smile.stat.hypothesis.TTest Maven / Gradle / Ivy
/*******************************************************************************
* Copyright (c) 2010 Haifeng Li
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*******************************************************************************/
package smile.stat.hypothesis;
import smile.math.special.Beta;
import smile.math.Math;
/**
* Student's t test. A t-test is any statistical hypothesis test in which the test statistic has
* a Student's t distribution if the null hypothesis is true. It is applied
* when the population is assumed to be normally distributed but the sample
* sizes are small enough that the statistic on which inference is based is
* not normally distributed because it relies on an uncertain estimate of
* standard deviation rather than on a precisely known value.
*
* Among the most frequently used t tests are:
*
* - A test of whether the mean of a normally distributed population has
* a value specified in a null hypothesis.
*
- A test of the null hypothesis that the means of two normally
* distributed populations are equal. Given two data sets, each characterized
* by its mean, standard deviation and number of data points, we can use some
* kind of t test to determine whether the means are distinct, provided that
* the underlying distributions can be assumed to be normal. All such tests
* are usually called Student's t tests, though strictly speaking that name
* should only be used if the variances of the two populations are also assumed
* to be equal; the form of the test used when this assumption is dropped is
* sometimes called Welch's t test. There are different versions of the t test
* depending on whether the two samples are
*
* - unpaired, independent of each other (e.g., individuals randomly
* assigned into two groups, measured after an intervention and compared
* with the other group),
*
- or paired, so that each member of one sample has a unique relationship
* with a particular member of the other sample (e.g., the same people measured
* before and after an intervention).
*
* If the calculated p-value is below the threshold chosen for
* statistical significance (usually 0.05 or 0.01 level), then
* the null hypothesis which usually states that the two groups do not differ
* is rejected in favor of an alternative hypothesis, which typically states
* that the groups do differ.
* - A test of whether the slope of a regression line differs significantly
* from 0.
*
*
* @author Haifeng Li
*/
public class TTest {
/**
* The degree of freedom of t-statistic.
*/
public double df;
/**
* t-statistic
*/
public double t;
/**
* p-value
*/
public double pvalue;
/**
* Constructor.
*/
private TTest(double t, double df, double pvalue) {
this.t = t;
this.df = df;
this.pvalue = pvalue;
}
/**
* Independent one-sample t-test whether the mean of a normally distributed
* population has a value specified in a null hypothesis. Small values of
* p-value indicate that the array has significantly different mean.
*/
public static TTest test(double[] x, double mean) {
int n = x.length;
double mu = Math.mean(x);
double var = Math.var(x);
int df = n - 1;
double t = (mu - mean) / Math.sqrt(var/n);
double p = Beta.regularizedIncompleteBetaFunction(0.5 * df, 0.5, df / (df + t * t));
return new TTest(t, df, p);
}
/**
* Test if the arrays x and y have significantly different means. The data
* arrays are assumed to be drawn from populations with unequal variances.
* Small values of p-value indicate that the two arrays have significantly
* different means.
*/
public static TTest test(double[] x, double[] y) {
return test(x, y, false);
}
/**
* Test if the arrays x and y have significantly different means. Small
* values of p-value indicate that the two arrays have significantly
* different means.
* @param equalVariance true if the data arrays are assumed to be
* drawn from populations with the same true variance. Otherwise, The data
* arrays are allowed to be drawn from populations with unequal variances.
*/
public static TTest test(double[] x, double[] y, boolean equalVariance) {
if (equalVariance) {
int n1 = x.length;
int n2 = y.length;
double mu1 = Math.mean(x);
double var1 = Math.var(x);
double mu2 = Math.mean(y);
double var2 = Math.var(y);
int df = n1 + n2 - 2;
double svar = ((n1 - 1) * var1 + (n2 - 1) * var2) / df;
double t = (mu1 - mu2) / Math.sqrt(svar * (1.0 / n1 + 1.0 / n2));
double p = Beta.regularizedIncompleteBetaFunction(0.5 * df, 0.5, df / (df + t * t));
return new TTest(t, df, p);
} else {
int n1 = x.length;
int n2 = y.length;
double mu1 = Math.mean(x);
double var1 = Math.var(x);
double mu2 = Math.mean(y);
double var2 = Math.var(y);
double df = Math.sqr(var1 / n1 + var2 / n2) / (Math.sqr(var1 / n1) / (n1 - 1) + Math.sqr(var2 / n2) / (n2 - 1));
double t = (mu1 - mu2) / Math.sqrt(var1 / n1 + var2 / n2);
double p = Beta.regularizedIncompleteBetaFunction(0.5 * df, 0.5, df / (df + t * t));
return new TTest(t, df, p);
}
}
/**
* Given the paired arrays x and y, test if they have significantly
* different means. Small values of p-value indicate that the two arrays
* have significantly different means.
*/
public static TTest pairedTest(double[] x, double[] y) {
if (x.length != y.length) {
throw new IllegalArgumentException("Input vectors have different size");
}
double mu1 = Math.mean(x);
double var1 = Math.var(x);
double mu2 = Math.mean(y);
double var2 = Math.var(y);
int n = x.length;
int df = n - 1;
double cov = 0.0;
for (int j = 0; j < n; j++) {
cov += (x[j] - mu1) * (y[j] - mu2);
}
cov /= df;
double sd = Math.sqrt((var1 + var2 - 2.0 * cov) / n);
double t = (mu1 - mu2) / sd;
double p = Beta.regularizedIncompleteBetaFunction(0.5 * df, 0.5, df / (df + t * t));
return new TTest(t, df, p);
}
/**
* Test whether the Pearson correlation coefficient, the slope of
* a regression line, differs significantly from 0. Small values of p-value
* indicate a significant correlation.
* @param r the Pearson correlation coefficient.
* @param df the degree of freedom. df = n - 2, where n is the number of samples
* used in the calculation of r.
*/
public static TTest test(double r, int df) {
final double TINY = 1.0e-20;
double t = r * Math.sqrt(df / ((1.0 - r + TINY) * (1.0 + r + TINY)));
double p = Beta.regularizedIncompleteBetaFunction(0.5 * df, 0.5, df / (df + t * t));
return new TTest(t, df, p);
}
}