Table of Contents

Class MannWhitneyUTestResult<T>

Namespace
AiDotNet.Models.Results
Assembly
AiDotNet.dll

Represents the results of a Mann-Whitney U test, which is a non-parametric statistical test used to determine whether two independent samples come from the same distribution.

public class MannWhitneyUTestResult<T>

Type Parameters

T

The numeric type used for statistical values, typically float or double.

Inheritance
MannWhitneyUTestResult<T>
Inherited Members

Remarks

The Mann-Whitney U test (also known as the Wilcoxon rank-sum test) is a non-parametric alternative to the independent samples t-test. It's used when the data doesn't meet the assumptions required for the t-test, particularly when the data isn't normally distributed. This class stores the results of such a test, including the U statistic, Z-score, p-value, and whether the result is statistically significant. The test is commonly used to compare the medians of two groups, though it technically tests whether one sample tends to have larger values than the other. The class uses generic type parameter T to support different numeric types for the statistical values, such as float, double, or decimal.

For Beginners: The Mann-Whitney U test helps determine if two groups are different when you can't use a regular t-test.

For example, you might use this test to answer questions like:

  • Do patients on treatment A have different recovery times than those on treatment B?
  • Do students using one learning method score differently than those using another method?
  • Are customer satisfaction ratings different between two product versions?

The test works by:

  1. Ranking all values from both groups together
  2. Calculating how much the ranks differ between groups
  3. Determining if this difference is statistically significant

This test is particularly useful when:

  • Your data doesn't follow a normal distribution
  • You have ordinal data (rankings) rather than continuous measurements
  • You have outliers that might skew the results of a t-test

This class stores all the information about the test results, helping you interpret whether the observed difference between groups is likely due to chance or represents a real difference.

Constructors

MannWhitneyUTestResult(T, T, T, T)

Initializes a new instance of the MannWhitneyUTestResult class with the specified test statistics and parameters.

public MannWhitneyUTestResult(T uStatistic, T zScore, T pValue, T significanceLevel)

Parameters

uStatistic T

The U statistic value.

zScore T

The Z-score associated with the U statistic.

pValue T

The p-value associated with the Mann-Whitney U test.

significanceLevel T

The significance level used for the test.

Remarks

This constructor creates a new MannWhitneyUTestResult instance with the specified test statistics and parameters. It initializes all properties of the class and calculates the IsSignificant property by comparing the p-value to the significance level. The IsSignificant property is set to true if the p-value is less than the significance level, indicating that the test result is statistically significant. This constructor provides a convenient way to create a complete MannWhitneyUTestResult object in a single step after performing a Mann-Whitney U test.

For Beginners: This constructor creates a new result object with all the Mann-Whitney U test statistics and parameters.

When a new MannWhitneyUTestResult is created with this constructor:

  • All the test statistics and parameters are set to the values you provide
  • The IsSignificant property is automatically calculated by comparing the p-value to the significance level

This constructor is useful because:

  • It creates a complete result object in one step
  • It automatically determines statistical significance
  • It ensures all the related test information is kept together

The IsSignificant calculation uses MathHelper to compare the p-value to the significance level, which works regardless of what numeric type T is (float, double, decimal, etc.).

Properties

IsSignificant

Gets or sets a value indicating whether the Mann-Whitney U test result is statistically significant.

public bool IsSignificant { get; set; }

Property Value

bool

True if the result is statistically significant; otherwise, false.

Remarks

This property indicates whether the Mann-Whitney U test result is statistically significant at the specified significance level. A result is considered significant if the p-value is less than or equal to the significance level. Statistical significance suggests that the observed difference between the samples is unlikely to have occurred by chance alone, leading to the rejection of the null hypothesis that the samples come from the same distribution. This property provides a convenient boolean indicator of significance without requiring the user to compare the p-value to the significance level.

For Beginners: This tells you whether your result is statistically significant or not.

The significance indicator:

  • Provides a simple yes/no answer about statistical significance
  • True means the groups are significantly different
  • False means the difference between groups could reasonably be due to random chance

This is determined by:

  • Comparing the p-value to the significance level

This value makes it easy to interpret your results without having to manually check if the p-value is below the significance threshold. It's particularly useful when automating analysis or presenting results to non-statisticians.

PValue

Gets or sets the p-value associated with the Mann-Whitney U test.

public T PValue { get; set; }

Property Value

T

The calculated p-value.

