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parte 3 (di 3)

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parte 2 (di 3)

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i comandi di R

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parte 1 (di 3)

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i comandi di R

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parte 3 (di 3)

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parte 2 (di 3)

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i comandi di R

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parte 1 (di 3)

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perche' il t test non funziona

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l'indice R.O.M.A.

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ecco come selezionare un modello in base al criterio di Akaike

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in base al criterio di informazione di Akaike: "small is beautiful"

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The Akaike information criterion ( AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model.

In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting.

The Akaike information criterion is named after the Japanese statistician Hirotsugu Akaike, who formulated it. It now forms the basis of a paradigm for the foundations of statistics and is also widely used for statistical inference.

Suppose that we have a statistical model of some data. Let k be the number of estimated parameters in the model. Let be the maximized value of the likelihood function for the model. Then the AIC value of the model is the following.[4][5]

Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. Thus, AIC rewards goodness of fit (as assessed by the likelihood function), but it also includes a penalty that is an increasing function of the number of estimated parameters. The penalty discourages overfitting, which is desired because increasing the number of parameters in the model almost always improves the goodness of the fit.

AIC is founded in information theory. Suppose that the data is generated by some unknown process f. We consider two candidate models to represent f: g ₁ and g ₂. If we knew f, then we could find the information lost from using g ₁ to represent f by calculating the Kullback-Leibler divergence, ; similarly, the information lost from using g ₂ to represent f could be found by calculating . We would then, generally, choose the candidate model that minimized the information loss.

We cannot choose with certainty, because we do not know f. Akaike (1974) showed, however, that we can estimate, via AIC, how much more (or less) information is lost by g ₁ than by g ₂. The estimate, though, is only valid asymptotically; if the number of data points is small, then some correction is often necessary (see AICc, below).

Note that AIC tells nothing about the absolute quality of a model, only the quality relative to other models. Thus, if all the candidate models fit poorly, AIC will not give any warning of that. Hence, after selecting a model via AIC, it is usually good practice to validate the absolute quality of the model. Such validation commonly includes checks of the model's residuals (to determine whether the residuals seem like random) and tests of the model's predictions. For more on this topic, see statistical model validation.

To apply AIC in practice, we start with a set of candidate models, and then find the models' corresponding AIC values. There will almost always be information lost due to using a candidate model to represent the "true model," i.e. the process that generated the data. We wish to select, from among the candidate models, the model that minimizes the information loss. We cannot choose with certainty, but we can minimize the estimated information loss.

Suppose that there are R candidate models. Denote the AIC values of those models by AIC ₁, AIC ₂, AIC ₃, ..., AICR. Let AIC min be the minimum of those values. Then the quantity exp((AIC min − AICi)/2) can be interpreted as being proportional to the probability that the i th model minimizes the (estimated) information loss.

As an example, suppose that there are three candidate models, whose AIC values are 100, 102, and 110. Then the second model is exp((100 − 102)/2) = 0.368 times as probable as the first model to minimize the information loss. Similarly, the third model is exp((100 − 110)/2) = 0.007 times as probable as the first model to minimize the information loss.

In this example, we would omit the third model from further consideration. We then have three options: (1) gather more data, in the hope that this will allow clearly distinguishing between the first two models; (2) simply conclude that the data is insufficient to support selecting one model from among the first two; (3) take a weighted average of the first two models, with weights proportional to 1 and 0.368, respectively, and then do statistical inference based on the weighted multimodel.

The quantity is known as the relative likelihood of model i. It is closely related to the likelihood ratio used in the likelihood-ratio test. Indeed, if all the models in the candidate set have the same number of parameters, then using AIC might at first appear to be very similar to using the likelihood-ratio test. There are, however, important distinctions. In particular, the likelihood-ratio test is valid only for nested models, whereas AIC (and AICc) has no such restriction.

Every statistical hypothesis test can be formulated as a comparison of statistical models. Hence, every statistical hypothesis test can be replicated via AIC. Two examples are briefly described in the subsections below. Details for those examples, and many more examples, are given by Sakamoto, Ishiguro & Kitagawa (1986, Part II) and Konishi & Kitagawa (2008, ch. 4).

Replicating Student's t-test

As an example of a hypothesis test, consider the t-test to compare the means of two normally-distributed populations. The input to the t-test comprises a random sample from each of the two populations.

