# Statistics

## Overview

The statistics package provides basic descriptive statistics, frequency distributions, linear regression, analysis of variance, correlation and a variety of inference tests.

## Descriptive statistics

The stat package includes the following Descriptive statistics in the `descriptive` subpackage:

- arithmetic and geometric means
- variance and standard deviation
- sum, product, log sum, sum of squared values
- minimum, maximum, median, and percentiles
- skewness and kurtosis
- first, second, third and fourth moments

With the exception of percentiles and the median, all of these statistics can be computed without maintaining the full list of input data values in memory. The stat package provides interfaces and implementations that do not require value storage as well as implementations that operate on arrays of stored values. There are two classes, `PSquarePercentile` and `RandomPercentile` class that approximate percentiles, using the PSquare and RANDOM algorithms, respectively.

The top level interface is UnivariateStatistic. This interface, implemented by all statistics, consists of `evaluate()` methods that take double[] arrays as arguments and return the value of the statistic. This interface is extended by StorelessUnivariateStatistic , which adds `increment()`, `getResult()` and associated methods to support streaming implementations that maintain counters, sums or other state information as values are added using the `increment()` method. Statistics that implement this interface can be assumed to use bounded storage, regardless of the length of the data stream injested by their `increment()` methods.

Abstract implementations of the top level interfaces are provided in AbstractUnivariateStatistic and AbstractStorelessUnivariateStatistic respectively.

Each statistic is implemented as a separate class, in one of the subpackages (`moment`, `rank`, `summary`) and each extends one of the abstract classes above (depending on whether or not value storage is required to compute the statistic). There are several ways to instantiate and use statistics. Statistics can be instantiated and used directly, but it is generally more convenient (and efficient) to access them using the provided aggregates, DescriptiveStatistics and StreamingStatistics..

`DescriptiveStatistics` maintains the input data in memory and has the capability of producing “rolling” statistics computed from a “window” consisting of the most recently added values.

`StreamingStatistics` does not store the full set of input data values in memory. It includes a RandomPercentile instance that maintains a bounded sample of data from the stream (see the class javadoc for `RandomPercentile` for details).

Aggregate | Statistics Included | Values stored? | “Rolling” capability? |
---|---|---|---|

DescriptiveStatistics | min, max, mean, geometric mean, n, sum, sum of squares, standard deviation, variance, percentiles, skewness, kurtosis, median | Yes | Yes |

StreamingStatistics | min, max, mean, geometric mean, n, sum, sum of squares, standard deviation, variance, percentiles | No | No |

`StreamingStatistics` supports aggregation of results using various `aggregate` methods.

`MultivariateSummaryStatistics` is similar to `StreamingStatistics` but handles n-tuple values instead of scalar values. It can also compute the full covariance matrix for the input data.

Neither `DescriptiveStatistics` nor `StreamingStatistics` is thread-safe.

There is also a utility class, StatUtils, that provides static methods for computing statistics directly from double[] arrays.

Here are some examples showing how to compute Descriptive statistics.

**Compute summary statistics for a list of double values**

Using the `DescriptiveStatistics` aggregate (values are stored in memory):

// Get a DescriptiveStatistics instance DescriptiveStatistics stats = new DescriptiveStatistics(); // Add the data from the array for( int i = 0; i < inputArray.length; i++) { stats.addValue(inputArray[i]); } // Compute some statistics double mean = stats.getMean(); double std = stats.getStandardDeviation(); double median = stats.getPercentile(50);

Using the `StreamingStatistics` aggregate to handle data from a stream:

// Get a StreamingStatistics instance StreamingStatistics stats = new StreamingStatistics(); // Read data from an input stream, // adding values and updating sums, counters, etc. while (line != null) { line = in.readLine(); stats.addValue(Double.parseDouble(line.trim())); } in.close(); // Compute the statistics double mean = stats.getMean(); double std = stats.getStandardDeviation();

Using the `StatUtils` utility class:

// Compute statistics directly from the array // assume values is a double[] array double mean = StatUtils.mean(values); double std = FastMath.sqrt(StatUtils.variance(values)); double median = StatUtils.percentile(values, 50); // Compute the mean of the first three values in the array mean = StatUtils.mean(values, 0, 3);

**Maintain a “rolling mean” of the most recent 100 values from an input stream**

Use a `DescriptiveStatistics` instance with window size set to 100

// Create a DescriptiveStats instance and set the window size to 100 DescriptiveStatistics stats = new DescriptiveStatistics(); stats.setWindowSize(100); // Read data from an input stream, // displaying the mean of the most recent 100 observations // after every 100 observations long nLines = 0; while (line != null) { line = in.readLine(); stats.addValue(Double.parseDouble(line.trim())); if (nLines == 100) { nLines = 0; System.out.println(stats.getMean()); } } in.close();

