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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
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*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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package org.apache.commons.statistics.inference;
import java.util.Arrays;
import java.util.Objects;
import java.util.function.Consumer;
import java.util.function.DoublePredicate;
import java.util.function.DoubleUnaryOperator;
import java.util.function.IntToDoubleFunction;
import org.apache.commons.numbers.combinatorics.LogBinomialCoefficient;
import org.apache.commons.statistics.inference.BrentOptimizer.PointValuePair;
/**
* Implements an unconditioned exact test for a contingency table.
*
* <p>Performs an exact test for the statistical significance of the association (contingency)
* between two kinds of categorical classification. A 2x2 contingency table is:
*
* <p>\[ \left[ {\begin{array}{cc}
* a &amp; b \\
* c &amp; d \\
* \end{array} } \right] \]
*
* <p>This test applies to the case of a 2x2 contingency table with one margin fixed. Note that
* if both margins are fixed (the row sums and column sums are not random)
* then Fisher's exact test can be applied.
*
* <p>This implementation fixes the column sums \( m = a + c \) and \( n = b + d \).
* All possible tables can be created using \( 0 \le a \le m \) and \( 0 \le b \le n \).
* The random values \( a \) and \( b \) follow a binomial distribution with probabilities
* \( p_0 \) and \( p_1 \) such that \( a \sim B(m, p_0) \) and \( b \sim B(n, p_1) \).
* The p-value of the 2x2 table is the product of two binomials:
*
* <p>\[ \begin{aligned}
* p &amp;= Pr(a; m, p_0) \times Pr(b; n, p_1) \\
* &amp;= \binom{m}{a} p_0^a (1-p_0)^{m-a} \times \binom{n}{b} p_1^b (1-p_1)^{n-b} \end{aligned} \]
*
* <p>For the binomial model, the null hypothesis is the two nuisance parameters are equal
* \( p_0 = p_1 = \pi\), with \( \pi \) the probability for equal proportions, and the probability
* of any single table is:
*
* <p>\[ p = \binom{m}{a} \binom{n}{b} \pi^{a+b} (1-\pi)^{m+n-a-b} \]
*
* <p>The p-value of the observed table is calculated by maximising the sum of the as or more
* extreme tables over the domain of the nuisance parameter \( 0 \lt \pi \lt 1 \):
*
* <p>\[ p(a, b) = \sum_{i,j} \binom{m}{i} \binom{n}{j} \pi^{i+j} (1-\pi)^{m+n-i-j} \]
*
* <p>where table \( (i,j) \) is as or more extreme than the observed table \( (a, b) \). The test
* can be configured to select more extreme tables using various {@linkplain Method methods}.
*
* <p>Note that the sum of the joint binomial distribution is a univariate function for
* the nuisance parameter \( \pi \). This function may have many local maxima and the
* search enumerates the range with a configured {@linkplain #withInitialPoints(int)
* number of points}. The best candidates are optionally used as the start point for an
* {@linkplain #withOptimize(boolean) optimized} search for a local maxima.
*
* <p>References:
* <ol>
* <li>
* Barnard, G.A. (1947).
* <a href="https://doi.org/10.1093/biomet/34.1-2.123">Significance tests for 2x2 tables.</a>
* Biometrika, 34, Issue 1-2, 123–138.
* <li>
* Boschloo, R.D. (1970).
* <a href="https://doi.org/10.1111/j.1467-9574.1970.tb00104.x">Raised conditional level of
* significance for the 2 × 2-table when testing the equality of two probabilities.</a>
* Statistica neerlandica, 24(1), 1–9.
* <li>
* Suisaa, A and Shuster, J.J. (1985).
* <a href="https://doi.org/10.2307/2981892">Exact Unconditional Sample Sizes
* for the 2 × 2 Binomial Trial.</a>
* Journal of the Royal Statistical Society. Series A (General), 148(4), 317-327.
* </ol>
*
* @see FisherExactTest
* @see <a href="https://en.wikipedia.org/wiki/Boschloo%27s_test">Boschloo&#39;s test (Wikipedia)</a>
* @see <a href="https://en.wikipedia.org/wiki/Barnard%27s_test">Barnard&#39;s test (Wikipedia)</a>
* @since 1.1
*/
public final class UnconditionedExactTest {
/**
* Default instance.
*
* <p>SciPy's boschloo_exact and barnard_exact tests use 32 points in the interval [0,
* 1) The R Exact package uses 100 in the interval [1e-5, 1-1e-5]. Barnards 1947 paper
* describes the nuisance parameter in the open interval {@code 0 < pi < 1}. Here we
* respect the open-interval for the initial candidates and ignore 0 and 1. The
* initial bounds used are the same as the R Exact package. We closely match the inner
* 31 points from SciPy by using 33 points by default.