To formulate the test as a comparison of models, we construct two different models. The first model models the two populations as having potentially different means and standard deviations. The likelihood function for the first model is thus the product of the likelihoods for two distinct normal distributions; so it has four parameters: . To be explicit, the likelihood function is as follows (denoting the sample sizes by and ).

The second model models the two populations as having the same means but potentially different standard deviations. The likelihood function for the second model thus sets in the above equation; so it has three parameters.

We then maximize the likelihood functions for the two models (in practice, we maximize the log-likelihood functions); after that, it is easy to calculate the AIC values of the models. We next calculate the relative likelihood. For instance, if the second model was only 0.01 times as likely as the first model, then we would omit the second model from further consideration: so we would conclude that the two populations have different means.

The t-test assumes that the two populations have identical standard deviations; the test tends to be unreliable if the assumption is false and the sizes of the two samples are very different ( Welch's t-test would be better). Comparing the means of the populations via AIC, as in the example above, has an advantage by not making such assumptions.

For another example of a hypothesis test, suppose that we have two populations, and each member of each population is in one of two categories-category #1 or category #2. Each population is binomially distributed. We want to know whether the distributions of the two populations are the same. We are given a random sample from each of the two populations.

Let m be the size of the sample from the first population. Let be the number of observations (in the sample) in category #1; so the number of observations in category #2 is . Similarly, let n be the size of the sample from the second population. Let be the number of observations (in the sample) in category #1.

Let p be the probability that a randomly-chosen member of the first population is in category #1. Hence, the probability that a randomly-chosen member of the first population is in category #2 is 1 − p. Note that the distribution of the first population has one parameter. Let q be the probability that a randomly-chosen member of the second population is in category #1. Note that the distribution of the second population also has one parameter.

To compare the distributions of the two populations, we construct two different models. The first model models the two populations as having potentially different distributions. The likelihood function for the first model is thus the product of the likelihoods for two distinct binomial distributions; so it has two parameters: p, q. To be explicit, the likelihood function is as follows.

The second model models the two populations as having the same distribution. The likelihood function for the second model thus sets in the above equation; so the second model has one parameter.

We then maximize the likelihood functions for the two models (in practice, we maximize the log-likelihood functions); after that, it is easy to calculate the AIC values of the models. We next calculate the relative likelihood. For instance, if the second model was only 0.01 times as likely as the first model, then we would omit the second model from further consideration: so we would conclude that the two populations have different distributions.

Statistical inference is generally regarded as comprising hypothesis testing and estimation. Hypothesis testing can be done via AIC, as discussed above. Regarding estimation, there are two types: point estimation and interval estimation. Point estimation can be done within the AIC paradigm: it is provided by maximum likelihood estimation. Interval estimation can also be done within the AIC paradigm: it is provided by likelihood intervals. Hence, statistical inference generally can be done within the AIC paradigm.

The most commonly used paradigms for statistical inference are frequentist inference and Bayesian inference. AIC, though, can be used to do statistical inference without relying on either the frequentist paradigm or the Bayesian paradigm: because AIC can be interpreted without the aid of significance levels or Bayesian priors. In other words, AIC can be used to form a foundation of statistics that is distinct from both frequentism and Bayesianism.

Modification for small sample size

When the sample size is small, there is a substantial probability that AIC will select models that have too many parameters, i.e. that AIC will overfit. To address such potential overfitting, AICc was developed: AICc is AIC with a correction for small sample sizes.

The formula for AICc depends upon the statistical model. Assuming that the model is univariate, is linear in its parameters, and has normally-distributed residuals (conditional upon regressors), then the formula for AICc is as follows.

-where n denotes the sample size and k denotes the number of parameters. Thus, AICc is essentially AIC with an extra penalty term for the number of parameters. Note that as n → ∞, the extra penalty term converges to 0, and thus AICc converges to AIC.

If the assumption that the model is univariate and linear with normal residuals does not hold, then the formula for AICc will generally be different from the formula above. For some models, the formula can be difficult to determine. For every model that has AICc available, though, the formula for AICc is given by AIC plus terms that includes both k and k ². In comparison, the formula for AIC includes k but not k ². In other words, AIC is a first-order estimate (of the information loss), whereas AICc is a second-order estimate.

