Georg Rasch comments on "The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data," a paper presented by Per Martin-Löf, at the Conference on Foundational Questions in Statistical Inference, Aarhus, Denmark, May 7-12, 1973.
Courtesy of Svend Kreiner.
"I wish to make it quite explicit, that the reason for using both significance and redundancy lies in the contention that every model is basically wrong [emphasis author's], i.e., it is bound to fail, given enough data.
When you are in the possession of a set of data you may then either be in the position that your significance test tells you that the model fails, or you may not have got enough observations for that purpose. In the latter case you cannot yet reject the model on statistical grounds - which of course should not be construed as meaning that you really accept it. In the former case you have to realize that the model fails - and I have no sympathy for relaxing the significance requirement for the reason that the data are substantial enough to show it - but that does not mean that the model is too bad to be applied in the actual case. To take a parallel from elementary physics: A "mathematical pendulum" is defined as "a heavy point, swinging frictionless in a weightless string in vacuum". A contraption like that was never seen; thus as a model for the motion of a real pendulum it is "unrealistic". Notwithstanding, it works quite well for a short time interval, but it begins soon to show a systematic decrease of the oscillation angle. To the model - a second order differential equation - thus requiring an amendment, a Friction term is added, and now it works perfectly well for a long time, even during a few days, until another systematic deviation shows. If needed, a further correction, for air resistance, say, should be attempted - but as a matter of fact, this is not needed, because it has worked well enough for the purpose of the geo-physicist, which was to measure the gravity constant ("g") with 7 decimal places !
It is exactly at this point Martin-Löf's redundancy sets in: the model fails - that being demonstrated by some significance test - but does it matter for its purposes ? Taking his cue from Information Theory, Martin-Löf uses the redundancy, as there defined, for measuring the deviation of the model from the data, in the sense of determining the relative decrease of the amount of information in the data which is caused by the departure from the null hypothesis.
Taken literally, the redundancy as a tool may be a rather gross evaluation of the loss suffered by replacing the data by the model. Even if it seems small the parts lost may affect some of the use of the model quite appreciably. Therefore it may be necessary to undertake a careful analysis in order to localize the losses and consider what to do about them.
In this connection I may touch upon Weldon's dice throwing experiment with a redundancy of 0.000024. But what if we on several repetitions found the same result and it turned out, that the deviations of the observed distributions from the model distributions persisted in the same parts of them ?
I do not know of any repetition of the experiment, neither of any detailed report on fractions of it as they were produced during some years, but I do happen to know (see Steffensen (1923)) that in a similar case the deviations were taken sufficiently seriously by statisticians to attempt fitting them with a number of alternative distributions, any particular justification of which I do not recall having seen.
Let me end up with the scale of redundancies presented by the speaker. It did leave me with the notion of new horrors of conventional limits ! In this connection we may, however, have a chance of doing it more rationally by analyzing just which sort of damage and how much of it is invoked by using the model for specified purposes.
I do look forward to the contribution of the redundancy concept to articulating my vague thesis, that we should never succumb to the illusion that any of our models are correct, but we should certainly aim at making them adequate for our purposes - the redundancy possibly being a useful measuring instrument in that connection."
References:
Johan Frederik Steffensen, Factorial moments and discontinuous frequency functions, Skandinavisk Aktuarietidsskrift, vol. 6 (1923), pp. 73-89
Walter Frank Raphael Weldon, Letter to Francis Galton, Feb.2, 1894, reporting 26,306 rolls of 12 dice.
Svend Kreiner adds:
Let me also point out that David Cox in his book on "Applied Statistics" with E.J. Snell (1981, Chapman and Hall, p. 42) does not talk about model fitting or model testing. He talks about how to examine the adequacy (my italics) of models. That, I think, is the way we should understand what we are doing, when we test the fit of the Rasch model. If we want to use the Rasch model, despite the fact that items do not fit the model, we are obliged to provide some evidence that it is adequate for the purpose we have with the items. It is not enough to say that we want to disregard the statistical test results because models are always rejected if we have enough data.
All Statistical Models are Wrong!, G. Rasch ... Rasch Measurement Transactions, 2011, 24:4, 1309
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