*/
private static final UnconditionedExactTest DEFAULT = new UnconditionedExactTest(
AlternativeHypothesis.TWO_SIDED, Method.BOSCHLOO, 33, true);
/** Lower bound for the enumerated interval. The upper bound is {@code 1 - lower}. */
private static final double LOWER_BOUND = 1e-5;
/** Relative epsilon for the Brent solver. This is limited for a univariate function
* to approximately sqrt(eps) with eps = 2^-52. */
private static final double SOLVER_RELATIVE_EPS = 1.4901161193847656E-8;
/** Fraction of the increment (interval between enumerated points) to initialise the bracket
* for the minima. Note the minima should lie between x +/- increment. The bracket should
* search within this range. Set to 1/8 and so the initial point of the bracket is
* approximately 1.61 * 1/8 = 0.2 of the increment away from initial points a or b. */
private static final double INC_FRACTION = 0.125;
/** Maximum number of candidate to optimize. This is a safety limit to avoid excess
* optimization. Only candidates within a relative tolerance of the best candidate are
* stored. If the number of candidates exceeds this value then many candidates have a
* very similar p-value and the top candidates will be optimized. Using a value of 3
* allows at least one other candidate to be optimized when there is two-fold
* symmetry in the energy function. */
private static final int MAX_CANDIDATES = 3;
/** Relative distance of candidate minima from the lowest candidate. Used to exclude
* poor candidates from optimization. */
private static final double MINIMA_EPS = 0.02;
/** The maximum number of tables. This is limited by the maximum number of indices that
* can be maintained in memory. Potentially up to this number of tables must be tracked
* during computation of the p-value for as or more extreme tables. The limit is set
* using the same limit for maximum capacity as java.util.ArrayList. In practice any
* table anywhere near this limit can be computed using an alternative such as a chi-squared
* or g test. */
private static final int MAX_TABLES = Integer.MAX_VALUE - 8;
/** Error message text for zero column sums. */
private static final String COLUMN_SUM = "Column sum";
/** Alternative hypothesis. */
private final AlternativeHypothesis alternative;
/** Method to identify more extreme tables. */
private final Method method;
/** Number of initial points. */
private final int points;
/** Option to optimize the best initial point(s). */
private final boolean optimize;
/**
* Define the method to determine the more extreme tables.
*
* @since 1.1
*/
public enum Method {
/**
* Uses the test statistic from a Z-test using a pooled variance.
*
* <p>\[ T(X) = \frac{\hat{p}_0 - \hat{p}_1}{\sqrt{\hat{p}(1 - \hat{p}) (\frac{1}{m} + \frac{1}{n})}} \]
*
* <p>where \( \hat{p}_0 = a / m \), \( \hat{p}_1 = b / n \), and
* \( \hat{p} = (a+b) / (m+n) \) are the estimators of \( p_0 \), \( p_1 \) and the
* pooled probability \( p \) assuming \( p_0 = p_1 \).
*
* <p>The more extreme tables are identified using the {@link AlternativeHypothesis}:
* <ul>
* <li>greater: \( T(X) \ge T(X_0) \)
* <li>less: \( T(X) \le T(X_0) \)
* <li>two-sided: \( | T(X) | \ge | T(X_0) | \)
* </ul>
*
* <p>The use of the Z statistic was suggested by Suissa and Shuster (1985).
* This method is uniformly more powerful than Fisher's test for balanced designs
* (\( m = n \)).
*/
Z_POOLED,
/**
* Uses the test statistic from a Z-test using an unpooled variance.
*
* <p>\[ T(X) = \frac{\hat{p}_0 - \hat{p}_1}
* {\sqrt{ \frac{\hat{p}_0(1 - \hat{p}_0)}{m} + \frac{\hat{p}_1(1 - \hat{p}_1)}{n}} } \]
*
* <p>where \( \hat{p}_0 = a / m \) and \( \hat{p}_1 = b / n \).
*
* <p>The more extreme tables are identified using the {@link AlternativeHypothesis} as
* per the {@link #Z_POOLED} method.
*/
Z_UNPOOLED,
/**
* Uses the p-value from Fisher's exact test. This is also known as Boschloo's test.
*
* <p>The p-value for Fisher's test is computed using using the
* {@link AlternativeHypothesis}. The more extreme tables are identified using
* \( p(X) \le p(X_0) \).
*
* <p>This method is always uniformly more powerful than Fisher's test.
