A Random Variable (random quantity or stochastic variable) is a set of possible values from a random experiment. The domain of a random variable is called sample space. For example, in the case of a coin toss experiment, there are only two possible outcomes, namely heads or tails. A random variable can be either discrete or continuous. The discrete random variable takes only certain values such as 1, 2, 3, etc., and a continuous random variable can take any value within a range such as the height of persons.
By using random variables, one can use the tools of probability and statistics to analyze the outcomes of the experiment. One can calculate such as the probability of getting a certain result, the average outcome, or how spread out the results are.
The Wilcoxon Signed Rank test assumes that the population of interest is both continuous and symmetric (not necessarily normal). Since the mean and median are the same (for symmetrical distribution), the hypothesis tests on the median are the same as the hypothesis test on the mean.
The Wilcoxon test is performed by ranking the non-zero deviations in order of increasing magnitude (that is, the smallest non-zero deviation has a rank of 1 and the largest deviation has a rank of $n$). The ranks of the deviations with positive and negative values are summed.
These sums are used to determine whether or not the deviations are significantly different from zero. Wilcoxon Signed Rank Test is an alternative to the Paired Sample t-test.
Test Statistics: $T^-$: an absolute value of the sum of the negative ranks
Two-tailed Test
$H_0: \mu = \mu_0 \quad$ vs $\quad H_1:\mu \ne \mu_0$
Test Statistics: $min(T^+, T^-)$
Because the underlying population is assumed to be continuous, ties are theoretically impossible, however, in practice ties can exist, especially if the data has only a couple of significant digits.
Two or more deviations having the same magnitude are all given the same average rank. The deviations of zero are theoretically impossible but practically possible. Any deviations of exactly zero are simply thrown out and the value of $n$ is reduced accordingly.
Single Sample Wilcoxon Signed Rank Test
The Wilcoxon Signed Rank Test is important for researchers as it fills a critical gap in statistical analysis.
Non-normal data: Most of the statistical tests, like the dependent samples t-test, assume that the data follows a normal distribution (bell curve). The Wilcoxon Signed Rank Test supersede the assumption of normality, making it ideal for analyzing data that is skewed, ranked, or ordinal (like survey responses on a Likert scale Questions).
Robust against outliers: Outliers (very large or small observations in the data) can significantly skew the results of some statistical tests. The Wilcoxon Signed Rank Test focuses on the ranks of the differences, making it less sensitive to extreme values (outliers) in the data compared to tests that rely on raw numbers.
Focuses on changes within subjects: The Wilcoxon Signed Rank Test is designed for paired data (dependent samples), to look at the same subjects before and after situation (like a treatment) or under two different conditions.
Nonparametric tests are experiments that do not require the underlying population for assumptions. It does not rely on data referring to any particular parametric group of probability distributions. Nonparametric methods are also called distribution-free tests since they do not have any underlying population.
Nonparametric Tests/ Statistics are Helpful when
Inferences must be made on categorical or ordinal data
The assumption of normality is not appropriate
The sample size is small
Advantages of NonParametric Methods
Easy application (does not even need a calculator in many cases)
It can serve as a quick check to determine whether or not further analysis is required
Many assumptions concerning the population of the data source can be relaxed
Can be used to test categorical (yes/ no) data
Can be used to test ordinal (1, 2, 3) data
Disadvantages of NonParametric Methods
Nonparametric procedures are less efficient than parametric procedures. It means that nonparametric tests require a larger sample size to have the same probability of a type-I error as the equivalent parametric procedure.
Nonparametric procedures often discard helpful information. That is, the magnitudes of the actual data values are lost. As a result, nonparametric procedures are typically less powerful.
That is they produce conclusions that have a higher probability of being incorrect. Examples of widely used Parametric Tests: include the paired and unpaired t-test, Pearson’s product-moment correlation, Analysis of Variance (ANOVA), and multiple regression.
Note: Do not use nonparametric procedures if parametric procedures can be used.
Some widely used Non-Parametric Tests are:
Sign Test
Runs Test
Wilcoxon Signed Rank Test
Wilcoxon Rank Sum Test
Spearman’s Rank Correlation
Kruskal Wallis Test
Chi-Square Goodness of Fit Test
Nonparametric tests are crucial tools in statistics because they offer valuable analysis even when the data doesn’t meet the strict assumptions of parametric tests. NonParametric tests provide a valuable alternative for researchers who encounter data that doesn’t fit the mold of parametric tests. They ensure that valuable insights can still be extracted from the data without compromising the reliability of the analysis.
However, it is important to remember that nonparametric tests can sometimes be less powerful than the related parametric tests. This means non-parametric tests might be less likely to detect a true effect, especially with smaller datasets.
In summary, nonparametric tests are valuable because these kinds of tests offer flexibility in terms of data assumptions and data types. They are particularly useful for small samples, skewed data, and situations where data normality is uncertain. These tests also ensure researchers draw statistically sound conclusions from a wider range of data types and situations. But, it is always a good practice to consider both parametric and non-parametric approaches when appropriate.
