Non-parametric tests

The range of values that lie between the first and third quartiles and, therefore, represent 50°/o of the data points. Used in non-parametric tests. 

The data obtained was analyzed using SPSS-9.0 software. Any differences with a significance level p<0.05 were considered valid. To assess the validity of the differences and correlation, non-parametric tests were used as indicated vide infra). The data are displayed as mean SE. 

Non-parametric tests, as seen in the two procedures outlined earlier in Section 11.5, are based on some form of ranking of the data. Once the data are ranked then the test is based entirely on those ranks the original data play no further part. It is therefore the behaviour of ranks that determines the properties of these tests and it is this element that gives them their robustness. Whatever the original data looks like, once the rank transformation is performed then the data become well-behaved. 

In terms of summary statistics, means are less relevant because of the inevitable skewness of the original data (otherwise we would not be using non-parametric tests). This skewness frequently produces extremes, which then tend to dominate the calculation of the mean. Medians are usually a better, more stable, description of the average . 

Extending non-parametric tests to more complex settings, such as regression, ANOVA and ANCOVA is not straightforward and this is one aspect of these methods that limits their usefulness. 

Non-parametric methods reduce power. Therefore if the data are normally distributed, either on the original scale or following a transformation, the non-parametric test will be less able to detect differences should they exist. 

There are two main families of statistical tests parametric tests, which are based on the hypothesis that data are distributed according to a normal curve (on which the values in Student s table are based), and non-parametric tests, for more liberally distributed data (robust statistics). In analytical chemistry, large sets of data are often not available. Therefore, statistical tests must be applied with judgement and must not be abused. In chemistry, acceptable margins of precision are 10, 5 or 1%. Greater values than this can only be endorsed depending on the problem concerned. 

The Wilcoxon s Rank-Sum Test (WRST) is a non-parametric alternative. The WRST is robust to the normal distribution assumption, but not to the assumption of equal variance. Furthermore, this test requires that the two groups of data under comparison have similarly shaped distributions. Non-parametric tests typically suffer from having less statistical power than their parametric counterparts. Similar to the /-test, the WRST will exhibit false positive rate inflation across a microarray dataset. It is possible to use the Wilcoxon test statistic as the single filtering mechanism however calculation of the false positive rate is challenging (48). 

The two-sample f-test is robust, i.e. it will still function reasonably well with samples that are somewhat non-normally distributed. However, where data is severely nonnormal, even this test will start to produce inappropriate conclusions. In that case, either transform the data to normality or use a non-parametric test. 

Ordinal and non-normally distributed data. Transformations and non-parametric tests... 

Discuss the appropriate wording of conclusions where a non-parametric test has proved significant... 

Instead of transforming the data to normality, we could employ one of a range of procedures referred to as non-parametric tests . These partially duplicate the functionality of tests we have already met, but use a method of calculation that does not depend upon a normal distribution. The non-parametric test that is generally substituted for the two sample f-test goes by a variety of names, but is most commonly called the Mann-Whitney test. 

In non-parametric tests the data are transformed into rank values and then all further calculations are based solely upon these rankings. 

When a /-test produces a significant outcome, its interpretation is quite straightforward- there is evidence that the two population mean values differ. Unfortunately, with non-parametric tests such as Mann-Whitney, it is not so simple. The conclusion... 

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. 

A good general rule is to use a parametric test whenever possible even if that necessitates a transformation of the data. However, some data sets are such a mess that no amount of jiggery-pokery will render them normal and in these cases a non-parametric test is our ultimate fall-back. 

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. 

An alternative approach is to use a non-parametric test that does the same job as one of the parametric tests. These tests convert the data to rankings and the distribution of the data (largely) ceases to be an issue. They are sometimes called distribution-free tests . 

Where data are reasonably normally distributed, non-parametric methods are a little less powerful than their parametric equivalents, but where the data are severely nonnormal, the non-parametric test may be much more powerful. With non-normally distributed interval scale data, the best solution is transformation to normality, but failing that, non-parametric methods can be used with only modest loss of power. 

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