Wilcoxon Signed Rank Test Made Easy

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.

One-Tailed Test

$H_0: \mu = \mu_0\quad $ vs $\quad H_1: \mu < \mu_0$

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: $min(T^+, T^-)$

Wilcoxon Signed Ranked Test

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

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.

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NonParametric Tests: Introduction Easy Version (2023)

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.

nonparametric-tests

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.

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Important MCQs on Chi-Square Test Quiz – 3

The post is about Online MCQs on Chi-Square Test Quiz with Answers. The Quiz MCQs on Chi-Square Test cover the topics of attributes, Chi-Square Distribution, Coefficient of Association, Contingency Table, and Hypothesis Testing on Association between attributes, etc. Let us start with MCQs on Chi-Square Test Quiz.

The quiz about Chi-Square Association between attributes.

1. If for a contingency table, $df=12$ and the number of rows is 4 then the number of columns will be

 
 
 
 

2. If $\chi^2_c=5.8$ and $df=1$, we make the following decision ———-.

 
 
 
 

3. Association measures the strength of the relationship between

 
 
 
 

4. The presence of an attribute is denoted by

 
 
 
 

5. The range of $\chi^2$ is

 
 
 
 

6. If $(AB) < \frac{(A)(B)}{n}$ then association between two attributes $A$ and $B$ is

 
 
 
 

7. A $4 \times 5$ contingency table consists of ———.

 
 
 
 

8. The value of $\chi^2$ cannot be ———.

 
 
 
 

9. Two attributes $A$ and $B$ are said to be positively associated if

 
 
 
 

10. Two attributes $A$ and $B$ are said to be independent if

 
 
 
 

11. For the $3\times 3$ contingency table, the degrees of freedom is

 
 
 
 

12. A characteristic which varies in quality from one individual to another is called

 
 
 
 

13. The process of dividing the objects into two mutually exclusive classes is called

 
 
 
 

14. A contingency table with $r$ rows and $c$ columns is called

 
 
 
 

15. If $(AB) > \frac{(A)(B)}{n}$ then association is

 
 
 
 

16. There are ———– parameters of Chi-Square distribution.

 
 
 
 

17. The eye colour of 100 men is

 
 
 
 

18. The coefficient of association $Q$ lies between

 
 
 
 

19. For $r\times c$ contingency table, the Chi-Square test has $df=$ ———-.

 
 
 
 

20. The parameter of the Chi-Square distribution is ———–.

 
 
 
 

The relationship/ dependency between the attributes is called association and the measure of degrees of relationship between the attributes is called the coefficient of association. The Chi-Square Statistic is used to test the association between the attributes. The Chi-Square Association is defined as

$$\chi^2 = \sum \frac{(of_i – ef_i)^2}{ef_i}\sim \chi^2_{v},$$

where $v$ denotes the degrees of freedom

MCQs on Chi-Square Test quiz

A population can be divided into two or more mutually exclusive and exhaustive classes according to their characteristics. It is called dichotomy or twofold division if, it is divided into two mutually exclusive classes. A contingency table is a two-way table in which the data is classified according to two attributes, each having two or more levels. A measure of the degree of association between attributes expressed in a contingency table is known as the coefficient of contingency. Pearson’s mean square coefficient of contingency is

\[C=\sqrt{\frac{\chi^2}{n+\chi^2}}\]

MCQs on Chi-Square Test Quiz with Answers

  • A characteristic which varies in quality from one individual to another is called
  • The eye colour of 100 men is
  • Association measures the strength of the relationship between
  • The presence of an attribute is denoted by
  • The process of dividing the objects into two mutually exclusive classes is called
  • There are ———– parameters of Chi-Square distribution.
  • The parameter of the Chi-Square distribution is ———–.
  • The value of $\chi^2$ cannot be ———.
  • The range of $\chi^2$ is
  • Two attributes $A$ and $B$ are said to be independent if
  • Two attributes $A$ and $B$ are said to be positively associated if
  • If $(AB) > \frac{(A)(B)}{n}$ then association is
  • If $(AB) < \frac{(A)(B)}{n}$ then association between two attributes $A$ and $B$ is
  • The coefficient of association $Q$ lies between
  • If $\chi^2_c=5.8$ and $df=1$, we make the following decision ———-.
  • A contingency table with $r$ rows and $c$ columns is called
  • A $4 \times 5$ contingency table consists of ———.
  • If for a contingency table, $df=12$ and the number of rows is 4 then the number of columns will be
  • For $r\times c$ contingency table, the Chi-Square test has $df=$ ———-.
  • For the $3\times 3$ contingency table, the degrees of freedom is

Attributes are said to be independent if there is no association between them. Independence means the presence or absence of one attribute does not affect the other. The association is positive if the observed frequency of attributes is greater than the expected frequency and negative association or disassociation (negative association) is if the observed frequency is less than the expected frequency.

Important MCQs on Chi-Square Test Quiz - 3

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