Further discussion of the formula, with examples of other assumptions, is given by Burnham & Anderson (2002, ch. 7) and by Konishi & Kitagawa (2008, ch. 7-8). In particular, with other assumptions, bootstrap estimation of the formula is often feasible.

To summarize, AICc has the advantage of tending to be more accurate than AIC (especially for small samples), but AICc also has the disadvantage of sometimes being much more difficult to compute than AIC. Note that if all the candidate models have the same k and the same formula for AICc, then AICc and AIC will give identical (relative) valuations; hence, there will be no disadvantage in using AIC, instead of AICc. Furthermore, if n is many times larger than k ², then the extra penalty term will be negligible; hence, the disadvantage in using AIC, instead of AICc, will be negligible.

Hirotugu Akaike

The Akaike information criterion was formulated by the statistician Hirotsugu Akaike. It was originally named "an information criterion". It was first announced in English by Akaike at a 1971 symposium; the proceedings of the symposium were published in 1973. The 1973 publication, though, was only an informal presentation of the concepts. The first formal publication was a 1974 paper by Akaike. As of October 2014, the 1974 paper had received more than 14,000 citations in the Web of Science: making it the 73rd most-cited research paper of all time.

Nowadays, AIC has become common enough that it is often used without citing Akaike's 1974 paper. Indeed, there are over 150,000 scholarly articles/books that use AIC (as assessed by Google Scholar).[when? ]

The initial derivation of AIC relied upon some strong assumptions. Takeuchi (1976) showed that the assumptions could be made much weaker. Takeuchi's work, however, was in Japanese and was not widely known outside Japan for many years.

AICc was originally proposed for linear regression (only) by Sugiura (1978). That instigated the work of Hurvich & Tsai (1989), and several further papers by the same authors, which extended the situations in which AICc could be applied.

The first general exposition of the information-theoretic approach was the volume by Burnham & Anderson (2002). It includes an English presentation of the work of Takeuchi. The volume led to far greater use of AIC, and it now has more than 48,000 citations on Google Scholar.

Akaike called his approach an "entropy maximization principle", because the approach is founded on the concept of entropy in information theory. Indeed, minimizing AIC in a statistical model is effectively equivalent to maximizing entropy in a thermodynamic system; in other words, the information-theoretic approach in statistics is essentially applying the Second Law of Thermodynamics. As such, AIC has roots in the work of Ludwig Boltzmann on entropy. For more on these issues, see Akaike (1985) and Burnham & Anderson (2002, ch. 2).

A statistical model must account for random errors. A straight line model might be formally described as y i = b ₀ + b ₁ x i + ε i. Here, the ε i are the residuals from the straight line fit. If the ε i are assumed to be i.i.d. Gaussian (with zero mean), then the model has three parameters: b ₀, b ₁, and the variance of the Gaussian distributions. Thus, when calculating the AIC value of this model, we should use k=3. More generally, for any least squares model with i.i.d. Gaussian residuals, the variance of the residuals' distributions should be counted as one of the parameters.

As another example, consider a first-order autoregressive model, defined by x i = c + φx i−1 + ε i, with the ε i being i.i.d. Gaussian (with zero mean). For this model, there are three parameters: c, φ, and the variance of the ε i. More generally, a pth-order autoregressive model has p + 2 parameters. (If, however, c is not estimated from the data, but instead given in advance, then there are only p + 1 parameters.)

The AIC values of the candidate models must all be computed with the same data set. Sometimes, though, we might want to compare a model of the log( response variable, y, with a model of the logarithm of the response variable, y). More generally, we might want to compare a model of the data with a model of transformed data. Following is an illustration of how to deal with data transforms (adapted from Burnham & Anderson (2002, §2.11.3): "Investigators should be sure that all hypotheses are modeled using the same response variable").

Suppose that we want to compare two models: one with a log(1/ normal distribution of y and one with a normal distribution of y). We should not directly compare the AIC values of the two models. Instead, we should transform the normal cumulative distribution function to first take the logarithm of y. To do that, we need to perform the relevant integration by substitution: thus, we need to multiply by the derivative of the (natural) logarithm function, which is y. Hence, the transformed distribution has the following probability density function:

-which is the probability density function for the log-normal distribution. We then compare the AIC value of the normal model against the AIC value of the log-normal model.

