Module org.bzdev.math

Package org.bzdev.math.stats

Class StudentsTStat.Slope

java.lang.Object

org.bzdev.math.stats.Statistic

org.bzdev.math.stats.StudentsTStat

org.bzdev.math.stats.StudentsTStat.Slope

Enclosing class:

StudentsTStat

public static class StudentsTStat.Slope
extends StudentsTStat

Class providing Student's t-test to determine if the slope of a linear regression has a specific value. Using a value of 0 for the specified slope tests whether the x and y values are independent of each other.

Nested Class Summary

Nested classes/interfaces inherited from class org.bzdev.math.stats.StudentsTStat

StudentsTStat.Mean1, StudentsTStat.Mean2, StudentsTStat.PairedDiff, StudentsTStat.Slope

Nested classes/interfaces inherited from class org.bzdev.math.stats.Statistic

Statistic.PValueMode

Constructor Summary

Constructors

Constructor	Description
Slope(double beta0, double[] x, double[] y)	Constructor.
Slope(double beta0, double beta, double variance, double ssq, int n)	Constructor given a description of the statistic.

Method Summary

Modifier and Type	Method	Description
LeastSquaresFit.Polynomial	getFit()	Get the least squares fit (a polynomial fit of degree 1)
double	getNCParameter(double diff)	Get the noncentrality parameter given a difference in mean values.
double	getValue()	Get the value of this statistic.

Methods inherited from class org.bzdev.math.stats.StudentsTStat

getDegreesOfFreedom, getDistribution, getDistribution, setDegreesOfFreedom

Methods inherited from class org.bzdev.math.stats.Statistic

getBeta, getBeta, getCriticalValue, getNCParameter, getPower, getPower, getPValue, optimalValue

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

Slope

public Slope(double beta0,
 double[] x,
 double[] y)

Constructor. The x and y values must be arrays of equal lengths and each data point consists of an x and y value with the same index.

Parameters:

beta0 - the specified slope

x - the X values

y - the Y values

Slope

public Slope(double beta0,
 double beta,
 double variance,
 double ssq,
 int n)

Constructor given a description of the statistic.

Parameters:

beta0 - the specified slope

beta - the slope determined by a least squares fit

variance - the population variance of the x values used in the least-squares fit.

ssq - the sum of the squares of the residuals

n - the number of points used to create the least squares fit

Method Details

getFit

public LeastSquaresFit.Polynomial getFit()

Get the least squares fit (a polynomial fit of degree 1)

Returns:

the least squares fit for this statistic's data set; null if not available

getValue

public double getValue()

Description copied from class: Statistic

Get the value of this statistic.

Specified by:

getValue in class Statistic

Returns:

the value of this statistic

getNCParameter

public double getNCParameter(double diff)

Description copied from class: StudentsTStat

Get the noncentrality parameter given a difference in mean values. A Student's t-test can typically be written as $$ T = \frac{\mbox{x̅} - \mu_0}{ \frac{\sigma}{\sqrt{n}} \sqrt{\frac{S}{\sigma^2\nu}} }$$ where ν is the number of degrees of freedom, σ is the population standard deviation of a random variable for which x is the sample mean of X, μ₀ is the population mean of X, n is the sample size, and S is a sum of squares. Note that $\sqrt{\frac{S}{\sigma^2\nu}}$ will be the sample standard deviation for a suitable choice of ν.

If we set θ = μ₁ - μ₀, we can express T as $$T = \frac{\mbox{x̅} - \mu_0 +\theta - \theta }{ \frac{\sigma}{\sqrt{n}} \frac{S/\sigma^2}{\sqrt{\nu}} }$$ or $$T = \frac{\mbox{x̅} - \mu_1 + \theta }{ \frac{\sigma}{\sqrt{n}} \frac{S/\sigma^2}{\sqrt{\nu}} } \ .$$ If we set $Z = \frac{\mbox{x̅} - \mu_1}{\sigma/\sqrt{n}}$ and $\mu = \frac{\theta}{\sigma/\sqrt{n}}$, then T can be written as $\frac{Z+\mu}{\sqrt{V/\nu}}$. If the actual mean is μ₁, then Z has a normal distribution with a mean of zero and a variance of 1. Meanwhile V has a χ² distribution with ν degrees of freedom, and Z and V are independent. Consequently, the random variable $\frac{Z + \mu}{\sqrt{V/\nu}}$ has a noncentral t distribution characterized by the number of degrees of freedom ν and the noncentrality parameter μ.

For citations for specific cases, please see

• Sample Size & Power Calculations and Sample size and power calculations using the noncentral t-distribution for two-sample case.
• Statistical power of the t-tests for the one-sample case.

The notation in this documentation will differ slightly due to the use of a lower-case x instead of an upper-case X as part of a work-around for a mathjax bug involving the LaTeX 'overline' command.

Specified by:

getNCParameter in class StudentsTStat

Parameters:

diff - the difference of the H1 mean value and the H0 mean value.

Returns:

the noncentrality parameter