**Compute statistics for multiple samples and aggregate results**

Use multiple `StreamingStatistics` instances and aggregate them into a final result:

// Create individual StreamingStatistics instances to accumulate // statistics for the subsamples StreamingStatistics setOneStats = new StreamingStatistics(); StreamingStatistics setTwoStats = new StreamingStatistics(); // Add values to the subsample aggregates setOneStats.addValue(2); setOneStats.addValue(3); setTwoStats.addValue(2); setTwoStats.addValue(4); ... // Aggregate the results StreamingStatistics aggregate = new StreamingStatistics(); aggregate.aggregate(setOneStats, setTwoStats); // Full sample data is reported by the aggregate double totalSampleSum = aggregate.getSum();

Additionally, `StatisticalSummary` instances can be aggregated as well:

// Create StreamingStatistic instances for the subsample data StreamingStatistics setOneStats = new StreamingStatistics(); StreamingStatistics setTwoStats = new StreamingStatistics(); // Add values to the subsample StreamingStatistic instances setOneStats.addValue(2); setOneStats.addValue(3); setTwoStats.addValue(2); setTwoStats.addValue(4); ... // Aggregate the subsample statistics StatisticalSummary aggregatedStats = StatisticalSummary.aggregate(setOneStats, setTwoStats); // Full sample data is reported by aggregatedStats double totalSampleSum = aggregatedStats.getSum();

## Frequency distributions

Frequency provides a simple interface for maintaining counts and percentages of discrete values.

Strings, integers, longs and chars are all supported as value types, as well as instances of any class that implements `Comparable.` The ordering of values used in computing cumulative frequencies is by default the *natural ordering,* but this can be overridden by supplying a `Comparator` to the constructor.

Here are some examples.

**Compute a frequency distribution based on integer values**

Mixing integers, longs, Integers and Longs:

LongFrequency f = new LongFrequency(); f.addValue(1); f.addValue(new Integer(1)); f.addValue(new Long(1)); f.addValue(2); f.addValue(new Integer(-1)); System.out.prinltn(f.getCount(1)); // displays 3 System.out.println(f.getCumPct(0)); // displays 0.2 System.out.println(f.getPct(new Integer(1))); // displays 0.6 System.out.println(f.getCumPct(-2)); // displays 0 System.out.println(f.getCumPct(10)); // displays 1

**Count string frequencies**

Using case-sensitive comparison, alpha sort order (natural comparator):

Frequency<String> f = new Frequency<>(); f.addValue("one"); f.addValue("One"); f.addValue("oNe"); f.addValue("Z"); System.out.println(f.getCount("one")); // displays 1 System.out.println(f.getCumPct("Z")); // displays 0.5 System.out.println(f.getCumPct("Ot")); // displays 0.25

Using case-insensitive comparator:

Frequency<String> f = new Frequency<>(String.CASE_INSENSITIVE_ORDER); f.addValue("one"); f.addValue("One"); f.addValue("oNe"); f.addValue("Z"); System.out.println(f.getCount("one")); // displays 3 System.out.println(f.getCumPct("z")); // displays 1

## Simple regression

SimpleRegression provides ordinary least squares regression with one independent variable estimating the linear model:

y = intercept + slope * x

or

y = slope * x

Standard errors for `intercept` and `slope` are available as well as ANOVA, r-square and Pearson’s r statistics.

Observations (x,y pairs) can be added to the model one at a time or they can be provided in a 2-dimensional array. The observations are not stored in memory, so there is no limit to the number of observations that can be added to the model.

**Usage Notes:**

- When there are fewer than two observations in the model, or when there is no variation in the x values (i.e. all x values are the same) all statistics return
`NaN`. At least two observations with different x coordinates are required to estimate a bivariate regression model. - getters for the statistics always compute values based on the current set of observations – i.e., you can get statistics, then add more data and get updated statistics without using a new instance. There is no “compute” method that updates all statistics. Each of the getters performs the necessary computations to return the requested statistic.
- The intercept term may be suppressed by passing false to the SimpleRegression(boolean) constructor. When the
`hasIntercept`property is`false`, the model is estimated without a constant term and`getIntercept()`returns 0. - The
`SimpleRegression`class is not thread-safe. If multiple threads concurrently access a single instance of this class, their access to methods that add data or compute statistics must be externally synchronized.