*
* @see FisherExactTest
*/
BOSCHLOO;
}
/**
* Result for the unconditioned exact test.
*
* <p>This class is immutable.
*
* @since 1.1
*/
public static final class Result extends BaseSignificanceResult {
/** Nuisance parameter. */
private final double pi;
/**
* Create an instance where all tables are more extreme, i.e. the p-value
* is 1.0.
*
* @param statistic Test statistic.
*/
Result(double statistic) {
super(statistic, 1);
this.pi = 0.5;
}
/**
* @param statistic Test statistic.
* @param pi Nuisance parameter.
* @param p Result p-value.
*/
Result(double statistic, double pi, double p) {
super(statistic, p);
this.pi = pi;
}
/**
* {@inheritDoc}
*
* <p>The value of the statistic is dependent on the {@linkplain Method method}
* used to determine the more extreme tables.
*/
@Override
public double getStatistic() {
// Note: This method is here for documentation
return super.getStatistic();
}
/**
* Gets the nuisance parameter that maximised the probability sum of the as or more
* extreme tables.
*
* @return the nuisance parameter.
*/
public double getNuisanceParameter() {
return pi;
}
}
/**
* An expandable list of (x,y) values. This allows tracking 2D positions stored as
* a single index.
*/
private static class XYList {
/** The maximum size of array to allocate. */
private final int max;
/** Width, or maximum x value (exclusive). */
private final int width;
/** The size of the list. */
private int size;
/** The list data. */
private int[] data = new int[10];
/**
* Create an instance. It is assumed that (maxx+1)*(maxy+1) does not exceed the
* capacity of an array.
*
* @param maxx Maximum x-value (inclusive).
* @param maxy Maximum y-value (inclusive).
*/
XYList(int maxx, int maxy) {
this.width = maxx + 1;
this.max = width * (maxy + 1);
}
/**
* Gets the width.
* (x, y) values are stored using y * width + x.
*
* @return the width
*/
int getWidth() {
return width;
}
/**
* Gets the maximum X value (inclusive).
*
* @return the max X
*/
int getMaxX() {
return width - 1;
}
/**
* Gets the maximum Y value (inclusive).
*
* @return the max Y
*/
int getMaxY() {
return max / width - 1;
}
/**
* Adds the value to the list.
*
* @param x X value.
* @param y Y value.
*/
void add(int x, int y) {
if (size == data.length) {
// Overflow safe doubling of the current size.
data = Arrays.copyOf(data, (int) Math.min(max, size * 2L));
}
data[size++] = width * y + x;
}
/**
* Gets the 2D index at the specified {@code index}.
* The index is y * width + x:
* <pre>
* x = index % width
* y = index / width
* </pre>
*
* @param index Element index.
* @return the 2D index
*/
int get(int index) {
return data[index];
}
/**
* Gets the number of elements in the list.
*
* @return the size
*/
int size() {
return size;
}
/**
* Checks if the list size is zero.
*
* @return true if empty
*/
boolean isEmpty() {
return size == 0;
}
/**
* Checks if the list is the maximum capacity.
*
* @return true if full
*/
boolean isFull() {
return size == max;
}
}
/**
* A container of (key,value) pairs to store candidate minima. Encapsulates the
* logic of storing multiple initial search points for optimization.
*
* <p>Stores all pairs within a relative tolerance of the lowest minima up to a set
* capacity. When at capacity the worst candidate is replaced by addition of a
* better candidate.
*
* <p>Special handling is provided to store only a single NaN value if no non-NaN
* values have been observed. This prevents storing a large number of NaN
* candidates.
*/
static class Candidates {
/** The maximum size of array to allocate. */
private final int max;
/** Relative distance from lowest candidate. */
private final double eps;
/** Candidate (key,value) pairs. */
private double[][] data;
/** Current size of the list. */
private int size;
/** Current minimum. */
private double min = Double.POSITIVE_INFINITY;
/** Current threshold for inclusion. */
private double threshold = Double.POSITIVE_INFINITY;
/**
* Create an instance.
*
* @param max Maximum number of allowed candidates (limited to at least 1).
* @param eps Relative distance of candidate minima from the lowest candidate
* (assumed to be positive and finite).
*/
Candidates(int max, double eps) {
this.max = Math.max(1, max);
this.eps = eps;
// Create the initial storage
data = new double[Math.min(this.max, 4)][];
}
/**
* Adds the (key, value) pair.
*
* @param k Key.
* @param v Value.
*/
void add(double k, double v) {
// Store only a single NaN
if (Double.isNaN(v)) {
if (size == 0) {
// No requirement to check capacity
data[size++] = new double[] {k, v};
}
return;
}
// Here values are non-NaN.
// If higher then do not store.
if (v > threshold) {
return;
}
// Check if lower than the current minima.
if (v < min) {
min = v;
// Get new threshold
threshold = v + Math.abs(v) * eps;
// Remove existing entries above the threshold
int s = 0;
for (int i = 0; i < size; i++) {
// This will filter NaN values
if (data[i][1] <= threshold) {
data[s++] = data[i];
}
}
size = s;
// Caution: This does not clear stale data
// by setting all values in [newSize, oldSize) = null
}
addPair(k, v);
}
/**
* Add the (key, value) pair to the data.
* It is assumed the data satisfy the conditions for addition.
*
* @param k Key.
* @param v Value.
*/
private void addPair(double k, double v) {
if (size == data.length) {
if (size == max) {
// At capacity.
replaceWorst(k, v);
return;
}
// Expand
data = Arrays.copyOfRange(data, 0, (int) Math.min(max, size * 2L));
}
data[size++] = new double[] {k, v};
}
/**
* Replace the worst candidate.
*
* @param k Key.
* @param v Value.
*/
private void replaceWorst(double k, double v) {
// Note: This only occurs when NaN values have been removed by addition
// of non-NaN values.
double[] worst = data[0];
for (int i = 1; i < size; i++) {
if (worst[1] < data[i][1]) {
worst = data[i];
}
}
worst[0] = k;
worst[1] = v;
}
/**
* Return the minimum (key,value) pair.
*
* @return the minimum (or null)
*/
double[] getMinimum() {
// This will handle size=0 as data[0] will be null
double[] best = data[0];
for (int i = 1; i < size; i++) {
if (best[1] > data[i][1]) {
best = data[i];
}
}
return best;
}
/**
* Perform the given action for each (key, value) pair.
*
* @param action Action.
*/
void forEach(Consumer<double[]> action) {
for (int i = 0; i < size; i++) {
action.accept(data[i]);
}
}
}
/**
* Compute the statistic for Boschloo's test.
*/
private interface BoschlooStatistic {
/**
* Compute Fisher's p-value for the 2x2 contingency table with the observed
* value {@code x} in position [0][0]. Note that the table margins are fixed
* and are defined by the population size, number of successes and sample
* size of the specified hypergeometric distribution.
*
* @param dist Hypergeometric distribution.
* @param x Value.
* @return Fisher's p-value
*/
double value(Hypergeom dist, int x);
}
/**
* @param alternative Alternative hypothesis.
* @param method Method to identify more extreme tables.
* @param points Number of initial points.
* @param optimize Option to optimize the best initial point(s).
*/
private UnconditionedExactTest(AlternativeHypothesis alternative,
Method method,
int points,
boolean optimize) {
this.alternative = alternative;
this.method = method;
this.points = points;
this.optimize = optimize;
}
/**
* Return an instance using the default options.
*
* <ul>
* <li>{@link AlternativeHypothesis#TWO_SIDED}
* <li>{@link Method#BOSCHLOO}
* <li>{@linkplain #withInitialPoints(int) points = 33}
* <li>{@linkplain #withOptimize(boolean) optimize = true}
* </ul>
*
* @return default instance
*/
public static UnconditionedExactTest withDefaults() {
return DEFAULT;
}
/**
* Return an instance with the configured alternative hypothesis.
*
* @param v Value.
* @return an instance
*/
public UnconditionedExactTest with(AlternativeHypothesis v) {
return new UnconditionedExactTest(Objects.requireNonNull(v), method, points, optimize);
}
/**
* Return an instance with the configured method.
*
* @param v Value.
* @return an instance
*/
public UnconditionedExactTest with(Method v) {
return new UnconditionedExactTest(alternative, Objects.requireNonNull(v), points, optimize);
}
/**
* Return an instance with the configured number of initial points.
*
* <p>The search for the nuisance parameter will use \( v \) points in the open interval
* \( (0, 1) \). The interval is evaluated by including start and end points approximately
* equal to 0 and 1. Additional internal points are enumerated using increments of
* approximately \( \frac{1}{v-1} \). The minimum number of points is 2. Increasing the
* number of points increases the precision of the search at the cost of performance.
*
* <p>To approximately double the number of points so that all existing points are included
* and additional points half-way between them are sampled requires using {@code 2p - 1}
* where {@code p} is the existing number of points.
*
* @param v Value.
* @return an instance
* @throws IllegalArgumentException if the value is {@code < 2}.
*/
public UnconditionedExactTest withInitialPoints(int v) {
if (v <= 1) {
throw new InferenceException(InferenceException.X_LT_Y, v, 2);
}
return new UnconditionedExactTest(alternative, method, v, optimize);
}
/**
* Return an instance with the configured optimization of initial search points.
*
* <p>If enabled then the initial point(s) with the highest probability is/are used as the start
* for an optimization to find a local maxima.
*
* @param v Value.
* @return an instance
* @see #withInitialPoints(int)
*/
public UnconditionedExactTest withOptimize(boolean v) {
return new UnconditionedExactTest(alternative, method, points, v);
}
/**
* Compute the statistic for the unconditioned exact test. The statistic returned
* depends on the configured {@linkplain Method method}.
*
* @param table 2-by-2 contingency table.
* @return test statistic
* @throws IllegalArgumentException if the {@code table} is not a 2-by-2 table; any
* table entry is negative; any column sum is zero; the table sum is zero or not an
* integer; or the number of possible tables exceeds the maximum array capacity.
* @see #with(Method)
* @see #test(int[][])
*/
public double statistic(int[][] table) {
checkTable(table);
final int a = table[0][0];
final int b = table[0][1];
final int c = table[1][0];
final int d = table[1][1];
final int m = a + c;
final int n = b + d;
switch (method) {
case Z_POOLED:
return statisticZ(a, b, m, n, true);
case Z_UNPOOLED:
return statisticZ(a, b, m, n, false);
case BOSCHLOO:
return statisticBoschloo(a, b, m, n);
default:
throw new IllegalStateException(String.valueOf(method));
}
}
/**
* Performs an unconditioned exact test on the 2-by-2 contingency table. The statistic and
* p-value returned depends on the configured {@linkplain Method method} and
* {@linkplain AlternativeHypothesis alternative hypothesis}.
*
* <p>The search for the nuisance parameter that maximises the p-value can be configured to:
* start with a number of {@linkplain #withInitialPoints(int) initial points}; and
* {@linkplain #withOptimize(boolean) optimize} the best points.
*
* @param table 2-by-2 contingency table.
* @return test result
* @throws IllegalArgumentException if the {@code table} is not a 2-by-2 table; any
* table entry is negative; any column sum is zero; the table sum is zero or not an
* integer; or the number of possible tables exceeds the maximum array capacity.
* @see #with(Method)
* @see #with(AlternativeHypothesis)
* @see #statistic(int[][])
*/
public Result test(int[][] table) {
checkTable(table);
final int a = table[0][0];
final int b = table[0][1];
final int c = table[1][0];
final int d = table[1][1];
final int m = a + c;
final int n = b + d;
// Used to track more extreme tables
final XYList tableList = new XYList(m, n);
final double statistic = findExtremeTables(a, b, tableList);
if (tableList.isEmpty() || tableList.isFull()) {
// All possible tables are more extreme, e.g. a two-sided test where the
// z-statistic is zero.
return new Result(statistic);
}
final double[] opt = computePValue(tableList);
return new Result(statistic, opt[0], opt[1]);
}
/**
* Find all tables that are as or more extreme than the observed table.
*
* <p>If the list of tables is full then all tables are more extreme.
* Some configurations can detect this without performing a search
* and in this case the list of tables is returned as empty.
*
* @param a Observed value for a.
* @param b Observed value for b.
* @param tableList List to track more extreme tables.
* @return the test statistic
*/
private double findExtremeTables(int a, int b, XYList tableList) {
final int m = tableList.getMaxX();
final int n = tableList.getMaxY();
switch (method) {
case Z_POOLED:
return findExtremeTablesZ(a, b, m, n, true, tableList);
case Z_UNPOOLED:
return findExtremeTablesZ(a, b, m, n, false, tableList);
case BOSCHLOO:
return findExtremeTablesBoschloo(a, b, m, n, tableList);
default:
throw new IllegalStateException(String.valueOf(method));
}
}
/**
* Compute the statistic from a Z-test.
*
* @param a Observed value for a.
* @param b Observed value for b.
* @param m Column sum m.
* @param n Column sum n.
* @param pooled true to use a pooled variance.
* @return z
*/
private static double statisticZ(int a, int b, int m, int n, boolean pooled) {
final double p0 = (double) a / m;
final double p1 = (double) b / n;
// Avoid NaN generation 0 / 0 when the variance is 0
if (p0 != p1) {
double variance;
if (pooled) {
// Integer sums will not overflow
final double p = (double) (a + b) / (m + n);
variance = p * (1 - p) * (1.0 / m + 1.0 / n);
} else {
variance = p0 * (1 - p0) / m + p1 * (1 - p1) / n;
}
return (p0 - p1) / Math.sqrt(variance);
}
return 0;
}
/**
* Find all tables that are as or more extreme than the observed table using the Z statistic.
*
* @param a Observed value for a.
* @param b Observed value for b.
* @param m Column sum m.
* @param n Column sum n.
* @param pooled true to use a pooled variance.
* @param tableList List to track more extreme tables.
* @return observed z
*/
private double findExtremeTablesZ(int a, int b, int m, int n, boolean pooled, XYList tableList) {
final double statistic = statisticZ(a, b, m, n, pooled);
// Identify more extreme tables using the alternate hypothesis
DoublePredicate test;
if (alternative == AlternativeHypothesis.GREATER_THAN) {
test = z -> z >= statistic;
} else if (alternative == AlternativeHypothesis.LESS_THAN) {
test = z -> z <= statistic;
} else {
// two-sided
if (statistic == 0) {
// Early exit: all tables are as extreme
return 0;
}
final double za = Math.abs(statistic);
test = z -> Math.abs(z) >= za;
}
// Precompute factors
final double mn = (double) m + n;
final double norm = 1.0 / m + 1.0 / n;
double z;
// Process all possible tables
for (int i = 0; i <= m; i++) {
final double p0 = (double) i / m;
final double vp0 = p0 * (1 - p0) / m;
for (int j = 0; j <= n; j++) {
final double p1 = (double) j / n;
// Avoid NaN generation 0 / 0 when the variance is 0
if (p0 == p1) {
z = 0;
} else {
double variance;
if (pooled) {
// Integer sums will not overflow
final double p = (i + j) / mn;
variance = p * (1 - p) * norm;
} else {
variance = vp0 + p1 * (1 - p1) / n;
}
z = (p0 - p1) / Math.sqrt(variance);
}
if (test.test(z)) {
tableList.add(i, j);
}
}
}
return statistic;
}
/**
* Compute the statistic using Fisher's p-value (also known as Boschloo's test).
*
* @param a Observed value for a.
* @param b Observed value for b.
* @param m Column sum m.
* @param n Column sum n.
* @return p-value
*/
private double statisticBoschloo(int a, int b, int m, int n) {
final int nn = m + n;
final int k = a + b;
// Re-use the cached Hypergeometric implementation to allow the value
// to be identical for the statistic and test methods.
final Hypergeom dist = new Hypergeom(nn, k, m);
if (alternative == AlternativeHypothesis.GREATER_THAN) {
return dist.sf(a - 1);
} else if (alternative == AlternativeHypothesis.LESS_THAN) {
return dist.cdf(a);
}
// two-sided: Find all i where Pr(X = i) <= Pr(X = a) and sum them.
return statisticBoschlooTwoSided(dist, a);
}
/**
* Compute the two-sided statistic using Fisher's p-value (also known as Boschloo's test).
*
* @param distribution Hypergeometric distribution.
* @param k Observed value.
* @return p-value
*/
private static double statisticBoschlooTwoSided(Hypergeom distribution, int k) {
// two-sided: Find all i where Pr(X = i) <= Pr(X = k) and sum them.
// Logic is the same as FisherExactTest but using the probability (PMF), which
// is cached, rather than the logProbability.
final double pk = distribution.pmf(k);
final int m1 = distribution.getLowerMode();
final int m2 = distribution.getUpperMode();
if (k < m1) {
// Lower half = cdf(k)
// Find upper half. As k < lower mode i should never
// reach the lower mode based on the probability alone.
// Bracket with the upper mode.
final int i = Searches.searchDescending(m2, distribution.getSupportUpperBound(), pk,
distribution::pmf);
return distribution.cdf(k) +
distribution.sf(i - 1);
} else if (k > m2) {
// Upper half = sf(k - 1)
// Find lower half. As k > upper mode i should never
// reach the upper mode based on the probability alone.
// Bracket with the lower mode.
final int i = Searches.searchAscending(distribution.getSupportLowerBound(), m1, pk,
distribution::pmf);
return distribution.cdf(i) +
distribution.sf(k - 1);
}
// k == mode
// Edge case where the sum of probabilities will be either
// 1 or 1 - Pr(X = mode) where mode != k
final double pm = distribution.pmf(k == m1 ? m2 : m1);
return pm > pk ? 1 - pm : 1;
}
/**
* Find all tables that are as or more extreme than the observed table using the
* Fisher's p-value as the statistic (also known as Boschloo's test).
*
* @param a Observed value for a.
* @param b Observed value for b.
* @param m Column sum m.
* @param n Column sum n.
* @param tableList List to track more extreme tables.
* @return observed p-value
*/
private double findExtremeTablesBoschloo(int a, int b, int m, int n, XYList tableList) {
final double statistic = statisticBoschloo(a, b, m, n);
// Function to compute the statistic
BoschlooStatistic func;
if (alternative == AlternativeHypothesis.GREATER_THAN) {
func = (dist, x) -> dist.sf(x - 1);
} else if (alternative == AlternativeHypothesis.LESS_THAN) {
func = Hypergeom::cdf;
} else {
func = UnconditionedExactTest::statisticBoschlooTwoSided;
}
// All tables are: 0 <= i <= m by 0 <= j <= n
// Diagonal (upper-left to lower-right) strips of the possible
// tables use the same hypergeometric distribution
// (i.e. i+j == number of successes). To enumerate all requires
// using the full range of all distributions: 0 <= i+j <= m+n.
// Note the column sum m is fixed.
final int mn = m + n;
for (int k = 0; k <= mn; k++) {
final Hypergeom dist = new Hypergeom(mn, k, m);
final int lo = dist.getSupportLowerBound();
final int hi = dist.getSupportUpperBound();
for (int i = lo; i <= hi; i++) {
if (func.value(dist, i) <= statistic) {
// j = k - i
tableList.add(i, k - i);
}
}
}
return statistic;
}
/**
* Compute the nuisance parameter and p-value for the binomial model given the list
* of possible tables.
*
* <p>The current method enumerates an initial set of points and stores local
* extrema as candidates. Any candidate within 2% of the best is optionally
* optimized; this is limited to the top 3 candidates. These settings
* could be exposed as configurable options. Currently only the choice to optimize
* or not is exposed.
*
* @param tableList List of tables.
* @return [nuisance parameter, p-value]
*/
private double[] computePValue(XYList tableList) {
final DoubleUnaryOperator func = createBinomialModel(tableList);
// Enumerate the range [LOWER, 1-LOWER] and save the best points for optimization
final Candidates minima = new Candidates(MAX_CANDIDATES, MINIMA_EPS);
final int n = points - 1;
final double inc = (1.0 - 2 * LOWER_BOUND) / n;
// Moving window of 3 values to identify minima.
// px holds the position of the previous evaluated point.
double v2 = 0;
double v3 = func.applyAsDouble(LOWER_BOUND);
double px = LOWER_BOUND;
for (int i = 1; i < n; i++) {
final double x = LOWER_BOUND + i * inc;
final double v1 = v2;
v2 = v3;
v3 = func.applyAsDouble(x);
addCandidate(minima, v1, v2, v3, px);
px = x;
}
// Add the upper bound
final double x = 1 - LOWER_BOUND;
final double vn = func.applyAsDouble(x);
addCandidate(minima, v2, v3, vn, px);
addCandidate(minima, v3, vn, 0, x);
final double[] min = minima.getMinimum();
// Optionally optimize the best point(s) (if not already optimal)
if (optimize && min[1] > -1) {
final BrentOptimizer opt = new BrentOptimizer(SOLVER_RELATIVE_EPS, Double.MIN_VALUE);
final BracketFinder bf = new BracketFinder();
minima.forEach(candidate -> {
double a = candidate[0];
double fa;
// Attempt to bracket the minima. Use an initial second point placed relative to
// the size of the interval: [x - increment, x + increment].
// if a < 0.5 then add a small delta ; otherwise subtract the delta.
final double b = a - Math.copySign(inc * INC_FRACTION, a - 0.5);
if (bf.search(func, a, b, 0, 1)) {
// The bracket a < b < c must have f(b) < min(f(a), f(b))
final PointValuePair p = opt.optimize(func, bf.getLo(), bf.getHi(), bf.getMid(), bf.getFMid());
a = p.getPoint();
fa = p.getValue();
} else {
// Mid-point is at one of the bounds (i.e. is 0 or 1)
a = bf.getMid();
fa = bf.getFMid();
}
if (fa < min[1]) {
min[0] = a;
min[1] = fa;
}
});
}
// Reverse the sign of the p-value to create a maximum.
// Note that due to the summation the p-value can be above 1 so we clip the final result.
// Note: Apply max then reverse sign. This will pass through spurious NaN values if
// the p-value computation produced all NaNs.
min[1] = -Math.max(-1, min[1]);
return min;
}
/**
* Creates the binomial model p-value function for the nuisance parameter.
* Note: This function computes the negative p-value so is suitable for
* optimization by a search for a minimum.
*
* @param tableList List of tables.
