/**
* Computes p-value for 2-sided, 2-sample t-test, under the assumption of equal subpopulation
* variances.
*
* <p>The sum of the sample sizes minus 2 is used as degrees of freedom.
*
* @param m1 first sample mean
* @param m2 second sample mean
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return p-value
* @throws MathException if an error occurs computing the p-value
*/
protected double homoscedasticTTest(
double m1, double m2, double v1, double v2, double n1, double n2) throws MathException {
double t = Math.abs(homoscedasticT(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = n1 + n2 - 2;
distribution.setDegreesOfFreedom(degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Computes p-value for 2-sided, 2-sample t-test.
*
* <p>Does not assume subpopulation variances are equal. Degrees of freedom are estimated from the
* data.
*
* @param m1 first sample mean
* @param m2 second sample mean
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return p-value
* @throws MathException if an error occurs computing the p-value
*/
protected double tTest(double m1, double m2, double v1, double v2, double n1, double n2)
throws MathException {
double t = Math.abs(t(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = 0;
degreesOfFreedom = df(v1, v2, n1, n2);
distribution.setDegreesOfFreedom(degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
@Override
public final void compute() {
if (input[0].isDefined() && input[1].isDefined()) {
double param = a.getDouble();
double val = b.getDouble();
try {
TDistribution t = getTDistribution(param);
num.setValue(t.inverseCumulativeProbability(val));
} catch (Exception e) {
num.setUndefined();
}
} else num.setUndefined();
}
/**
* Modify the distribution used to compute inference statistics.
*
* @param value the new distribution
* @since 1.2
*/
public void setDistribution(TDistribution value) {
distribution = value;
// modify degrees of freedom
if (n > 2) {
distribution.setDegreesOfFreedom(n - 2);
}
}
/**
* Returns a matrix of p-values associated with the (two-sided) null hypothesis that the
* corresponding correlation coefficient is zero.
*
* <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability that a random variable
* distributed as <code>t<sub>n-2</sub></code> takes a value with absolute value greater than or
* equal to <br>
* <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code>
*
* <p>The values in the matrix are sometimes referred to as the <i>significance</i> of the
* corresponding correlation coefficients.
*
* @return matrix of p-values
* @throws MathException if an error occurs estimating probabilities
*/
public RealMatrix getCorrelationPValues() throws MathException {
TDistribution tDistribution = new TDistributionImpl(nObs - 2);
int nVars = correlationMatrix.getColumnDimension();
double[][] out = new double[nVars][nVars];
for (int i = 0; i < nVars; i++) {
for (int j = 0; j < nVars; j++) {
if (i == j) {
out[i][j] = 0d;
} else {
double r = correlationMatrix.getEntry(i, j);
double t = Math.abs(r * Math.sqrt((nObs - 2) / (1 - r * r)));
out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
}
}
}
return new BlockRealMatrix(out);
}
/**
* Adds the observation (x,y) to the regression data set.
*
* <p>Uses updating formulas for means and sums of squares defined in "Algorithms for Computing
* the Sample Variance: Analysis and Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
* 1983, American Statistician, vol. 37, pp. 242-247, referenced in Weisberg, S. "Applied Linear
* Regression". 2nd Ed. 1985.
*
* @param x independent variable value
* @param y dependent variable value
*/
public void addData(double x, double y) {
if (n == 0) {
xbar = x;
ybar = y;
} else {
double dx = x - xbar;
double dy = y - ybar;
sumXX += dx * dx * (double) n / (double) (n + 1.0);
sumYY += dy * dy * (double) n / (double) (n + 1.0);
sumXY += dx * dy * (double) n / (double) (n + 1.0);
xbar += dx / (double) (n + 1.0);
ybar += dy / (double) (n + 1.0);
}
sumX += x;
sumY += y;
n++;
if (n > 2) {
distribution.setDegreesOfFreedom(n - 2);
}
}
/**
* Returns the significance level of the slope (equiv) correlation.
*
* <p>Specifically, the returned value is the smallest <code>alpha</code> such that the slope
* confidence interval with significance level equal to <code>alpha</code> does not include <code>
* 0</code>. On regression output, this is often denoted <code>Prob(|t| > 0)</code>
*
* <p><strong>Usage Note</strong>:<br>
* The validity of this statistic depends on the assumption that the observations included in the
* model are drawn from a <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
* Bivariate Normal Distribution</a>.
*
* <p>If there are fewer that <strong>three</strong> observations in the model, or if there is no
* variation in x, this returns <code>Double.NaN</code>.
*
* @return significance level for slope/correlation
* @throws MathException if the significance level can not be computed.
*/
public double getSignificance() throws MathException {
return 2d * (1.0 - distribution.cumulativeProbability(Math.abs(getSlope()) / getSlopeStdErr()));
}
/**
* Returns the half-width of a (100-100*alpha)% confidence interval for the slope estimate.
*
* <p>The (100-100*alpha)% confidence interval is
*
* <p><code>(getSlope() - getSlopeConfidenceInterval(),
* getSlope() + getSlopeConfidenceInterval())</code>
*
* <p>To request, for example, a 99% confidence interval, use <code>alpha = .01</code>
*
* <p><strong>Usage Note</strong>:<br>
* The validity of this statistic depends on the assumption that the observations included in the
* model are drawn from a <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
* Bivariate Normal Distribution</a>.
*
* <p><strong> Preconditions:</strong>
*
* <ul>
* <li>If there are fewer that <strong>three</strong> observations in the model, or if there is
* no variation in x, this returns <code>Double.NaN</code>.
* <li><code>(0 < alpha < 1)</code>; otherwise an <code>IllegalArgumentException</code> is
* thrown.
* </ul>
*
* @param alpha the desired significance level
* @return half-width of 95% confidence interval for the slope estimate
* @throws MathException if the confidence interval can not be computed.
*/
public double getSlopeConfidenceInterval(double alpha) throws MathException {
if (alpha >= 1 || alpha <= 0) {
throw new IllegalArgumentException();
}
return getSlopeStdErr() * distribution.inverseCumulativeProbability(1d - alpha / 2d);
}
/**
* Computes p-value for 2-sided, 1-sample t-test.
*
* @param m sample mean
* @param mu constant to test against
* @param v sample variance
* @param n sample n
* @return p-value
* @throws MathException if an error occurs computing the p-value
*/
protected double tTest(double m, double mu, double v, double n) throws MathException {
double t = Math.abs(t(m, mu, v, n));
distribution.setDegreesOfFreedom(n - 1);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* @param param degrees of freedom
* @return T-distribution
*/
protected TDistribution getTDistribution(double param) {
if (t == null || t.getDegreesOfFreedom() != param) t = new TDistributionImpl(param);
return t;
}