**Implementation Notes:**

- As observations are added to the model, the sum of x values, y values, cross products (x times y), and squared deviations of x and y from their respective means are updated using updating formulas 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. All regression statistics are computed from these sums.
- Inference statistics (confidence intervals, parameter significance levels) are based on on the assumption that the observations included in the model are drawn from a Bivariate Normal Distribution

Here are some examples.

**Estimate a model based on observations added one at a time**

Instantiate a regression instance and add data points

regression = new SimpleRegression(); regression.addData(1d, 2d); // At this point, with only one observation, // all regression statistics will return NaN regression.addData(3d, 3d); // With only two observations, // slope and intercept can be computed // but inference statistics will return NaN regression.addData(3d, 3d); // Now all statistics are defined.

Compute some statistics based on observations added so far

// displays intercept of regression line System.out.println(regression.getIntercept()); // displays slope of regression line System.out.println(regression.getSlope()); // displays slope standard error System.out.println(regression.getSlopeStdErr());

Use the regression model to predict the y value for a new x value

// displays predicted y value for x = 1.5 System.out.println(regression.predict(1.5d)

More data points can be added and subsequent `getXxx` calls will incorporate additional data in statistics.

**Estimate a model from a double[][] array of data points**

Instantiate a regression object and load dataset

double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }}; SimpleRegression regression = new SimpleRegression(); regression.addData(data);

Estimate regression model based on data

// displays intercept of regression line System.out.println(regression.getIntercept()); // displays slope of regression line System.out.println(regression.getSlope()); // displays slope standard error System.out.println(regression.getSlopeStdErr());

More data points – even another `double[][]` array – can be added and subsequent `getXxx` calls will incorporate additional data in statistics.

**Estimate a model from a double[][] array of data points, excluding the intercept**

Instantiate a regression object and load dataset

double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }}; // the argument, false, tells the class not to include a constant SimpleRegression regression = new SimpleRegression(false); regression.addData(data);

Estimate regression model based on data

// displays intercept of regression line, since we have // constrained the constant, 0.0 is returned System.out.println(regression.getIntercept()); // displays slope of regression line System.out.println(regression.getSlope()); // displays slope standard error System.out.println(regression.getSlopeStdErr()); // will return Double.NaN, since we constrained the parameter to zero System.out.println(regression.getInterceptStdErr() );

Caution must be exercised when interpreting the slope when no constant is being estimated. The slope may be biased.

## Multiple linear regression

OLSMultipleLinearRegression, GLSMultipleLinearRegression and MillerUpdatingRegression provide least squares regression to fit the linear model:

Y = X * b + u

where Y is an n-vector **regressand**, X is a `[n,k]` matrix whose k columns are called **regressors**, b is k-vector of **regression parameters** and u is an n-vector of **error terms** or **residuals**.

OLSMultipleLinearRegression provides Ordinary Least Squares (OLS) Regression, GLSMultipleLinearRegression implements Generalized Least Squares and MillerUpdatingRegression provides a streaming implemnentation of OLS regression. See the javadoc for these classes for details on the algorithms and formulas used.

Data for `OLSMultipleLinearRegression` models can be loaded in a single double[] array, consisting of concatenated rows of data, each containing the regressand (Y) value, followed by regressor values; or using a `double[][]` array with rows corresponding to observations.

`GLSMultipleLinearRegression` models also require a `double[][]` array representing the covariance matrix of the error terms. See AbstractMultipleLinearRegression#newSampleData(double[],int,int), OLSMultipleLinearRegression#newSampleData(double[], double[][]) and GLSMultipleLinearRegression#newSampleData(double[],double[][],double[][]) for details.

`MillerUpdatingRegression` models implement the `UpdatingMultipleLinearRegression` interface, which includes methods similar to those provided by `OLSMultipleLinearRegression` for adding data. Simlarly to `StorelessUnivariateStatistics`, the contract for `UpdatingMultipleLinearRegression` is that implementations use bounded storage, so there is no limit to the number of observations streamed to them via the `addObservation` methods.

**Usage Notes:**

- Data are validated when invoking any of the newSample, newX, newY or newCovariance methods and
`IllegalArgumentException`is thrown when input data arrays do not have matching dimensions or do not contain sufficient data to estimate the model. - By default, regression models are estimated with intercept terms. In the notation above, this implies that the X matrix contains an initial row identically equal to 1. X data supplied to the newX or newSample methods should not include this column - the data loading methods will create it automatically. To estimate a model without an intercept term, set the
`noIntercept`property to true. - None of the multiple regression classes are thread-safe. If multiple threads concurrently access a single instance of one of these classes, their access to methods that add data or compute statistics must be externally synchronized.

Here are some examples.

