Following are the non-parametric tests used for analysis of different types of data.
The Chi-square test is a non-parametric test of proportions. This test is not based on any assumption or distribution of any variable. This test, though different, follows a specific distribution known as Chi-square distribution, which is very useful in research. It is most commonly used when data are in frequencies such as number of responses in two or more categories. This test involves the calculations of a quantity called Chi-square (x2) from Greek letter ‘Chi’(x) and pronounced as ‘Kye.’ It was developed by Karl Pearson.
a) Test of proportion: This test is used to find the significance of difference in two or more than two proportions.
b) Test of association: The test of association between two events in binomial or multinomial samples is the most important application of the test in statistical methods. Itmeasures the probabilities of association between two discrete attributes. Two events can often be studied for their association such as smoking and cancer, treatment and outcome of disease, level of cholesterol and coronary heart disease. In these cases, there are two possibilities, either they influence or affect each other or they do not. In other words, you can say that they are dependent or independent of each other. Thus, the test measures the probability (P) or relative frequency of association due to chance and also if two events are associated or dependent on each other. Varieties used are generally dichotomous e.g. improved / not improved. If data are not in that format, investigator can transform data into dichotomous data by specifying above and below limit. Multinomial sample is also useful to find out association between two discrete attributes. For example, to test the association between numbers of cigarettes equal to 10, 11-20, 21-30, and more than 30 smoked per day and the incidence of lung cancer. Since, the table presents joint occurrence of two sets of events, the treatment and outcome of disease, it is called contingency table (Con- together, tangle- to touch).
When there are only two samples, each divided into two classes, it is called as four cell or 2 × 2 contingency table. In contingency table, we need to enter the actual number of subjects in each category. We cannot enter fractions or percentage or mean. Most contingency tables have two rows (two groups) and two columns (two possible outcomes). The top row usually represents exposure to a risk factor or treatment, and bottom row is mainly for control. The outcome is entered as column on the right side with the positive outcome as the first column and the negative outcome as the second column. A particular subject or patient can be only in one column but not in both. The following table explains it in more detail: Even if sample size is small (< 30), this test is used by using Yates correction, but frequency in each cell should not be less than 5. Though, Chi-square test tells an association between two events or characters, it does not measure the strength of association. This is the
limitation of this test. It only indicates the probability (P) of occurrence of association by chance. Yate's correction is not applicable to tables larger than 2 X 2. When total number of items in 2 X 2 table is less than 40 or number in any cell is less than 5, Fischer's test is more reliable than the Chi-square test.
This is a non-parametric test. This test is used when data are not normally distributed in a paired design. It is also called Wilcoxon-Matched Pair test. It analyses only the difference between the paired measurements for each subject. If P value is small, we can reject the idea that the difference is coincidence and conclude that the populations have different medians.
It is a Student’s‘t’ test performed on ranks. For large numbers, it is almost as sensitive as Student’s‘t’ test. For small numbers with unknown distribution, this test is more sensitive than Student’s‘t’ test. This test is generally used when two unpaired groups are to be compared and the scale is ordinal (i.e. ranks and scores), which are not normally distributed.
This is a non-parametric test, which compares three or more paired groups. In this, we have to rank the values in each row from low to high. The goal of using a matched test is to control experimental variability between subjects, thus increasing the power of the test.
It is a non-parametric test, which compares three or more unpaired groups. Non-parametric tests are less powerful than parametric tests. Generally, P values tend to be higher, making it harder to detect real differences. Therefore, first of all, try to transform the data. Sometimes, simple transformation will convert non-Gaussian data to a Gaussian distribution. Non-parametric test is considered only if outcome variable is in rank or scale with only a few categories. In this case, population is far from Gaussian or one or few values are off scale, too high, or too low to measure.