1 Non-parametric tests

Sometimes it is possible to measure a quantity and express the measurements numerically in such a way that meaningful arithmetic can be done. For example, if you measure three spacers and determine that they are 1 mm 2 mm and 3 mm spacers you can certainly assert that $1+2=3$ in the sense that the combination of the 1 mm and 2 mm spacers are effectively the same as the 3 mm spacer. There are occasions when data may be expressed numerically but doing arithmetic leads to nonsensical conclusions. Suppose, for example that as a manager, you are asked to assess the work of three colleagues, John, Tony and George. You might come to the conclusion that overall George is the “best” worker, followed in order by John and the Tony. You may present the results as follows:

In this case, if you assert that $1+2=3$ you may be interpreted as saying that the combined work of George and John is equivalent to the work of Tony. This, of course, is in complete contradiction to the way you have rated the work of your colleagues! Remember that the appearance of numbers does not imply that you can do meaningful arithmetic. In fact, meaningless arithmetic, while giving a piece of work the appearance of careful analysis can (and almost certainly will) be totally misleading in any conclusions reached. In other statistical problems, the variable measured may allow meaningful arithmetic but we might not feel able to assume that it follows a probability distribution of any particular type. In particular, we might not be willing to assume that it has a normal distribution. In cases such as these we use tests which do not depend on the assumption of a particular distribution, unlike $t$ -tests, $F$ -tests etc., where a normal distribution is assumed. Tests which do not require such distributional assumptions are called non-parametric tests.

Very often, the non-parametric procedure described in this Workbook may be thought of as direct competitors of the $t$ -test and $F$ -test when normality can be assumed and we will compare the performance of parametric and non-parametric methods under conditions of normality and non-normality. In general terms, you will find the non-parametric methods fail to use all of the information that is available in a sample and as a consequence they may be though to as less efficient than parametric methods. Essentially, you should remember that in cases where it is difficult or impossible to justify normality but it is known that the underlying distribution is continuous, non-parametric methods remain valid while parametric methods may not. You should also bear in mind that in terms of practical application it may be difficult to decide whether to use parametric or non-parametric tests since both the $t$ -test (and the $F$ -test) are relatively insensitive to small departures from normality.

Our work concerning non-parametric tests begins with the sign test.