Other non-parametric methods

Both distributions are strongly skewed, but as one shows positive and the other negative skew, there is no way we are going to be able to transform them both to normality. 

With an unpaired design and measurements on an interval scale, we would have used a two-sample t-test to check for any difference. However, this data are ordinal and not remotely normally distributed, so we will have to move to its non-parametric equivalent - the Mann-Whitney test. 

The result of a Man-Whitney test on this data is an uncorrected P value of 0.0008. There are many tied values in this set of results and adjusting for ties does now slightly reduce P to 0.0006. Either way the result is strongly significant. 

As explained previously, the significant result can certainly be interpreted as evidence that the active preparation generally had a greater effect than the placebo and a more detailed claim that the active preparation had a greater median effect is pretty uncontroversial, even with these rather skew distributions. 

There are huge numbers of other non-parametric tests available. Three are especially useful as substitutes for parametric tests that have already been covered in this book. 

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It is fair to say that statisticians tend to disagree somewhat regarding the value of non-parametric methods. Some statisticians view them very favourably while others are reluctant to use them unless there is no other alternative. 

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. 

When non-parametric methods are applied to data that is normally distributed, they are slightly less powerful than their parametric equivalents, although the difference is not great. For the tests covered in this chapter, the non-parametric test has about 95 per cent of the power of its parametric equivalent. In other words, if our data is normally distributed then a sample of 19 tested by a parametric method would provide about the same power as a sample of 20 tested by the non-parametric equivalent. Since the power of the two types of test is so similar, it is not surprising that the P values generated [0.034 for the f-test (when applied to the transformed data) and 0.036 for the Mann-Whitney test] are barely different. 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

What is the significance of these different scales of measurement As was mentioned in Section 1.5, many of the well-known statistical methods are parametric, that is, they rely on assumptions concerning the distribution of the data. The computation of parametric tests involves arithmetic manipulation such as addition, multiplication, and division, and this should only be carried out on data measured on interval or ratio scales. When these procedures are used on data measured on other scales they introduce distortions into the data and thus cast doubt on any conclusions which may be drawn from the tests. Non-parametric or distribution-free methods, on the other hand, concentrate on an order or ranking of data and thus can be used with ordinal data. Some of the non-parametric techniques are also designed to operate with classified (nominal) data. Since interval and ratio scales of measurement have all the properties of ordinal scales it is possible to use non-parametric methods for data measured on these scales. Thus, the distribution-free techniques are the safest to use since they can be applied to most types of data. If, however, the data does conform to the distributional assumptions of the parametric techniques, these methods may well extract more information from the data. 

Repairable systems are those systems that can be restored to fully satisfactory performance by a method other than replacement of the entire system (Ascher and Feingold, 1984). Reliability analysis of repairable units can be classified into parametric and non-parametric methods. 

Efron, B. (1981) Non parametric estimates of standard error the jack-knife, the bootstrap and other methods, Biometrika 68, 589-599. 

Other methods alternative first-order, NPML, non-parametric Rich data, sparse data, or a mixture of both Properly accounts for data imbalance. Relatively good estimation Not widely used... 

Here, we discuss a simple and informative plot for analyzing recurrent repair data. The plotting method provides a non-parametric graphical estimate of the number/cost of repairs per unit versus age. This estimate can be used to evaluate the repair rate of the population with respect to age and predict future number/cost of repairs. The MCF plot reveals most of the information that we are searching for in recurrent data (Nelson, 2003). A key advantage of the MCF plot is that it provides a single summary plot for a group of units, which is easily compared with other MCF plots of different populations. The MCF plot is the population of staircase functions for all the studied units. This staircase is a consequence of... 

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