The critical difference between AIC and BIC (and their variants) is the asymptotic property under well-specified and misspecified model classes. Their fundamental differences have been well-studied in regression variable selection and autoregression order selection problems. In general, if the goal is prediction, AIC and leave-one-out cross-validations are preferred. If the goal is selection, inference, or interpretation, BIC or leave-many-out cross-validations are preferred. A comprehensive overview of AIC and other popular model selection methods is given by Ding et al. (2018)

The formula for the 2 Bayesian information criterion (BIC) is similar to the formula for AIC, but with a different penalty for the number of parameters. With AIC the penalty is k, whereas with BIC the penalty is .

A comparison of AIC/AICc and BIC is given by Burnham & Anderson (2002, §6.3-6.4), with follow-up remarks by Burnham & Anderson (2004). The authors show that AIC/AICc can be derived in the same Bayesian framework as BIC, just by using different prior probabilities. In the Bayesian derivation of BIC, though, each candidate model has a prior probability of 1/ R (where R is the number of candidate models). Additionally, the authors present a few simulation studies that suggest AICc tends to have practical/performance advantages over BIC.

A point made by several researchers is that AIC and BIC are appropriate for different tasks. In particular, BIC is argued to be appropriate for selecting the "true model" (i.e. the process that generated the data) from the set of candidate models, whereas AIC is not appropriate. To be specific, if the "true model" is in the set of candidates, then BIC will select the "true model" with probability 1, as n → ∞; in contrast, when selection is done via AIC, the probability can be less than 1. Proponents of AIC argue that this issue is negligible, because the "true model" is virtually never in the candidate set. Indeed, it is a common aphorism in statistics that " all models are wrong"; hence the "true model" (i.e. reality) cannot be in the candidate set.

Another comparison of AIC and BIC is given by Vrieze (2012). Vrieze presents a simulation study-which allows the "true model" to be in the candidate set (unlike with virtually all real data). The simulation study demonstrates, in particular, that AIC sometimes selects a much better model than BIC even when the "true model" is in the candidate set. The reason is that, for finite n, BIC can have a substantial risk of selecting a very bad model from the candidate set. This reason can arise even when n is much larger than k ². With AIC, the risk of selecting a very bad model is minimized.

If the "true model" is not in the candidate set, then the most that we can hope to do is select the model that best approximates the "true model". AIC is appropriate for finding the best approximating model, under certain assumptions. (Those assumptions include, in particular, that the approximating is done with regard to information loss.)

Comparison of AIC and BIC in the context of regression is given by Yang (2005). In regression, AIC is asymptotically optimal for selecting the model with the least mean squared error, under the assumption that the "true model" is not in the candidate set. BIC is not asymptotically optimal under the assumption. Yang additionally shows that the rate at which AIC converges to the optimum is, in a certain sense, the best possible.

Sometimes, each candidate model assumes that the residuals are distributed according to independent identical normal distributions (with zero mean). That gives rise to least squares model fitting.

With least squares fitting, the maximum likelihood estimate for the variance of a model's residuals distributions is the reduced chi-squared statistic,

where the residual sum of squares is

and is the degree of freedom (for n observations and m unknowns). Then, the maximum value of a model's log-likelihood function is (see Normal distribution#Log-likelihood):

where C is a constant independent of the model, and dependent only on the particular data points, i.e. it does not change if the data does not change.

Because only differences in AIC are meaningful, the constant C can be ignored, which allows us to conveniently take the following for model comparisons:

Note that if all the models have the same k, then selecting the model with minimum AIC is equivalent to selecting the model with minimum RSS-which is the usual objective of model selection based on least squares.

Leave-one-out cross-validation is asymptotically equivalent to AIC, for ordinary linear regression models. Asymptotic equivalence to AIC also holds for mixed-effects models.

Mallows's Cp is equivalent to AIC in the case of (Gaussian) linear regression.

1. ^ Stoica, P.; Selen, Y. (2004), "Model-order selection: a review of information criterion rules", IEEE Signal Processing Magazine (July): 36-47, doi:10.1109/MSP.2004.1311138, S2CID 17338979

2. ^ McElreath, Richard (2016). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. CRC Press. p. 189. ISBN 978-1-4822-5344-3. AIC provides a surprisingly simple estimate of the average out-of-sample deviance.