**OLS regression**

Instantiate an OLS regression object and load a dataset:

OLSMultipleLinearRegression regression = new OLSMultipleLinearRegression(); double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0}; double[][] x = new double[6][]; x[0] = new double[]{0, 0, 0, 0, 0}; x[1] = new double[]{2.0, 0, 0, 0, 0}; x[2] = new double[]{0, 3.0, 0, 0, 0}; x[3] = new double[]{0, 0, 4.0, 0, 0}; x[4] = new double[]{0, 0, 0, 5.0, 0}; x[5] = new double[]{0, 0, 0, 0, 6.0}; regression.newSampleData(y, x);

Get regression parameters and diagnostics:

double[] beta = regression.estimateRegressionParameters(); double[] residuals = regression.estimateResiduals(); double[][] parametersVariance = regression.estimateRegressionParametersVariance(); double regressandVariance = regression.estimateRegressandVariance(); double rSquared = regression.calculateRSquared(); double sigma = regression.estimateRegressionStandardError();

**GLS regression**

Instantiate a GLS regression object and load a dataset:

GLSMultipleLinearRegression regression = new GLSMultipleLinearRegression(); double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0}; double[][] x = new double[6][]; x[0] = new double[]{0, 0, 0, 0, 0}; x[1] = new double[]{2.0, 0, 0, 0, 0}; x[2] = new double[]{0, 3.0, 0, 0, 0}; x[3] = new double[]{0, 0, 4.0, 0, 0}; x[4] = new double[]{0, 0, 0, 5.0, 0}; x[5] = new double[]{0, 0, 0, 0, 6.0}; double[][] omega = new double[6][]; omega[0] = new double[]{1.1, 0, 0, 0, 0, 0}; omega[1] = new double[]{0, 2.2, 0, 0, 0, 0}; omega[2] = new double[]{0, 0, 3.3, 0, 0, 0}; omega[3] = new double[]{0, 0, 0, 4.4, 0, 0}; omega[4] = new double[]{0, 0, 0, 0, 5.5, 0}; omega[5] = new double[]{0, 0, 0, 0, 0, 6.6}; regression.newSampleData(y, x, omega);

**Streaming regression**

Instantiate a streaming OLS regression object and load a dataset:

// Create a streaming regression with 3 regressors // and an intercept term MillerUpdatingRegression regression = new MillerUpdatingRegression(3, true); // Add one observation to the model double[] x = {1.0, 1.0, 1.0}; double y = {1.0}; instance.addObservation(x, y); // Add two observations at once double[][] xMult = {{2.0, 4.0, 5.0}, {1.4, 2.4, 2.1}}; double[] yMult = {2.0, 8.0}; instance.addObservations(xMult, yMult); // Add more observations - not stored in memory...

Get regression parameters:

RegressionResults result = regression.regress(); double[] parameters = result.getParameterEstimates();

Since this model has an intercept, `parameters[0]` is the intercept estimate. `parameters[1]`, `[2]` and `[3]` are estimates of regression coefficients for the three independent variables.

Standard errors for parameter estimates (in the same order):

double[] stdErrs = result.getStdErrorOfEstimates();

R-square, SSE, MSE:

double Rsquare = result.getRSquared(); double MSE = result.getMeanSquareError(); double SSE = result.getErrorSumSquares();

Covariance of parameters 1 and 2:

double cov = result.getCovarianceOfParameters(1, 2)

## Rank transformations

Some statistical algorithms require that input data be replaced by ranks. The org.hipparchus.stat.ranking package provides rank transformation. RankingAlgorithm defines the interface for ranking. NaturalRanking provides an implementation that has two configuration options.

- Ties strategy determines how ties in the source data are handled by the ranking
- NaN strategy determines how NaN values in the source data are handled.

Examples:

NaturalRanking ranking = new NaturalRanking(NaNStrategy.MINIMAL, TiesStrategy.MAXIMUM); double[] data = { 20, 17, 30, 42.3, 17, 50, Double.NaN, Double.NEGATIVE_INFINITY, 17 }; double[] ranks = ranking.rank(exampleData);

results in `ranks` containing `{6, 5, 7, 8, 5, 9, 2, 2, 5}.`

new NaturalRanking(NaNStrategy.REMOVED,TiesStrategy.SEQUENTIAL).rank(exampleData);

returns `{5, 2, 6, 7, 3, 8, 1, 4}.`

The default `NaNStrategy` is NaNStrategy.MAXIMAL. This makes `NaN` values larger than any other value (including `Double.POSITIVE_INFINITY`). The default `TiesStrategy` is `TiesStrategy.AVERAGE,` which assigns tied values the average of the ranks applicable to the sequence of ties. See the NaturalRanking for more examples and TiesStrategy and NaNStrategy for details on these configuration options.

