Numerous statistical tests have been recently developed that do not require rigorous assumption about the population distribution nor do they require to have hypothesis stated in terms of specified parameters values. These tests are valid over a wide range of distributions of the parent population. Such tests are called non-parametric or distribution free because these do not depend upon the population parameters such as mean and variance. Distribution free because these tests do not depend upon the shape of the distribution. Therefore, chooses an appropriate analytical technique. The classifications of these tests are based upon the following.
A question that arises mostly is “Are the results statistically significant”? Or to determine whether there is significant difference between two groups of data. To answer these, one uses some kind of statistical test of significance. Quite often, we wasted to determine whether there is any association between two or more variables and it so, to investigate the strength of association and functional form of relationships. For instance, a training programme is developed for the sales force, we wanted to test the effectiveness of the training programme.
There are two scales used in measurement i.e., metric and non-metric. The metric scales measure the facts in number. This fact is an example of ratio scale having equal intervals and a true zero points. When data is qualitative and not measured with numbers, except when some synthetic numbering system is applied are called non-metric.
The number of variables to be analyzed together may range from one to several. On the basis of this the analytical techniques can be divided into.
Dependent, when the variables are associated with each other and independent when they are not associated with each other. Some non-parametric test namely sign test median test and rank correlation test.
Sign test can be applied in univariate and bivariate data. The one sample sign test is applicable, when sample is taken from a continuous symmetrical population. In this case, the probabilities that the sample value is greater than mean are both ½. The procedure adopted in sign test is the simplest one.
If two samples are compared to determine whether there is a significant difference between two population means, the signs of difference between paired observations from the two samples can be used for hypothesis testing. In order to apply the test the population from which random samples are taken must be normally distributed. For testing the difference between the two means of two populations, we assume that the population variances are identical. In many situations, however, either or both of these two assumptions cannot be applied. Under these circumstances, non-parametric tests are generally applied. The sign test, as discussed in one sample case, is based on signs, plus or minus of the difference between the paired observations, not taking into consideration the magnitude of the differences.
This is a procedure of testing whether two or more independent samples are taken from populations with the same median. The paired sign test discussed earlier cannot be applied when two samples are drawn from different populations. The paired sign test is applied mostly in before and after treatment and number of observations are in pairs. Whereas in median test, the samples selected from two populations may differ. In such situations median test is applied. The median test helps in testing whether two independent samples belong to the population with same median or not. The same procedure can be applied to more than two samples. The null hypothesis to be tested is that two more samples are taken from the population with same median. The test may be either one tail or two tailed.
When the two variables had a joint normal distribution and that the variance of one variable given the other was same. In situations where there assumptions are doubtful, we may use other technique which is known as rank correlation method.
Rank correlation was developed by Carl Spearman in 190. It is a non-parametric method, is carried out by ranking the values of each variable. Since the relationship between variables is analyzed on the basis of ranks, the method can be very effective when exact measurements are not available.
Situations arise in practice in which such assumptions may not be justified or in which there is doubt that they apply, as in the case where a population may be highly skewed. Because of this, statisticians have devised various tests and methods that are independent of population distribution and associated parametric tests. These are called non parametric tests.
Non-parametric tests can be used as shortcut replacements for more complicated tests. They are especially valuable in dealing with non numerical data, such as arise when consumers rank cereals or other products in order of preference. We shall concentrate on just two tests here.
and these tests depends on the ranks of the sample observations. We shall use the Mann-Whitney test only where two population are involved. Use of these tests will enable us to determine whether independent samples have been draw from the same population (or from different population having the same distribution).
The Mann-Whitney U Test employs the actual ranks of the various observations as a device for testing hypotheses about the identity of two population distributions. We assume that the underlying variable on which two samples are to be compared is continuously distributed. The null hypothesis to be tested is that the two population distributions are identical.
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