Class FTestResult<T>

Namespace: AiDotNet.Models.Results

Assembly: AiDotNet.dll

Represents the results of an F-test, which is used to compare the variances of two populations.

public class FTestResult<T>

Type Parameters

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

Inheritance: object

FTestResult<T>

Inherited Members: object.Equals(object)

object.Equals(object, object)

object.GetHashCode()

object.GetType()

object.MemberwiseClone()

object.ReferenceEquals(object, object)

object.ToString()

Remarks

The F-test is a statistical test used to determine whether two populations have equal variances. It is based on the ratio of two sample variances. This class stores the results of such a test, including the F-statistic, p-value, degrees of freedom, the variances being compared, confidence intervals, and whether the result is statistically significant. The F-test is commonly used in analysis of variance (ANOVA) and as a preliminary test before applying other statistical tests that assume equal variances. 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 F-test helps determine if two groups have similar or different amounts of variability.

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

Do men and women have the same variability in test scores?
Is the precision of one measurement method better than another?
Are the variances in two manufacturing processes comparable?

The test works by:

Calculating the ratio of the two sample variances
Comparing this ratio to what would be expected if the population variances were equal
Determining if the difference is statistically significant

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

Constructors

FTestResult(T, T, int, int, T, T, T, T, T)

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

public FTestResult(T fStatistic, T pValue, int numeratorDf, int denominatorDf, T leftVariance, T rightVariance, T lowerCI, T upperCI, T significanceLevel)

Parameters

fStatistic T: The F-statistic value.
pValue T: The p-value associated with the F-test.
numeratorDf int: The degrees of freedom for the numerator.
denominatorDf int: The degrees of freedom for the denominator.
leftVariance T: The variance of the left (or first) sample.
rightVariance T: The variance of the right (or second) sample.
lowerCI T: The lower bound of the confidence interval for the ratio of population variances.
upperCI T: The upper bound of the confidence interval for the ratio of population variances.
significanceLevel T: The significance level used for the test.

Remarks

This constructor creates a new FTestResult 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 FTestResult object in a single step after performing an F-test.

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

When a new FTestResult 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

DenominatorDegreesOfFreedom

Gets or sets the degrees of freedom for the denominator.

public int DenominatorDegreesOfFreedom { get; set; }

Property Value

int: An integer representing the denominator degrees of freedom.

Remarks

This property represents the degrees of freedom for the denominator of the F-statistic, which is typically one less than the sample size of the second group (n2 - 1). The denominator degrees of freedom is one of the parameters of the F-distribution used to calculate the p-value. It affects the shape of the F-distribution and thus the interpretation of the F-statistic.

For Beginners: This value is related to the sample size of the second group.

The denominator degrees of freedom:

Is typically calculated as (n2 - 1), where n2 is the sample size of the second group
Is a parameter needed to interpret the F-statistic
Helps determine the shape of the F-distribution used to calculate the p-value

For example, if your second sample has 15 observations, the denominator degrees of freedom would be 14.

FStatistic

Gets or sets the F-statistic value.

public T FStatistic { get; set; }

Property Value

T: The calculated F-statistic.

Remarks

This property represents the F-statistic, which is the ratio of the larger sample variance to the smaller sample variance. An F-statistic close to 1 suggests that the two populations have similar variances, while values much larger than 1 suggest that the variances are different. The F-statistic follows an F-distribution with degrees of freedom determined by the sample sizes. This statistic is used in conjunction with the degrees of freedom to calculate the p-value, which determines statistical significance.

For Beginners: This value is the ratio of the larger variance to the smaller variance.

The F-statistic:

Is calculated as the ratio of two sample variances
Is always positive
A value close to 1 suggests similar variances
Values much larger than 1 suggest different variances

For example, an F-statistic of 1.05 suggests the variances are very similar, while a value of 4.75 suggests one group has much more variability than the other.

IsSignificant

Gets or sets a value indicating whether the F-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 F-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 sample variances is unlikely to have occurred by chance alone, leading to the rejection of the null hypothesis that the population variances are equal. 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 variances are significantly different
False means the difference in variances 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.

LeftVariance

Gets or sets the variance of the left (or first) sample.

public T LeftVariance { get; set; }

Property Value

T: The calculated variance of the left sample.

Remarks

This property represents the variance of the left (or first) sample, which is a measure of the spread or dispersion of the data points around their mean. The variance is calculated as the average of the squared differences from the mean. This is one of the two variances being compared in the F-test. In the context of the F-test, the left variance is often the numerator in the F-statistic calculation, especially if it's the larger of the two variances.

For Beginners: This value measures how spread out the data is in the first group.

The left variance:

Quantifies the variability or dispersion in the first sample
Higher values indicate more spread-out data
Is one of the two variances being compared in the F-test

For example, if measuring test scores, a variance of 100 means the scores are more spread out than a variance of 25.

LowerConfidenceInterval

Gets or sets the lower bound of the confidence interval for the ratio of population variances.

public T LowerConfidenceInterval { get; set; }

Property Value

T: The lower bound of the confidence interval.