3. ^ Taddy, Matt (2019). Business Data Science: Combining Machine Learning and Economics to Optimize, Automate, and Accelerate Business Decisions. New York: McGraw-Hill. p. 90. ISBN 978-1-260-45277-8. The AIC is an estimate for OOS deviance.

4. ^ Ding, Jie; Tarokh, Vahid; Yang, Yuhong (November 2018). "Model Selection Techniques: An Overview". IEEE Signal Processing Magazine. 35 (6): 16-34. arXiv:. Bibcode:2018ISPM...35...16D. doi:10.1109/MSP.2018.2867638. ISSN 1053-5888. S2CID 53035396.

5. ^ Ding, J.; Tarokh, V.; Yang, Y. (June 2018). "Bridging AIC and BIC: A New Criterion for Autoregression". IEEE Transactions on Information Theory. 64 (6): 4024-4043. arXiv:. doi:10.1109/TIT.2017.2717599. ISSN 1557-9654. S2CID 5189440.

6. ^ Ding, Jie; Tarokh, Vahid; Yang, Yuhong (2018-11-14). "Model Selection Techniques: An Overview". IEEE Signal Processing Magazine. 35 (6): 16-34. arXiv:. Bibcode:2018ISPM...35f..16D. doi:10.1109/MSP.2018.2867638. S2CID 53035396.

Aho, K.; Derryberry, D.; Peterson, T. (2014), "Model selection for ecologists: the worldviews of AIC and BIC", , 95 (3): 631-636, doi:10.1890/13-1452.1, PMID 24804445.

Akaike, H. (1973), "Information theory and an extension of the maximum likelihood principle", in Petrov, B. N.; Csáki, F. (eds.), 2nd International Symposium on Information Theory, Tsahkadsor, Armenia, USSR, September 2-8, 1971, Budapest: Akadémiai Kiadó, pp. 267-281. Republished in Kotz, S.; Johnson, N. L., eds. (1992), Breakthroughs in Statistics, vol. I, Springer-Verlag, pp. 610-624.

Akaike, H. (1974), "A new look at the statistical model identification", IEEE Transactions on Automatic Control, 19 (6): 716-723, Bibcode:1974ITAC...19..716A, doi:10.1109/TAC.1974.1100705, MR 0423716.

Akaike, H. (1985), "Prediction and entropy", in Atkinson, A. C.; Fienberg, S. E. (eds.), A Celebration of Statistics, Springer, pp. 1-24.

Bandyopadhyay, P. S.; Forster, M. R., eds. (2011), Philosophy of Statistics, North-Holland Publishing.

Boisbunon, A.; Canu, S.; Fourdrinier, D.; Strawderman, W.; Wells, M. T. (2014), "Akaike's Information Criterion, and estimators of loss for elliptically symmetric distributions", International Statistical Review, 82 (3): 422-439, doi:10.1111/insr.12052, S2CID 119926327.

Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel Inference: A practical information-theoretic approach (2nd ed.), Springer-Verlag.

Burnham, K. P.; Anderson, D. R. (2004), "Multimodel inference: understanding AIC and BIC in Model Selection" (PDF), Sociological Methods & Research, 33: 261-304, doi:10.1177/0049124104268644, S2CID 121861644.

Cavanaugh, J. E. (1997), "Unifying the derivations of the Akaike and corrected Akaike information criteria", Statistics & Probability Letters, 31 (2): 201-208, doi:10.1016/s0167-7152(96)00128-9.

Claeskens, G.; Hjort, N. L. (2008), Model Selection and Model Averaging, Cambridge University Press. [Note: the AIC defined by Claeskens & Hjort is the negative of the standard definition-as originally given by Akaike and followed by other authors.]

deLeeuw, J. (1992), "Introduction to Akaike (1973) information theory and an extension of the maximum likelihood principle" (PDF), in Kotz, S.; Johnson, N. L. (eds.), Breakthroughs in Statistics I, Springer, pp. 599-609, archived from the original (PDF) on 2016-01-08.

Fang, Yixin (2011), "Asymptotic equivalence between cross-validations and Akaike Information Criteria in mixed-effects models" (PDF), Journal of Data Science, 9: 15-21, archived from the original (PDF) on 2016-03-04.

Findley, D. F.; Parzen, E. (1995), "A conversation with Hirotugu Akaike", Statistical Science, 10: 104-117, doi:10.1214/ss/1177010133 .