## Covariance and correlation

The org.hipparchus.stat.correlation package computes covariances and correlations for pairs of arrays or columns of a matrix. Covariance computes covariances, PearsonsCorrelation provides Pearson’s Product-Moment correlation coefficients, SpearmansCorrelation computes Spearman’s rank correlation and KendallsCorrelation computes Kendall’s tau rank correlation.

**Implementation Notes**

- Unbiased covariances are given by the formula
`\(cov(X, Y) = \sum{(x_i - E(X))(y_i - E(Y))} / (n - 1)\)`where`\(E(X)\)`is the mean of X and`\(E(Y)\)`is the mean of the Y values. Non-bias-corrected estimates use n in place of`\(n - 1\)`. Whether or not covariances are bias-corrected is determined by the optional parameter, “biasCorrected,” which defaults to true. `PearsonsCorrelation`computes correlations defined by the formula`\(cor(X, Y) = \sum{(x_i - E(X))(y_i - E(Y))} / (n - 1)s(X)s(Y)\)`where`\(E(X)\)`and`\(E(Y)\)`are means of X and Y and`\(s(X)\)`,`\(s(Y)\)`are standard deviations.`SpearmansCorrelation`applies a rank transformation to the input data and computes Pearson’s correlation on the ranked data. The ranking algorithm is configurable. By default, NaturalRanking with default strategies for handling ties and NaN values is used.`KendallsCorrelation`computes the association between two measured quantities. A tau test is a non-parametric hypothesis test for statistical dependence based on the tau coefficient.

Examples:

**Covariance of 2 arrays**

To compute the unbiased covariance between 2 double arrays, `x` and `y`, use:

new Covariance().covariance(x, y)

For non-bias-corrected covariances, use

covariance(x, y, false)

**Covariance matrix**

A covariance matrix over the columns of a source matrix `data` can be computed using

new Covariance().computeCovarianceMatrix(data)

The i-jth entry of the returned matrix is the unbiased covariance of the ith and jth columns of `data.` As above, to get non-bias-corrected covariances, use

computeCovarianceMatrix(data, false)

**Pearson’s correlation of 2 arrays**

To compute the Pearson’s product-moment correlation between two double arrays `x` and `y`, use:

new PearsonsCorrelation().correlation(x, y)

**Pearson’s correlation matrix**

A (Pearson’s) correlation matrix over the columns of a source matrix `data` can be computed using

new PearsonsCorrelation().computeCorrelationMatrix(data)

The i-jth entry of the returned matrix is the Pearson’s product-moment correlation between the ith and jth columns of `data.`

**Pearson’s correlation significance and standard errors**

To compute standard errors and/or significances of correlation coefficients associated with Pearson’s correlation coefficients, start by creating a `PearsonsCorrelation` instance

PearsonsCorrelation correlation = new PearsonsCorrelation(data);

where `data` is either a rectangular array or a `RealMatrix.` Then the matrix of standard errors is

correlation.getCorrelationStandardErrors();

The formula used to compute the standard error is `\(SE_r = \sqrt{(1 - r^2) / (n - 2)}\)`

where `\(r\)` is the estimated correlation coefficient and `\(n\)` is the number of observations in the source dataset.

**p-values** for the (2-sided) null hypotheses that elements of a correlation matrix are zero populate the RealMatrix returned by

correlation.getCorrelationPValues()

`getCorrelationPValues().getEntry(i,j)` is the probability that a random variable distributed as `\(t_{n-2}\)` takes a value with absolute value greater than or equal to `\(|r_{ij}|\sqrt{(n - 2) / (1 - r_{ij}^2)}\)`, where `\(r_{ij}\)` is the estimated correlation between the ith and jth columns of the source array or RealMatrix. This is sometimes referred to as the *significance* of the coefficient.

For example, if `data` is a RealMatrix with 2 columns and 10 rows, then

new PearsonsCorrelation(data).getCorrelationPValues().getEntry(0,1)

is the significance of the Pearson’s correlation coefficient between the two columns of `data`. If this value is less than .01, we can say that the correlation between the two columns of data is significant at the 99% level.

**Spearman’s rank correlation coefficient**

To compute the Spearman’s rank-moment correlation between two double arrays `x` and `y`:

new SpearmansCorrelation().correlation(x, y)

This is equivalent to

RankingAlgorithm ranking = new NaturalRanking(); new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))

**Kendalls’s tau rank correlation coefficient**

To compute the Kendall’s tau rank correlation between two double arrays `x` and `y`:

new KendallsCorrelation().correlation(x, y)

## Statistical tests

The org.hipparchus.stat.inference package provides implementations for Student’s t, Chi-Square, G Test, One-Way ANOVA, Mann-Whitney U, Wilcoxon signed rank and Binomial test statistics as well as p-values associated with `t-`, `Chi-Square`, `G`, `One-Way ANOVA`, `Mann-Whitney U`, `Wilcoxon signed rank`, and `Kolmogorov-Smirnov` tests.

The respective test classes are TTest, ChiSquareTest, GTest, OneWayAnova, MannWhitneyUTest, WilcoxonSignedRankTest, BinomialTest and KolmogorovSmirnovTest. The InferenceTestUtils class provides static methods to get test instances or to compute test statistics directly. The examples below all use the static methods in `TestUtils` to execute tests. To get test object instances, either use e.g., `InferenceTestUtils.getTTest()` or use the implementation constructors directly, e.g. `new TTest()`.

**Implementation Notes**

- Both one- and two-sample t-tests are supported. Two sample tests can be either paired or unpaired and the unpaired two-sample tests can be conducted under the assumption of equal subpopulation variances or without this assumption. When equal variances is assumed, a pooled variance estimate is used to compute the t-statistic and the degrees of freedom used in the t-test equals the sum of the sample sizes minus 2. When equal variances is not assumed, the t-statistic uses both sample variances and the Welch-Satterwaite approximation is used to compute the degrees of freedom. Methods to return t-statistics and p-values are provided in each case, as well as boolean-valued methods to perform fixed significance level tests. The names of methods or methods that assume equal subpopulation variances always start with “homoscedastic.” Test or test-statistic methods that just start with “t” do not assume equal variances. See the examples below and the API documentation for more details.
- The validity of the p-values returned by the t-test depends on the assumptions of the parametric t-test procedure, as discussed here
- p-values returned by t-, chi-square and ANOVA tests are exact, based on numerical approximations to the t-, chi-square and F distributions in the distributions package.
- The G test implementation provides two p-values: gTest(expected, observed), which is the tail probability beyond g(expected, observed) in the ChiSquare distribution with degrees of freedom one less than the common length of input arrays and gTestIntrinsic(expected, observed) which is the same tail probability computed using a ChiSquare distribution with one less degeree of freedom.
- p-values returned by t-tests are for two-sided tests and the boolean-valued methods supporting fixed significance level tests assume that the hypotheses are two-sided. One sided tests can be performed by dividing returned p-values (resp. critical values) by 2.
- Degrees of freedom for G- and chi-square tests are integral values, based on the number of observed or expected counts (number of observed counts - 1).
- The KolmogorovSmirnov test uses a statistic based on the maximum deviation of the empirical distribution of sample data points from the distribution expected under the null hypothesis. Specifically, what is computed is Dn=supx|Fn(x)−F(x)|, where F is the expected distribution and Fn is the empirical distribution of the n sample data points. Both one-sample tests against a RealDistribution and two-sample tests (comparing two empirical distributions) are supported. For one-sample tests, the distribution of Dn is estimated using the method in Evaluating Kolmogorov’s Distribution by George Marsaglia, Wai Wan Tsang, and Jingbo Wang, with quick decisions in some cases for extreme values using the method described in Computing the Two-Sided Kolmogorov-Smirnov Distribution by Richard Simard and Pierre L’Ecuyer. In the 2-sample case, estimation by default depends on the number of data points. For small samples, the distribution is computed exactly and for large samples a numerical approximation of the Kolmogorov distribution is used. Methods to perform each type of p-value estimation are also exposed directly. See the class javadoc for details.

Examples:

**One-sample t tests**

To compare the mean of a double[] array to a fixed value:

double[] observed = {1d, 2d, 3d}; double mu = 2.5d; System.out.println(InferenceTestUtils.t(mu, observed));

The code above will display the t-statistic associated with a one-sample t-test comparing the mean of the `observed` values against `mu`.

To compare the mean of a dataset described by a StatisticalSummary to a fixed value:

double[] observed ={1d, 2d, 3d}; double mu = 2.5d; StreamingStatistics sampleStats = new StreamingStatistics(); for (int i = 0; i < observed.length; i++) { sampleStats.addValue(observed[i]); } System.out.println(TestUtils.t(mu, sampleStats));

To compute the p-value associated with the null hypothesis that the mean of a set of values equals a point estimate, against the two-sided alternative that the mean is different from the target value:

double[] observed = {1d, 2d, 3d}; double mu = 2.5d; System.out.println(TestUtils.tTest(mu, observed));

The snippet above will display the p-value associated with the null hypothesis that the mean of the population from which the `observed` values are drawn equals `mu.`

To perform the test using a fixed significance level, use:

TestUtils.tTest(mu, observed, alpha);

where `0 < alpha < 0.5` is the significance level of the test. The boolean value returned will be `true` iff the null hypothesis can be rejected with confidence `1 - alpha`. To test, for example at the 95% level of confidence, use `alpha = 0.05`

**Two-Sample t-tests**

*Example 1:* Paired test evaluating the null hypothesis that the mean difference between corresponding (paired) elements of the `double[]` arrays `sample1` and `sample2` is zero. To compute the t-statistic:

InferenceTestUtils.pairedT(sample1, sample2);

To compute the p-value:

InferenceTestUtils.pairedTTest(sample1, sample2);

To perform a fixed significance level test with alpha = .05:

InferenceTestUtils.pairedTTest(sample1, sample2, .05);

The last example will return `true` iff the p-value returned by `InferenceTestUtils.pairedTTest(sample1, sample2)` is less than `.05`

*Example 2:* unpaired, two-sided, two-sample t-test using `StatisticalSummary` instances, without assuming that subpopulation variances are equal. First create the `StatisticalSummary` instances. Both `DescriptiveStatistics` and `StreamingStatistics` implement this interface. Assume that `summary1` and `summary2` are `StreamingStatistics` instances, each of which has had at least 2 values added to the (virtual) dataset that it describes. The sample sizes do not have to be the same – all that is required is that both samples have at least 2 elements.

**Note:** The `StreamingStatistics` class does not store the dataset that it describes in memory, but it does compute all statistics necessary to perform t-tests, so this method can be used to conduct t-tests with very large samples. One-sample tests can also be performed this way. (See Descriptive statistics for details on the `StreamingStatistics` class.)

To compute the t-statistic:

InferenceTestUtils.t(summary1, summary2);

To compute the p-value:

InferenceTestUtils.tTest(sample1, sample2);

To perform a fixed significance level test with alpha = .05:

InferenceTestUtils.tTest(sample1, sample2, .05);

In each case above, the test does not assume that the subpopulation variances are equal. To perform the tests under this assumption, replace “t” at the beginning of the method name with “homoscedasticT”

**Chi-square tests**

To compute a chi-square statistic measuring the agreement between a `long[]` array of observed counts and a `double[]` array of expected counts, use:

long[] observed = {10, 9, 11}; double[] expected = {10.1, 9.8, 10.3}; System.out.println(InferenceTestUtils.chiSquare(expected, observed));

the value displayed will be `\(\sum{(expected[i] - observed[i])^2 / expected[i])}\)`

To get the p-value associated with the null hypothesis that `observed` conforms to `expected` use:

InferenceTestUtils.chiSquareTest(expected, observed);

To test the null hypothesis that `observed` conforms to `expected` with `alpha` significance level (equiv. `100 * (1-alpha)%` confidence) where `0 < alpha < 1` use:

InferenceTestUtils.chiSquareTest(expected, observed, alpha);

The boolean value returned will be `true` iff the null hypothesis can be rejected with confidence `1 - alpha`.

To compute a chi-square statistic statistic associated with a chi-square test of independence based on a two-dimensional (`long[][]`) `counts` array viewed as a two-way table, use:

InferenceTestUtils.chiSquareTest(counts);

The rows of the 2-way table are `count[0], ... , count[count.length - 1].`

The chi-square statistic returned is `sum((counts[i][j] - expected[i][j])^2/expected[i][j])` where the sum is taken over all table entries and `expected[i][j]` is the product of the row and column sums at row `i`, column `j` divided by the total count.

To compute the p-value associated with the null hypothesis that the classifications represented by the counts in the columns of the input 2-way table are independent of the rows, use:

InferenceTestUtils.chiSquareTest(counts);

To perform a chi-square test of independence with `alpha` significance level (equiv. `100 * (1-alpha)%` confidence) where `0 < alpha < 1` use:

InferenceTestUtils.chiSquareTest(counts, alpha);

The boolean value returned will be `true` iff the null hypothesis can be rejected with confidence `1 - alpha`.

**G tests**

G tests are an alternative to chi-square tests that are recommended when observed counts are small and / or incidence probabilities for some cells are small. See Ted Dunning’s paper, Accurate Methods for the Statistics of Surprise and Coincidence for background and an empirical analysis showing now chi-square statistics can be misleading in the presence of low incidence probabilities. This paper also derives the formulas used in computing G statistics and the root log likelihood ratio provided by the `GTest` class.

