The sign test and Wilcoxon signed rank test

In this blog, what I want to talk about is the sign test and Wilcoxon signed rank test.

 

1. The sign test : It translates the direction of the difference between two observations, not necessarily a numerical difference, into a variable with two categories and subsequently applied a one-sample test for a proportion. It's called the sign-test. This test offers a nonparametric alternative to a one sample t test. And it's one of the simplest nonparametric tests, but also a versatile one. It's particularly useful for situations where quantitative measurement is impossible but where ranking the pair of observations is feasible.

 1-1) hypotheses : The null hypothesis tested by the sign test is that the probability that X is larger than Y is equal to a given proportion P. Here X could be the judgement or score under one condition, and Y the judgement or score under the other condition. Usually a valuoe of 0.5 is chosen for the proportion P, but it canbe any value between zero and one. Finally, you could also replace Y with a constant so that you would measure the proportion where X exceeds a constant.

 ** In any of these cases, you would use the cumulative binomial distribution to find the probability of finding your results if your null hypothesis were true. If this probability is small, you would reject your null hypothesis.

 ** If the number of cases is larger than 35, the binomial distribution can be approximated by a normal distribution, so then the z-test can be used instead.

 ** The beauty of the sign-test is that you don't have to make any assumptions about the measurment level of your data. Furthermore, there are no requirements on the sample size, and the power of this test is relatively high for small samples, say fewer than 20 cases. At the down-side, the power decreases rapidly for bigger sample sizes.

 

2. One sample - Wilcoxon signed rank test : This is seen as a nonparametric equivalent to the one sample t-test. While the t-test compares the sample mean to theoretical value, the signed-ranks test compares the median to thoeretical value. This makes fewer assumptions than the t-test, aprat from the usual assumptions of independence among samples, it only assumes that the population distribution is symmetric.