Giraud, C. (2015), Introduction to High-Dimensional Statistics, CRC Press.

Hurvich, C. M.; Tsai, C.-L. (1989), "Regression and time series model selection in small samples", Biometrika, 76 (2): 297-307, doi:10.1093/biomet/76.2.297.

Konishi, S.; Kitagawa, G. (2008), Information Criteria and Statistical Modeling, Springer.

McQuarrie, A. D. R.; Tsai, C.-L. (1998), Regression and Time Series Model Selection, World Scientific.

Sakamoto, Y.; Ishiguro, M.; Kitagawa, G. (1986), Akaike Information Criterion Statistics, D. Reidel.

Stone, M. (1977), "An asymptotic equivalence of choice of model by cross-validation and Akaike's criterion", Journal of the Royal Statistical Society, Series B, 39 (1): 44-47, doi:10.1111/j.2517-6161.1977.tb01603.x, JSTOR 2984877.

Sugiura, N. (1978), "Further analysis of the data by Akaike's information criterion and the finite corrections", Communications in Statistics - Theory and Methods, 7: 13-26, doi:10.1080/03610927808827599.

Takeuchi, K. (1976), " " [Distribution of informational statistics and a criterion of model fitting], Suri Kagaku [Mathematical Sciences] (in Japanese), 153: 12-18, ISSN 0386-2240.

Vrieze, S. I. (2012), "Model selection and psychological theory: a discussion of the differences between the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC)", Psychological Methods, 17 (2): 228-243, doi:10.1037/a0027127, PMC , PMID 22309957.

Yang, Y. (2005), "Can the strengths of AIC and BIC be shared?", Biometrika, 92: 937-950, doi:10.1093/biomet/92.4.937.

Akaike, H. (21 December 1981), "This Week's Citation Classic" (PDF), Current Contents Engineering, Technology, and Applied Sciences, 12 (51): 42 [Hirotogu Akaike comments on how he arrived at AIC]

Anderson, D. R. (2008), Model Based Inference in the Life Sciences, Springer

Arnold, T. W. (2010), "Uninformative parameters and model selection using Akaike's Information Criterion", Journal of Wildlife Management, 74 (6): 1175-1178, doi:10.1111/j.1937-2817.2010.tb01236.x

Burnham, K. P.; Anderson, D. R.; Huyvaert, K. P. (2011), "AIC model selection and multimodel inference in behavioral ecology" (PDF), Behavioral Ecology and Sociobiology, 65: 23-35, doi:10.1007/s00265-010-1029-6, S2CID 3354490, archived from the original (PDF) on 2017-08-09

Cavanaugh, J. E.; Neath, A. A. (2019), "The Akaike information criterion", WIREs Computational Statistics, 11 (3): e1460, doi:10.1002/wics.1460, S2CID 88491599

Ing, C.-K.; Wei, C.-Z. (2005), "Order selection for same-realization predictions in autoregressive processes", Annals of Statistics, 33 (5): 2423-2474, doi:10.1214/009053605000000525

Ko, V.; Hjort, N. L. (2019), "Copula information criterion for model selection with two-stage maximum likelihood estimation", Econometrics and Statistics, 12: 167-180, doi:10.1016/j.ecosta.2019.01.001, hdl:10852/74878, S2CID 126873625

Larski, S. (2012), The Problem of Model Selection and Scientific Realism (PDF) (Thesis), London School of Economics

Pan, W. (2001), "Akaike's Information Criterion in generalized estimating equations", Biometrics, 57 (1): 120-125, doi:10.1111/j.0006-341X.2001.00120.x, PMID 11252586, S2CID 7862441

Parzen, E.; Tanabe, K.; Kitagawa, G., eds. (1998), Selected Papers of Hirotugu Akaike, Springer Series in Statistics, Springer, doi:10.1007/978-1-4612-1694-0, ISBN 978-1-4612-7248-9

Saefken, B.; Kneib, T.; van Waveren, C.-S.; Greven, S. (2014), "A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models", Electronic Journal of Statistics, 8: 201-225, doi:10.1214/14-EJS881

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Akaike information criterion - Wikipedia

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The Akaike information criterion ( AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models.

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Grazie, anche questa volta desidero registrare la tua email.

