11.1 Wilcoxon rank-sum and signed-rank tests
4 min read•july 30, 2024
Wilcoxon rank-sum and signed-rank tests are powerful nonparametric tools for comparing two samples without assuming normal distribution. They're perfect for analyzing biological data that doesn't fit typical parametric assumptions, like gene expression or species diversity.
These tests use ranking instead of actual values, making them robust against outliers and non-normal data. The rank-sum test compares independent samples, while the signed-rank test handles paired data. Both help biologists uncover significant differences in their research.
Nonparametric Tests for Two Samples
Principles and Assumptions
Top images from around the web for Principles and Assumptions
hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original
Is this image relevant?
Wilcoxon Signed Rank Test reports "U" instead of "Z" in SPSS - Cross Validated View original
Is this image relevant?
hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original
Is this image relevant?
Wilcoxon Signed Rank Test reports "U" instead of "Z" in SPSS - Cross Validated View original
Is this image relevant?
1 of 2
Top images from around the web for Principles and Assumptions
hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original
Is this image relevant?
Wilcoxon Signed Rank Test reports "U" instead of "Z" in SPSS - Cross Validated View original
Is this image relevant?
hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original
Is this image relevant?
Wilcoxon Signed Rank Test reports "U" instead of "Z" in SPSS - Cross Validated View original
Is this image relevant?
1 of 2
- Nonparametric tests are statistical methods that do not require assumptions about the underlying distribution of the data
- Suitable for analyzing data that violate the assumptions of parametric tests (normality, homogeneity of variance)
- The Wilcoxon rank-sum and signed-rank tests are nonparametric alternatives to the two-sample t-test and paired t-test, respectively
- The assumes that the two independent samples are randomly drawn from populations with the same shape and variability
- The assumes that the paired differences are symmetric about the
Ranking-Based Approach
- These tests are based on ranking the observations rather than using the actual values
- Less sensitive to outliers
- More robust to departures from normality
- Ranking involves assigning a rank to each observation based on its relative position in the combined dataset
- Smallest observation receives rank 1, second smallest rank 2, and so on
- Ties are assigned the average rank of the tied positions
Wilcoxon Rank-Sum Test
Test Procedure
- The Wilcoxon rank-sum test, also known as the Mann-Whitney U test, compares two independent samples
- Determines whether the samples come from the same population or if one sample tends to have larger values than the other
- To perform the test:
- Combine the observations from both samples
- Rank the combined observations from smallest to largest, assigning the average rank in case of ties
- Calculate the sum of the ranks for each sample
- Determine the U statistic for each sample by subtracting the sum of the ranks from the product of the sample size and the sum of the first n integers, where n is the sample size
- The test statistic is the smaller of the two U values
Interpretation and Significance
- The test statistic can be compared to a critical value or used to calculate a
- Determines the significance of the difference between the two samples
- A small p-value (typically < 0.05) indicates a significant difference between the two samples
- Suggests that the factor of interest has an effect on the measured variable
- Example: Comparing the weights of male and female mice in a study
- A significant result would suggest that gender has an effect on mouse weight
Wilcoxon Signed-Rank Test
Test Procedure
- The Wilcoxon signed-rank test compares two paired samples or tests whether the median difference between paired observations is zero
- To perform the test:
- Calculate the differences between each pair of observations
- Rank the absolute values of the differences, excluding any differences equal to zero
- Assign the average rank in case of ties and attach the sign of the original difference to each rank
- Calculate the sum of the positive ranks (W+) and the sum of the negative ranks (W-)
- The test statistic is the smaller of the two sums (W+ or W-)
Interpretation and Significance
- Compare the test statistic to a critical value or calculate a p-value to determine the significance of the difference between the paired samples
- A small p-value (typically < 0.05) indicates a significant difference between the paired observations
- Suggests that the treatment or intervention has a significant effect on the measured variable
- Example: Comparing the blood pressure of patients before and after taking a new medication
- A significant result would suggest that the medication has an effect on blood pressure
Interpreting Wilcoxon Tests in Biology
Biological Context
- A significant result in the Wilcoxon rank-sum test indicates that the two independent samples come from populations with different median values
- Suggests that the factor of interest (treatment, condition, etc.) has an effect on the measured variable
- A significant result in the Wilcoxon signed-rank test indicates that the median difference between the paired observations is significantly different from zero
- Suggests that the treatment or intervention has a significant effect on the measured variable
- Consider the biological context, the magnitude of the difference between the samples or paired observations, and the potential limitations of the study design or data collection methods
P-Values and Limitations
- The p-value represents the probability of observing a difference as extreme as or more extreme than the one observed, assuming the is true
- Nonparametric tests may have lower power than their parametric counterparts when the assumptions of the parametric tests are met
- Consider the trade-off between robustness and power when choosing between parametric and nonparametric methods
- Example: Comparing the expression levels of a gene in two different tissues
- A significant result would suggest that the gene is differentially expressed between the tissues, but the biological relevance of the difference should be considered
Key Terms to Review (18)
Alternative hypothesis: The alternative hypothesis is a statement that suggests there is an effect or a difference when conducting a statistical test, opposing the null hypothesis which posits no effect or difference. It serves as the research hypothesis that researchers aim to support, highlighting potential outcomes of an experiment or study.
