A non-normal statistical test

The Sign test belongs to the confirmatory types of statistical tests. Most statistical tests require data to follow the normal distribution. Many times, it is possible to transform data so that it follows the normal distribution; however sometimes it is not possible, or sometimes the statistical sample size is so small that it becomes difficult to identify if the data is normally distributed or not. In such scenarios, it becomes necessary to use a form of statistical testing, where it is not necessary for the data to be normally distributed. |

Statistical tests that do not require the normal distribution of the sample data are known as non-parametric tests. This is an example of one such test.

The Sign Test is a non-parametric statistical test whereby very few assumptions are taken about the nature of the distribution being tested. This statistics hypothesis test works on the theory that there are equal probabilities of two outcomes. Instead of magnitude, this non-parametric test works on the direction of + and - sign of the observation. Since, there are just two choices i.e. + or -, so the NULL proportion is taken as p=0.5.

This test is used for the testing of the null hypothesis as well as in the situation where you want to find out if two groups are sized equally or not. It is also used in the case of sample t-test analysis as well as paired t test statistics.

There are two types:

- One Sample Sign Test: In this test, the hypothesis is setup in a way that + and - signs are the values of the random variables that have equal size.
- Paired Sample Sign Test: It is also known as an alternative to the paired sample statistical t test. In this case, the + and - signs are used in the paired sample test or in the before/after study. The NULL hypothesis is setup in a way that either of the + and - signs are of equal size or the Population Mean values are equal to the Sample Mean value.

- First of all, we get the random sample of data pairs (d1, d2). Below data pertains to the efficacy of two drugs. Suppose, we want to look out for the chances if any of the drugs works better than the other drug or not.
- Calculate the differences (d2-d1) and record the + and - signs. If you have only one data pair then use the + sign for the values that are greater than the Mean value and the - sign for the values smaller than the Mean value. Use 0 for the values that are equal to the mean value. The statistical sample size should be large enough so that the total number of + and - signs is at least 12.
- Now compute the hypothesis H0: p=0.5, H1: p>0.5, p<0 or p?0.5.
- After that find the P-value for both H0 and H1.
- For testing the null hypothesis H0, find the proportion of the + signs.
- For H0 (p=0.5), assume that the Population Proportion of the + signs p = 0.5. The z-value corresponding to the Sample Proportion of the + signs x will be
- Find the P-value by using the standard normal distribution table.
- Repeat steps 5-7 for finding the P-value of the alternate hypothesis H1. The p-value will be 0.00939.
- Finally, conclude the test and interpret the results for it. If the P-value of H1 is less than the significance level ?=0.05 then reject the null hypothesis.

The Sign Test can be applied to both, the single observations or the pairs of related observations. However, the most useful application of this non-paremetric test is in the before-and-after scenarios where the effects of a certain phenomenon on the same population are evaluated before and after the phenomenon is administered. |

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