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approfondimento

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riprendiamo il Capitolo 1 di Sergio Invernizzi

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la retta di regressione

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quarto compito (aggiornato) per casa

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Grazie, anche questa volta desidero registrare la tua email.

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consigli per il futuro

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Guidelines for reporting statistics in journals published by the American Physiological Society | Journal of Applied Physiology

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The mere adherence to guidelines for reporting statistics can never substitute for an understanding of concepts and procedures in statistics. Nevertheless, we hope these guidelines, when used with other resources ( 4, 8, 9, 11, 12, 14), will help improve the caliber of statistical information reported in articles published by the American Physiological Society.

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tutto bene?

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una collezione di dataset per esercitarsi

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per le Scuole di Dottorato della Università Magna Græcia di Catanzaro

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Guinness brewer William S. Gosset's work is responsible for inspiring the concept of statistical significance, industrial quality control, efficient design of experiments and, not least of all, consistently great tasting beer.

But Gosset is certainly no household name. Books and articles about him are sparse, and he is rarely discussed among history's most important statisticians. Because he used a pseudonym, his name isn't even familiar to most people who frequently use his most famous discovery. Gosset is the "student" of the Student's T-Test, a method for interpreting what can be extrapolated from a small sample of data.

How did a brewer of dry stout revolutionize statistics? And why is he so little-known?

By all historical accounts, William S. Gosset was a pretty awesome guy.

Contemporary statisticians like W. Edward Deming , , and Florence Nightingale David , respectively called him a "very humble and pleasing personality," "very pleasant chap" and "A nice man [...] without a jealous bone in his body." Both Karl Pearson and R.A. Fisher, the two most famous statistical thinkers of the early 20th Century, who were known to hate each other, found common ground in their for Gosset.

Gosset would spend the rest of his life working for Guinness, and it was through working on Guinness products that he would develop his great statistical innovations.

Like most beers, Guinness is flavored with the flowers of the plant Humulus lupulus, also known as "hops". The brewery used nearly five million pounds of the stuff in 1898. They determined which hops to use based on qualitative measures such as " looks and fragrance ."

At the scale at which Guinness was brewing, the "looks and fragrance" method was not economical or even accurate. The scientific brewing team, of which Gosset was a part, would improve this selection process.

Gosset discovered Student's t-distribution; via Columbia University.

So Gosset set to work. His goal was to understand just how much less representative a sample is when the sample is small. In more technical terms, how much wider is the error distribution of an estimate when you only have a sample of two or ten, compared to when you have a sample of a thousand?

Gosset's first problem involved figuring out exactly how many observations of malt extract, a substance used in beer making, were necessary to be confident the "degrees saccharine" of the extract was within 0.5 degrees of a targeted goal of 133 degrees.

His initial approach was just to simulate a whole bunch of data. He had an extract for which he had a very large number of samples and could be relatively confident of the exact degrees saccharine. He then took many different two-observation samples from the extract in order to test the accuracy of such a small sample. He found that about 80% of the time, the measurement from just two observations was within 0.5 degrees of the true number.

He then tried the same thing with three measurements. This time, there was an approximate 87.5% chance of getting with 0.5 degrees. With four measurements, he found a better than 92% chance. With 82 measurements, the likelihood of getting within 0.5 degrees was "practically infinite".

His bosses at Guinness were thrilled with the findings. This would allow them to make intelligent decisions about which materials to use for their beer, in a way that no other business could.

Yet Gosset was not satisfied with his approximated method. He wanted to uncover the exact mathematics behind inference from small samples. He told Guinness that he wanted to consult with "some mathematical physicist" on the matter. The company obliged and sent Gosset to Karl Pearson's lab at the University College London. Pearson was one of the leading scientific figures of his time and the man later credited with establishing the field of statistics.

How Gosset Became the "Student"

But why is it the "Student's" t-distribution rather than "Gosset's"?

Upon completing his work on the t-distribution, Gosset was eager to make his work public. It was an important finding, and one he wanted to share with the wider world. The managers of Guinness were not so keen on this. They realized they had an advantage over the competition by using this method, and were not excited about relinquishing that leg up. If Gosset were to publish the paper, other breweries would be on to them.

So they came to a compromise. Guinness agreed to allow Gosset to publish the finding, as long as he used a pseudonym. This way, competitors would not be able to realize that someone on Guinness's payroll was doing such research, and figure out that the company's scientifically enlightened approach was key to their success.

