Quality Technology, 31: 16–29, doi:10.1080/00224065.1999.11979890
• Enderling, H.; Wolkenhauer, O. (2021), "Are all models wrong?", Computational and Systems Oncology, 1 (1): e1008, doi:10.1002/cso2.1008, PMC 7880041, PMID 33585835
• Saltelli, A.; Funtowicz, S. (Winter 2014), "When all models are wrong", Issues in Science and Technology, 30

• "All Models are Right, Most are Useless"—Andrew Gelman blog
• All models are wrong—Peter Coles blog