Remarks

This property represents the p-value of the Mann-Whitney U test, which is the probability of observing a U statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. The null hypothesis typically states that the two samples come from the same distribution or have the same median. A small p-value (typically = 0.05) suggests that the observed data is unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis that the distributions differ. The p-value is calculated from the Z-score using the cumulative distribution function of the standard normal distribution for large samples, or from tables of critical values for small samples.

For Beginners: This value tells you how likely your results could occur by random chance.

The p-value:

  • Ranges from 0 to 1
  • Smaller values indicate stronger evidence against the null hypothesis
  • Represents the probability of seeing your results (or more extreme) if the groups are actually the same

Common interpretation:

  • p = 0.05: Results are statistically significant (commonly used threshold)
  • p = 0.01: Results are highly significant
  • p > 0.05: Results are not statistically significant

For example, a p-value of 0.03 means there's only a 3% chance of seeing your results if the groups truly come from the same distribution, suggesting the difference is likely real and not due to random chance.

SignificanceLevel

Gets or sets the significance level used for the test.

public T SignificanceLevel { get; set; }

Property Value

T

The significance level, typically 0.05.

Remarks

This property represents the significance level used for the Mann-Whitney U test, which is the threshold p-value below which the result is considered statistically significant. Common values are 0.05 (5%), 0.01 (1%), and 0.001 (0.1%). The significance level represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A lower significance level reduces the risk of Type I errors but increases the risk of Type II errors (failing to reject a false null hypothesis).

For Beginners: This value is the threshold that determines when a result is considered statistically significant.

The significance level:

  • Is typically set to 0.05 (5%) by convention
  • Represents your tolerance for false positives
  • Lower values (like 0.01) make the test more conservative
  • Higher values (like 0.10) make the test more lenient

This value is important because:

  • It directly determines whether a result is considered "significant"
  • It represents the risk you're willing to take of finding a difference when none exists

For example, with a significance level of 0.05, you're accepting a 5% chance of incorrectly concluding the groups are different when they're actually the same.

UStatistic

Gets or sets the U statistic value.

public T UStatistic { get; set; }

Property Value

T

The calculated U statistic.

Remarks

This property represents the Mann-Whitney U statistic, which is calculated by counting the number of times a value in the first sample exceeds a value in the second sample, and vice versa. The U statistic is the smaller of these two counts. When the null hypothesis is true (the distributions are the same), U tends to be around half the product of the two sample sizes. Values of U that deviate significantly from this expected value suggest that the samples come from different distributions. The U statistic is used to calculate the Z-score and p-value, which determine statistical significance.

For Beginners: This value measures how different the rankings are between your two groups.

The U statistic:

  • Is calculated based on the rankings of values in both groups
  • Measures how much the two groups overlap or differ
  • Has a range from 0 to the product of the two sample sizes

When the groups are similar:

  • U will be close to half the product of the sample sizes

When the groups are different:

  • U will be closer to either 0 or the maximum possible value

For example, with two groups of 10 samples each, U would be around 50 if the groups are similar, but might be closer to 20 or 80 if they're substantially different.

ZScore

Gets or sets the Z-score associated with the U statistic.

public T ZScore { get; set; }

Property Value

T

The calculated Z-score.

Remarks

This property represents the Z-score, which is a standardized version of the U statistic. The Z-score is calculated by subtracting the expected value of U (under the null hypothesis) from the observed U statistic, and then dividing by the standard deviation of U. For large sample sizes, the Z-score approximately follows a standard normal distribution, which allows for the calculation of the p-value. A Z-score with a large absolute value (typically > 1.96 for a two-tailed test at a = 0.05) suggests that the observed U statistic is significantly different from what would be expected if the null hypothesis were true.

For Beginners: This value standardizes the U statistic to make it easier to interpret.

The Z-score:

  • Converts the U statistic to a standardized form
  • Follows a standard normal distribution for large samples
  • Makes it easier to calculate the p-value

Interpretation:

  • Z-scores close to 0 suggest the groups are similar
  • Z-scores with absolute values greater than 1.96 (for a = 0.05) suggest significant differences
  • Negative Z-scores indicate the first group tends to have lower values
  • Positive Z-scores indicate the first group tends to have higher values

For example, a Z-score of -2.5 suggests that the first group has significantly lower values than the second group.

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