**4. Misspecification testing**

Mayo and Spanos differentiate between model specification and model selection. An adequate model specification licenses primary statistical inference, and with it statistical model selection from the specified family. Serial feature detection in any time series is a form of model selection from a family of related models, reliant on model specification. It must be noted that for our work a series of tests are performed, a single detection test and multiple probative tests, but that as each is against an independent null, this does not involve a multiple-testing issue, instead increasing the overall power of the testing regime.

Chapter 2 of [25] defines experimental error as all extraneous variation outside experimental treatments, and states "Neither the presence of experimental errors or their causes need concern the investigator, provided his [sic] results are sufficiently accurate to permit definite conclusions to be reached". This definition still dominates statistical climatology. Climate data are not generally experimental, but often a feature of interest in climate data is investigated by treating natural variability as extraneous variation. Experimental design requires that statistical models are properly specified, however complex systems being observed may align to many different statistical models and have multiple features of interest, leading to the possibility of misspecification.

Mayo and Spanos [2] (MS2004) introduce a methodology for testing misspecifications in statistical models (M-S testing). Taking this as a point of departure we then propose that a full understanding of the assumptions of statistical models allows one to probe data for features even when available tests are misspecified.

Model specification delineates families of statistical models. For physical problems, the family would be misspecified if the available parameters do not properly reflect the physical processes [13].

*M*1 : *yi* ¼ *β*<sup>0</sup> þ *β*1*xi* þ *ui*, *ui* ¼ *ρui*�<sup>1</sup> þ *εi:* (4)

*<sup>M</sup>*<sup>2</sup> : *yi* <sup>¼</sup> *<sup>β</sup>*<sup>0</sup> <sup>þ</sup> *<sup>β</sup>*1*yi*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*2*yi*�<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*^*<sup>i</sup>* (5)

However an alternative AR(2) model is then shown to explain more variance

*Severe Testing and Characterization of Change Points in Climate Time Series*

Probing the model M0 shows it to be misspecified due to an irrelevant variable.

The secret variable *xi* is the number of shoes owned by Spanos's grandmother!

In some papers step-like changes are introduced *en passant*, on the way to revealing or locating in time various phenomena. For instance the delineation of the Pacific Decadal Oscillation [26–28], or reduction in South-Western Western Australian rainfall [29]. In the last decade an astonishing number of papers addressed the so-called hiatus, many purporting to show that it never happened [30] or was simply routine variability [31, 32], or a methodological/statistical error [33], or suggesting that natural variability, internal variability and extrinsic factors combined with forced warming [34]. However others, one way or another, simply

From these and other papers and some personal communication, the objections/ challenges to the existence of abrupt changes (including but not limited to the so-

1.Physical implausibility of step like changes in average temperatures.

2.Overcooking. In general, that warming is in fact more or less constant and positive, and more or less smoothly changing natural variability is imposed on it, with the result that a test for shifts is deceived by increases and decreases in

3.Overcooking worsened by autocorrelation. As above but with at least some

4.Model misspecification by virtue [sic] of step methods applied to trending

5.Non-determinism. Red noise/unit root processes masquerading as natural variability and/or as one off deterministic events. Non-determinism implies that detected events cannot be attributed to a deterministic physical model.

6.Presence of one or more sub-detection threshold deterministic events. This is a particularly nasty issue because (a) it affects detection of many phenomena, (b) it may deceive autocorrelation tests and unit-root tests as well as trend

Not all of these concern statistical M-S. Objection 1, physically implausibility of discontinuities in surface temperatures [37] can only result from an underlying

natural components following an autocorrelation model.

7.Conflict with objectors favoured model/approach.

*without* the *β*1*xi* term.

**4.2 Application to climate data**

incorporate it as fact [35, 36].

called hiatus) appear to be

data.

tests.

**215**

the derivative of the sum.

*4.2.1 Abrupt changes in previous literature*

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

In MS2004 the authors use an example of a linear regression model to address a problem of validation in regression models. Three general forms of M-S are recognized:

