Glossary

The is a powerful tool for comparing two when data isn't normally distributed or is ordinal. It's like a nonparametric version of the t-test, helping us spot differences between groups without assuming normal distributions.

This test combines and ranks data from both samples, calculates , and uses the to determine if there's a . For larger samples, we can use z-scores, while smaller samples rely on .

Nonparametric Tests for Two Independent Samples

Appropriateness of Mann-Whitney U Test

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  • Compares two independent samples not related or paired in any way ()
  • Serves as a to the independent samples t-test when data is ordinal or assumptions of normality are violated (, presence of outliers)
  • Tests the that the two populations have the same distribution against the that they have different distributions (location, shape, or variability)

Execution of Mann-Whitney U Test

  • Combines observations from both samples into a single ordered array and assigns ranks from lowest to highest (1, 2, 3...)
    • Averages the ranks for (2.5 for two observations tied for 2nd and 3rd rank)
  • Calculates the for each sample (R1R_1 for sample 1 and R2R_2 for sample 2)
  • Determines the of the two groups (n1n_1 for sample 1 and n2n_2 for sample 2)
  • Calculates the U statistic for each sample using the formulas:
    1. U1=n1n2+n1(n1+1)2−R1U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1
    2. U2=n1n2+n2(n2+1)2−R2U_2 = n_1n_2 + \frac{n_2(n_2+1)}{2} - R_2
  • Selects the smaller U value (UminU_{min}) for further analysis

Calculation of U statistic

  • Follows a known distribution for sample sizes greater than 20, allowing for
    • Relies on tables to find critical values for smaller sample sizes (n < 20)
  • Calculates the : μU=n1n22\mu_U = \frac{n_1n_2}{2}
  • Calculates the : σU=n1n2(n1+n2+1)12\sigma_U = \sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}
  • For large sample sizes, calculates the : z=Umin−μUσUz = \frac{U_{min} - \mu_U}{\sigma_U}
  • Compares the calculated z-score or UminU_{min} to the critical value at the desired (0.05, 0.01)

Interpretation of Mann-Whitney results

  • Rejects the null hypothesis if the calculated z-score or UminU_{min} is less than the critical value, concluding a significant difference between the two populations ()
  • Fails to reject the null hypothesis if the calculated z-score or UminU_{min} is greater than the critical value, concluding to suggest a significant difference (no treatment effect)
  • Reports results using appropriate language and statistical terminology, including the U statistic, sample sizes, significance level, and conclusion drawn from the test
  • Discusses the implications of the findings in the context of the research question or problem being addressed (, limitations, future research)

Key Terms to Review (26)