* @return the function
*/
private static DoubleUnaryOperator createBinomialModel(XYList tableList) {
final int m = tableList.getMaxX();
final int n = tableList.getMaxY();
final int mn = m + n;
// Compute the probability using logs
final double[] c = new double[tableList.size()];
final int[] ij = new int[tableList.size()];
final int width = tableList.getWidth();
// Compute the log binomial dynamically for a small number of values
IntToDoubleFunction binomM;
IntToDoubleFunction binomN;
if (tableList.size() < mn) {
binomM = k -> LogBinomialCoefficient.value(m, k);
binomN = k -> LogBinomialCoefficient.value(n, k);
} else {
// Pre-compute all values
binomM = createLogBinomialCoefficients(m);
binomN = m == n ? binomM : createLogBinomialCoefficients(n);
}
// Handle special cases i+j == 0 and i+j == m+n.
// These will occur only once, if at all. Mark if they occur.
int flag = 0;
int j = 0;
for (int i = 0; i < c.length; i++) {
final int index = tableList.get(i);
final int x = index % width;
final int y = index / width;
final int xy = x + y;
if (xy == 0) {
flag |= 1;
} else if (xy == mn) {
flag |= 2;
} else {
ij[j] = xy;
c[j] = binomM.applyAsDouble(x) + binomN.applyAsDouble(y);
j++;
}
}
final int size = j;
final boolean ij0 = (flag & 1) != 0;
final boolean ijmn = (flag & 2) != 0;
return pi -> {
final double logp = Math.log(pi);
final double log1mp = Math.log1p(-pi);
double sum = 0;
for (int i = 0; i < size; i++) {
// binom(m, i) * binom(n, j) * pi^(i+j) * (1-pi)^(m+n-i-j)
sum += Math.exp(ij[i] * logp + (mn - ij[i]) * log1mp + c[i]);
}
// Add the simplified terms where the binomial is 1.0 and one power is x^0 == 1.0.
// This avoids 0 * log(x) generating NaN when x is 0 in the case where pi was 0 or 1.
// Reuse exp (not pow) to support pi approaching 0 or 1.
if (ij0) {
// pow(1-pi, mn)
sum += Math.exp(mn * log1mp);
}
if (ijmn) {
// pow(pi, mn)
sum += Math.exp(mn * logp);
}
// The optimizer minimises the function so this returns -p.
return -sum;
};
}
/**
* Create the natural logarithm of the binomial coefficient for all {@code k = [0, n]}.
*
* @param n Limit N.
* @return ln binom(n, k)
*/
private static IntToDoubleFunction createLogBinomialCoefficients(int n) {
final double[] binom = new double[n + 1];
// Exploit symmetry.
// ignore: binom(n, 0) == binom(n, n) == 1
int j = n - 1;
for (int i = 1; i <= j; i++, j--) {
binom[i] = binom[j] = LogBinomialCoefficient.value(n, i);
}
return k -> binom[k];
}
/**
* Add point 2 to the list of minima if neither neighbour value is lower.
* <pre>
* !(v1 < v2 || v3 < v2)
* </pre>
*
* @param minima Candidate minima.
* @param v1 First point function value.
* @param v2 Second point function value.
* @param v3 Third point function value.
* @param x2 Second point.
*/
private void addCandidate(Candidates minima, double v1, double v2, double v3, double x2) {
final double min = v1 < v3 ? v1 : v3;
if (min < v2) {
// Lower neighbour(s)
return;
}
// Add the candidate. This could be NaN but the candidate list handles this by storing
// NaN only when no non-NaN values have been observed.
minima.add(x2, v2);
}
/**
* Check the input is a 2-by-2 contingency table.
*
* @param table Contingency table.
* @throws IllegalArgumentException if the {@code table} is not a 2-by-2 table; any
* table entry is negative; any column sum is zero; the table sum is zero or not an
* integer; or the number of possible tables exceeds the maximum array capacity.
*/
private static void checkTable(int[][] table) {
Arguments.checkTable(table);
// Must all be positive
final int a = table[0][0];
final int c = table[1][0];
// checkTable has validated the total sum is < 2^31
final int m = a + c;
if (m == 0) {
throw new InferenceException(InferenceException.ZERO_AT, COLUMN_SUM, 0);
}
final int b = table[0][1];
final int d = table[1][1];
final int n = b + d;
if (n == 0) {
throw new InferenceException(InferenceException.ZERO_AT, COLUMN_SUM, 1);
}
// Total possible tables must be a size we can track in an array (to compute the p-value)
final long size = (m + 1L) * (n + 1L);
if (size > MAX_TABLES) {
throw new InferenceException(InferenceException.X_GT_Y, size, MAX_TABLES);
}
}
}