To compute a G-test statistic measuring the agreement between a `long[]` array of observed counts and a `double[]` array of expected counts, use:

double[] expected = new double[]{0.54d, 0.40d, 0.05d, 0.01d}; long[] observed = new long[]{70, 79, 3, 4}; System.out.println(TestUtils.g(expected, observed));

the value displayed will be `2 * sum(observed[i]) * log(observed[i]/expected[i])`

To get the p-value associated with the null hypothesis that `observed` conforms to `expected` use:

InferenceTestUtils.gTest(expected, observed);

To test the null hypothesis that `observed` conforms to `expected` with `alpha` significance level (equiv. `100 * (1-alpha)%` confidence) where `0 < alpha < 1` use:

InferenceTestUtils.gTest(expected, observed, alpha);

The boolean value returned will be `true` iff the null hypothesis can be rejected with confidence `1 - alpha`.

To evaluate the hypothesis that two sets of counts come from the same underlying distribution, use long[] arrays for the counts and `gDataSetsComparison` for the test statistic

long[] obs1 = new long[]{268, 199, 42}; long[] obs2 = new long[]{807, 759, 184}; System.out.println(InferenceTestUtils.gDataSetsComparison(obs1, obs2)); // G statistic System.out.println(InferenceTestUtils.gTestDataSetsComparison(obs1, obs2)); // p-value

For 2 x 2 designs, the `rootLogLikelihoodRatio` method computes the signed root log likelihood ratio. For example, suppose that for two events A and B, the observed count of AB (both occurring) is 5, not A and B (B without A) is 1995, A not B is 0; and neither A nor B is 10000. Then

new GTest().rootLogLikelihoodRatio(5, 1995, 0, 100000);

returns the root log likelihood associated with the null hypothesis that A and B are independent.

**One-Way ANOVA tests**

double[] classA = {93.0, 103.0, 95.0, 101.0, 91.0, 105.0, 96.0, 94.0, 101.0 }; double[] classB = {99.0, 92.0, 102.0, 100.0, 102.0, 89.0 }; double[] classC = {110.0, 115.0, 111.0, 117.0, 128.0, 117.0 }; List classes = new ArrayList(); classes.add(classA); classes.add(classB); classes.add(classC);

Then you can compute ANOVA F- or p-values associated with the null hypothesis that the class means are all the same using a `OneWayAnova` instance or `TestUtils` methods:

double fStatistic = InferenceTestUtils.oneWayAnovaFValue(classes); // F-value double pValue = InferenceTestUtils.oneWayAnovaPValue(classes); // P-value

To test perform a One-Way ANOVA test with significance level set at 0.01 (so the test will, assuming assumptions are met, reject the null hypothesis incorrectly only about one in 100 times), use

InferenceTestUtils.oneWayAnovaTest(classes, 0.01); // returns a boolean // true means reject null hypothesis

**Kolmogorov-Smirnov tests**

Given a double[] array `data` of values, to evaluate the null hypothesis that the values are drawn from a unit normal distribution

final NormalDistribution unitNormal = new NormalDistribution(0d, 1d); InferenceTestUtils.kolmogorovSmirnovTest(unitNormal, sample, false)

returns the p-value and

InferenceTestUtils.kolmogorovSmirnovStatistic(unitNormal, sample)

returns the D-statistic.

If `y` is a double array, to evaluate the null hypothesis that `x` and `y` are drawn from the same underlying distribution, use

InferenceTestUtils.kolmogorovSmirnovStatistic(x, y)

to compute the D-statistic and

InferenceTestUtils.kolmogorovSmirnovTest(x, y)

for the p-value associated with the null hypothesis that `x` and `y` come from the same distribution. By default, here and above strict inequality is used in the null hypothesis - i.e., we evaluate `\(H_0 : D_{n,m} > d\)`. To make the inequality above non-strict, add `false` as an actual parameter above. For large samples, this parameter makes no difference. When the product of the sample sizes is less than 10,000, `KolmogorovSmirnnov` computes p-values exactly; otherwise the Kolmogorov approximation to the distribution of the D statistic is used. To force exact computation of the p-value (overriding the selection of estimation method), first compute the d-statistic and then use the `exactP` method

final double d = TestUtils.kolmogorovSmirnovStatistic(x, y); TestUtils.exactP(d, x.length, y.length, false)

assuming that the non-strict form of the null hypothesis is desired. Note, however, that exact computation for large samples takes a long time.

When there are ties in the data in a 2-sample Kolmogorov-Smirnov test, the p-value is strictly speaking undefined. If the combined sample size is less than 10,000 and there are ties in the data, random jitter is by default added to break the ties. If ties are known to be present in the data, the `bootstrap` method may be used as an alternative for estimating the p-value. See the javadoc for details on the p-value estimation algorithms used and how they are selected.