Non-parametric and robust methods

Finally we note that, just as in the treatment of outliers in replicate measurements, non-parametric and robust methods can be very effective in handling outliers in regression robust regression methods have proved particularly popular in recent years. These topics are covered in the next chapter. 

The outlier tests described above assume that the sample comes from a normal population. It is important to realize that a result that seems to be an outlier on the assumption of a normal population distribution may well not be an outlier if the sample actually comes from (for example) a log-normal distribution (Section 2.3). Therefore outlier tests should not be used if there is a suspicion that the population may not have a normal distribution. This difficulty, along with the extra complications arising in cases of multiple outliers, explains the increasing use of the non-parametric and robust statistical methods described in Chapter 6. Such methods are either insensitive to extreme values, or at least give them less weight in calculations, so the problem of whether or not to reject outliers is avoided. 

Overall a great variety of significance tests - parametric, non-parametric, and robust - are available, and often the most difficult task in practice is to decide which method is best suited to a particular problem. The diagram in Appendix 1 is designed to make such choices easier, though inevitably it cannot cover all possible practical situations. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

The problems caused by possible outliers in regression calculations have been outlined in Sections 5.13 and 6.9, where rejection using a specified criterion and non-parametric approaches respectively have been discussed. It is clear that robust approaches will be of value in regression statistics as well as in the statistics of repeated measurements, and there has indeed been a rapid growth of interest in robust regression methods amongst analytical scientists. A summary of two of the many approaches developed must suffice. 

In section Structural Parametric Identification by Extended Kalman Filter, online structural parametric identification using the EKF will be briefly reviewed. In section Online Identification of Noise Parameters, an online identification algorithm for the noise parameters in the EKF is introduced. Then, in section Outlier-Resistant Extended Kalman Filter, an online outlier detection algorithm is presented, and it is embedded into the EKF. This algorithm allows for robust structural identification in the presence of possible outliers. In section Online Bayesian Model Class Selection, a recursive Bayesian model class section method is presented for non-parametric identification problems. 

In Chapter 5 we saw how the calculation of the 95 per cent Cl for the mean can lead to nonsensical results if the data deviate severely from a normal distribution. This requirement for a normal distribution also applies to the t- tests, analyses of variance and correlation that we met in Chapters 6-14. These procedures are termed parametric methods and are quite robust, so moderate non-normality does little damage, but in more extreme cases, some pretty dumb conclusions can emerge. This chapter looks at steps that can be taken to allow the analysis of seriously non-normal data and also of ordinal scale data. 

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