Clinical trials: Clinical trials are systematic studies designed to evaluate the safety, efficacy, and effectiveness of medical interventions, such as drugs, devices, or treatment protocols, on human participants. These trials are crucial for determining whether new treatments work and should be approved for general use, as they provide rigorous evidence that helps inform medical practices and guidelines.
Continuous Data: Continuous data refers to numerical values that can take on an infinite number of possibilities within a given range. This type of data is crucial in biological research, as it allows for precise measurements of variables, such as weight, height, temperature, or time, which can vary continuously rather than in discrete steps.
Distribution-free tests: Distribution-free tests, also known as non-parametric tests, are statistical methods that do not assume a specific probability distribution for the data being analyzed. These tests are particularly useful when the assumptions required for parametric tests, such as normality, cannot be met. They rely on the rank or order of the data rather than the actual data values, making them robust and applicable in a wider range of situations.
Effect Size: Effect size is a quantitative measure that reflects the magnitude of a phenomenon or the strength of a relationship between variables. It helps to understand the practical significance of research findings beyond just statistical significance, indicating how meaningful or impactful the results are in real-world contexts.
Independence: In statistics, independence refers to the condition where two events or variables do not influence each other, meaning the occurrence of one does not affect the probability of the occurrence of the other. This concept is essential in various statistical methods, particularly in determining the relationships and associations among variables, ensuring that inferences drawn from data are valid and reliable.
Mean rank: Mean rank is the average of the ranks assigned to a set of values in a dataset, typically used in non-parametric statistics. This concept is essential for understanding how data is compared when the assumptions of parametric tests are not met. By calculating mean ranks, researchers can analyze the relative positions of values within groups, which plays a crucial role in non-parametric tests like the Wilcoxon rank-sum and signed-rank tests.
Median: The median is a measure of central tendency that represents the middle value in a dataset when the numbers are arranged in ascending order. It effectively divides the dataset into two equal halves, providing a robust indicator of the center of the data, particularly in skewed distributions or datasets with outliers.
Null hypothesis: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as a baseline for statistical testing. It is used to test the validity of an alternative hypothesis, providing a framework for evaluating whether observed data significantly deviates from what would be expected under the null scenario.
Ordinal data: Ordinal data is a type of categorical data where the values have a defined order or ranking, but the intervals between the values are not necessarily equal. This means that while you can say one value is greater or lesser than another, you cannot quantify how much greater or lesser it is. This unique property connects ordinal data to various statistical tests and methods that analyze differences in rankings and distributions.
P-value: A p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis in hypothesis testing. It quantifies the probability of obtaining an observed result, or one more extreme, assuming that the null hypothesis is true. This concept is crucial in evaluating the significance of findings in various areas, including biological research and data analysis.
Psychological studies: Psychological studies are research investigations that explore human thoughts, feelings, and behaviors to understand the underlying mental processes. These studies often employ various statistical methods to analyze data and draw conclusions about psychological phenomena, making them essential in the field of psychology and related disciplines. They can range from experimental designs to observational studies, providing insights into how different variables influence mental health and behavior.
Rank-based methods: Rank-based methods are statistical techniques that utilize the relative ranking of data points rather than their raw values to draw conclusions or make comparisons. These methods are particularly useful in non-parametric statistics, as they do not assume a normal distribution and are robust against outliers, making them ideal for analyzing ordinal data or when the assumptions of traditional parametric tests are not met.
Symmetry: Symmetry refers to the balanced and proportional arrangement of elements within a dataset, where patterns remain consistent across various axes or points. In statistical analyses, particularly in the context of non-parametric tests, symmetry plays a crucial role in understanding the distribution of data and validating the assumptions underlying different statistical tests.
W statistic: The w statistic is a test statistic used in non-parametric statistical methods, particularly in the context of the Wilcoxon rank-sum and signed-rank tests. It is a measure that helps to determine whether there is a significant difference between two independent samples or paired observations by analyzing the ranks of their data. This statistic is crucial because it provides an alternative to traditional parametric tests when the assumptions about normality are not met, allowing researchers to make valid inferences based on ranked data.
Wilcoxon rank-sum test: The Wilcoxon rank-sum test is a non-parametric statistical test used to determine whether there is a difference between the distributions of two independent samples. This test ranks all the observations from both groups and compares the sum of ranks between the groups, making it suitable for data that do not meet the assumptions of normality required for parametric tests.
Wilcoxon Signed-Rank Test: The Wilcoxon signed-rank test is a non-parametric statistical method used to determine whether there is a significant difference between the medians of two related samples. This test is particularly useful when the data does not meet the assumptions of normality required for parametric tests, making it ideal for analyzing matched pairs or repeated measurements.
Z score: A z score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It tells you how many standard deviations a data point is from the mean, which can help determine the relative position of that value within a distribution. In non-parametric tests like the Wilcoxon rank-sum and signed-rank tests, z scores can be used to assess the significance of differences between groups or conditions.