R.A. Fisher is the "Father of Modern Statistics" and, perhaps unfortunately, extended Gosset's ideas in a more stringent direction.

Fisher's Extensions and the Sanctification of .05

The British scientist and author Richard Dawkins R.A. Fisher "a genius who almost single-handedly created the foundations for modern statistical science." Some of Fisher's most important work include his theories of experimental design, analysis of variance, and introducing the concept of making guesses of unknowns based on maximum likelihood (the concept of approximating an unknown value based on the number that makes related data most likely). On top of all this, he was a hugely influential .

Fisher began studying mathematics at Cambridge University just one year after Gosset published "The Probable Error of Means." Fisher knew Gosset was on to something big.

Gosset's t-distribution and the idea of "statistical significance", discussed using slightly different terms by Gosset, became fundamental to Fisher's ideas about statistical methods. In 1925, Fisher published what would become arguably the most influential book in the history of statistics, Statistical Methods for Research Workers , which presented Gosset and Fisher's work to a wider public.

Some of Fisher's extensions of Gosset's ideas were controversial. In fact, Gosset himself objected to some of them.

The most controversial of these was Fisher's hallowing of a result that had a probability of less than 5% of occurring randomly (this probability is sometimes referred to as the p-value or P). For example, if a company surveyed people about which of two beers they preferred, they might find that 20 out of 25 surveyed preferred one beer. But how to decide whether this is enough evidence for the supremacy of that beer? Fisher suggests that if 20 people out of 25 choosing one beer is less than 5% likely to happen at random if people actually like the beers equally, then we can be comfortable using this as evidence. He wrote in Statistical Methods for Research Workers:

In the paper , the economist Stephen Ziliak demonstrates that Gosset found the .05 threshold to be arbitrary.

Today, many statisticians feel that that Fisher's assertion that .05 signifies significance has done serious damage to science. The doctor and researcher John Ioannidis demonstrated that a large proportion of research findings published in scientific journals are false, in large part because of the reliance on the .05 threshold. Given the huge number of studies that are conducted every year, and the fact that one in twenty studies will meet the .05 threshold by random chance, many studies that purport to have detected a "significant" finding are spurious.

The Original Six Sigma Blackbelt

What little fame Gosset has today is primarily among students of statistics. But perhaps his widest impact is as a of industrial quality control.

The industrial revolution and modern factory methods led to product creation at a scale and speed never before seen. Prior to this scaled production, it was generally possible to check the quality of your goods using qualitative methods. Bread makers, boat builders and brewers made so little product that they could generally check each individually for quality issues. Industrial production brought with it great benefits, but also the challenge of how to make sure you didn't ruin your brand's reputation by letting faulty product slip through the cracks.

Though the discovery of the t-distribution was Gosset's grand achievement, he also had several other influential statistical ideas. Gosset developed a theory of efficient field experiment design and methods for applying statistical techniques to experimental data. Gosset would publish all but one article under the "student" pseudonym, only once releasing an article under his own name. The economist Heinz Kohler , "For many years, an air of romanticism surrounded the appearance of 'Student's' papers, and only a few individuals knew his real identity, even for some time after his death."

Certainly, part of the reason for this is that Gosset worked in business rather than academia. Had he been a professor like Fisher and Pearson, he would have been able to use his own name in his work, and perhaps publish textbooks detailing his methods and ideas. But if he worked outside of industry, he may never have had the opportunity to take on the practical problems that seem to have most stimulated him.

Perhaps just as important to Gosset's absence of renown was his lack of ideology. Fisher, who would take his ideas and make them famous, was a man who believed in creating rules and structures for how quantitative research ought be done. Gosset was never stuck to any particular method. He was simply interested in solving the problems that would make sure each beer was as good as the last one.

For our next post, we examine how Auto-Tune changed the economics of the recording industry and ushered in a new era of music. To get notified when we post it →

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The Guinness Brewer Who Revolutionized Statistics

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Guinness brewer William S. Gosset's work is responsible for inspiring the concept of statistical significance, industrial quality control, efficient design of experiments and, not least of all, consistently great tasting beer. But Gosset is certainly no household name. Books and articles about him are sparse, and he is rarely discussed among history's most important statisticians.

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