Remarks

This property represents the lower bound of the confidence interval for the ratio of the two population variances. The confidence interval provides a range of plausible values for the true ratio of population variances, given the observed sample data. If this interval includes 1, then the null hypothesis that the population variances are equal cannot be rejected at the specified significance level. The width of the confidence interval depends on the significance level and the sample sizes.

For Beginners: This value is the lower end of the range where the true variance ratio likely falls.

The lower confidence interval:

Forms the lower bound of a range that likely contains the true ratio of population variances
Is calculated based on the F-statistic, degrees of freedom, and significance level
Helps assess the precision of your variance ratio estimate

This value is important because:

If the interval includes 1, the variances might be equal
If both the lower and upper bounds are above 1, the first population likely has higher variance
If both bounds are below 1, the second population likely has higher variance

For example, a lower bound of 1.2 suggests that the first population's variance is at least 1.2 times the second population's variance.

NumeratorDegreesOfFreedom

Gets or sets the degrees of freedom for the numerator.

public int NumeratorDegreesOfFreedom { get; set; }

Property Value

int: An integer representing the numerator degrees of freedom.

Remarks

This property represents the degrees of freedom for the numerator of the F-statistic, which is typically one less than the sample size of the first group (n1 - 1). The numerator degrees of freedom is one of the parameters of the F-distribution used to calculate the p-value. It affects the shape of the F-distribution and thus the interpretation of the F-statistic.

For Beginners: This value is related to the sample size of the first group.

The numerator degrees of freedom:

Is typically calculated as (n1 - 1), where n1 is the sample size of the first group
Is a parameter needed to interpret the F-statistic
Helps determine the shape of the F-distribution used to calculate the p-value

For example, if your first sample has 20 observations, the numerator degrees of freedom would be 19.

PValue

Gets or sets the p-value associated with the F-test.

public T PValue { get; set; }

Property Value

T: The calculated p-value.

Remarks

This property represents the p-value of the F-test, which is the probability of observing an F-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 populations have equal variances. 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 variances are different. The p-value is calculated from the F-statistic and the degrees of freedom using the cumulative distribution function of the F-distribution.

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 variances are actually equal

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 0.02 means there's only a 2% chance of seeing your results if the variances are truly equal, suggesting the difference in variability is likely real and not due to random chance.

RightVariance

Gets or sets the variance of the right (or second) sample.

public T RightVariance { get; set; }

Property Value

T: The calculated variance of the right sample.

Remarks

This property represents the variance of the right (or second) sample, which is a measure of the spread or dispersion of the data points around their mean. The variance is calculated as the average of the squared differences from the mean. This is one of the two variances being compared in the F-test. In the context of the F-test, the right variance is often the denominator in the F-statistic calculation, especially if it's the smaller of the two variances.

For Beginners: This value measures how spread out the data is in the second group.

The right variance:

Quantifies the variability or dispersion in the second sample
Higher values indicate more spread-out data
Is one of the two variances being compared in the F-test

For example, if measuring reaction times, a variance of 0.5 means the times are more consistent than a variance of 2.0.

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 F-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). The significance level also determines the width of the confidence interval, with lower significance levels resulting in wider intervals.

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 affects the width of the confidence interval
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 variances are different when they're actually equal.

UpperConfidenceInterval

Gets or sets the upper bound of the confidence interval for the ratio of population variances.

public T UpperConfidenceInterval { get; set; }

Property Value

T: The upper bound of the confidence interval.

Remarks

This property represents the upper bound of the confidence interval for the ratio of the two population variances. The confidence interval provides a range of plausible values for the true ratio of population variances, given the observed sample data. If this interval includes 1, then the null hypothesis that the population variances are equal cannot be rejected at the specified significance level. The width of the confidence interval depends on the significance level and the sample sizes.

For Beginners: This value is the upper end of the range where the true variance ratio likely falls.

The upper confidence interval:

Forms the upper bound of a range that likely contains the true ratio of population variances
Is calculated based on the F-statistic, degrees of freedom, and significance level
Helps assess the precision of your variance ratio estimate

This value is important because:

If the interval includes 1, the variances might be equal
A very high upper bound suggests uncertainty about how much larger the first variance might be
Together with the lower bound, it shows the precision of your estimate

For example, an upper bound of 3.5 suggests that the first population's variance could be as much as 3.5 times the second population's variance.

Table of Contents

Class FTestResult<T>

Type Parameters

Remarks

Constructors

FTestResult(T, T, int, int, T, T, T, T, T)

Parameters

Remarks

Properties

DenominatorDegreesOfFreedom

Property Value

Remarks

FStatistic

Property Value

Remarks

IsSignificant

Property Value

Remarks

LeftVariance

Property Value

Remarks

LowerConfidenceInterval

Property Value

Remarks

NumeratorDegreesOfFreedom

Property Value

Remarks

PValue

Property Value

Remarks

RightVariance

Property Value

Remarks

SignificanceLevel

Property Value

Remarks

UpperConfidenceInterval

Property Value

Remarks