Alternative Hypothesis: The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect, a difference, or a relationship in the population. It serves as the focus of research, aiming to provide evidence that supports its claim over the null hypothesis through statistical testing and analysis.
Critical Value Tables: Critical value tables are statistical tools that provide threshold values for various distributions used in hypothesis testing, determining the cut-off points beyond which we reject the null hypothesis. These tables are essential in understanding the significance of test statistics and help researchers make decisions based on their data. By comparing calculated statistics with values from critical value tables, one can assess whether to accept or reject the initial hypothesis based on pre-established significance levels.
Dallas Whitney: Dallas Whitney is an important figure in the development of non-parametric statistical methods, particularly known for his work related to the Mann-Whitney U Test. This test is a rank-based method used to compare differences between two independent groups, helping researchers analyze data that do not meet the assumptions of parametric tests. Whitney's contributions have been pivotal in providing a robust alternative for analyzing ordinal data or data that do not follow a normal distribution.
Henry Mann: Henry Mann is a significant figure in the development of nonparametric statistical methods, particularly known for the Mann-Whitney U Test, which he co-developed. This test is widely used to assess whether there are differences between two independent groups based on their ranks, making it a powerful tool for analyzing ordinal data or non-normally distributed interval data.
Independent Samples: Independent samples refer to two or more groups of observations that are collected separately and do not influence each other. This concept is crucial when performing statistical tests, as it ensures that the data points in one sample do not provide any information about the data points in another sample, allowing for valid comparisons between groups.
Insufficient evidence: Insufficient evidence refers to a situation where the data or statistical results do not provide enough support to reject the null hypothesis in a hypothesis test. This concept highlights the importance of having adequate data to draw meaningful conclusions, and it often indicates that the results are inconclusive or that further investigation is needed.
Mann-Whitney U Test: The Mann-Whitney U Test is a non-parametric statistical test used to compare differences between two independent groups. It assesses whether one group tends to have larger or smaller values than the other without assuming a normal distribution of the data, making it particularly useful for ordinal data or non-normally distributed continuous data.
Mean of the U Statistic: The mean of the U statistic is the average value calculated from the ranks assigned to data points in the Mann-Whitney U Test, which is a non-parametric test used to assess whether there is a difference between two independent groups. This statistic helps determine how much one group tends to rank higher than another and is crucial for understanding the distributions of two datasets without making assumptions about their underlying distributions. The mean of the U statistic allows researchers to interpret results in the context of rank differences rather than raw scores, providing insights into the relative positioning of data points across groups.
Non-normal distribution: A non-normal distribution refers to a probability distribution that does not follow the symmetric, bell-shaped curve characteristic of a normal distribution. This can manifest in various forms, such as skewness (where data leans towards one side) or kurtosis (where data has heavy tails), impacting statistical analysis and inference. Understanding non-normal distributions is crucial when applying tests that assume normality, especially for evaluating differences between groups.
Nonparametric alternative: A nonparametric alternative refers to statistical methods that do not assume a specific distribution for the data being analyzed, making them suitable for situations where traditional parametric tests may not be applicable. These alternatives are particularly useful when data do not meet the assumptions of normality or homogeneity of variance, and they often focus on ranking data rather than estimating parameters.
Null hypothesis: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as a default position that researchers aim to test against. It acts as a baseline to compare with the alternative hypothesis, which posits that there is an effect or a difference. This concept is foundational in statistical analysis and hypothesis testing, guiding researchers in determining whether observed data can be attributed to chance or if they suggest significant effects.
Ordinal Data: Ordinal data refers to a type of categorical data where the values can be ordered or ranked, but the differences between the ranks are not necessarily equal. This means that while you can say one value is greater or less than another, you can't measure how much greater or less it is in a meaningful way. It's important in various statistical methods as it allows for a comparison of relative positions, but with some limitations in mathematical operations.
Practical Significance: Practical significance refers to the real-world importance or relevance of a statistical finding, beyond just its statistical significance. It helps determine whether the results of a study have meaningful implications for decision-making or actions in real-life situations, emphasizing that a statistically significant result may not always translate into a substantial impact or change in practice.
Rank sum: Rank sum refers to the total of the ranks assigned to observations in a non-parametric statistical test, often used to compare two independent samples. In this context, the rank sum is crucial because it allows researchers to evaluate whether there are significant differences between the two groups without making assumptions about the underlying distributions. This method is particularly useful when data do not meet the requirements for parametric tests.
Rank sums: Rank sums refer to the total of the ranks assigned to observations from different groups in non-parametric statistical tests, such as the Mann-Whitney U Test. This method involves ranking all data points across groups and then summing these ranks for each group, which is essential for determining if there is a significant difference between the groups without making assumptions about the underlying distribution of the data.
Sample sizes: Sample sizes refer to the number of observations or data points collected from a population for analysis. The size of a sample is crucial because it affects the accuracy and reliability of statistical estimates, influencing the power of statistical tests and the precision of confidence intervals.
Significance Level: The significance level is a threshold in hypothesis testing that determines when to reject the null hypothesis. It is commonly denoted by the Greek letter alpha (\(\alpha\)) and represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected when it is true. This concept is essential for understanding the strength of evidence against the null hypothesis in various statistical tests.
Significant difference: A significant difference refers to a statistical term that indicates a noticeable variation between two or more groups that is unlikely to have occurred by chance alone. This concept is crucial in hypothesis testing, as it helps determine if the observed effects in data can be attributed to actual differences between groups rather than random fluctuations.
Standard Deviation of the U Statistic: The standard deviation of the U statistic measures the variability or dispersion of the Mann-Whitney U test statistic across different samples. This measure is crucial as it helps determine how far the observed U statistic is from the expected value under the null hypothesis, allowing for an assessment of statistical significance in non-parametric testing situations.
Tied observations: Tied observations occur when two or more data points in a dataset share the same value. This situation is particularly important in non-parametric tests, such as the Mann-Whitney U Test, where the rank of these tied values can affect the computation of the test statistic. Ties can complicate the analysis by necessitating special handling to ensure that the statistical properties of the test remain valid.
Treatment effect: The treatment effect refers to the impact that a specific intervention or treatment has on an outcome variable in a statistical analysis. It helps to measure how much a particular treatment changes the response compared to a control group or another treatment, allowing researchers to understand the effectiveness of that treatment. This concept is crucial in experiments and observational studies, especially when analyzing differences between groups.
Treatment vs control group: In experimental research, a treatment group is the subset of subjects that receives the intervention or treatment being tested, while a control group is the subset that does not receive the treatment, serving as a baseline for comparison. This distinction is crucial in determining the effectiveness of the treatment and in controlling for variables that could affect the outcome.
U Statistic: A U statistic is a type of non-parametric statistic used to compare two independent samples. It is particularly important in the context of the Mann-Whitney U test, which assesses whether one of two populations tends to yield larger values than the other. The U statistic helps determine if there is a significant difference between the distributions of the two groups being analyzed.
U_min: The term u_min refers to the minimum value of the Mann-Whitney U statistic, which is used to assess whether there are differences between two independent groups based on their ranks. It helps in determining if one group tends to have larger or smaller values than the other, providing a non-parametric alternative to the t-test when assumptions of normality are not met. Understanding u_min is crucial for interpreting the results of the Mann-Whitney U Test effectively.
Z-score: A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean, allowing for comparison between different datasets and understanding the relative position of a value within a distribution.
Z-score calculation: A z-score calculation is a statistical method used to determine how many standard deviations a data point is from the mean of a dataset. This score helps in understanding the relative position of a value within a distribution, enabling comparisons across different datasets. By converting raw scores into z-scores, one can easily assess whether a particular observation is typical or atypical, which is crucial for non-parametric tests like the Mann-Whitney U Test.
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