/*
* Copyright (c) 2012 The Broad Institute
*
* Permission is hereby granted, free of charge, to any person
* obtaining a copy of this software and associated documentation
* files (the "Software"), to deal in the Software without
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* copies of the Software, and to permit persons to whom the
* Software is furnished to do so, subject to the following
* conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR
* THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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package org.broadinstitute.gatk.utils;
import cern.jet.math.Arithmetic;
import cern.jet.random.Normal;
import com.google.java.contract.Ensures;
import com.google.java.contract.Requires;
import org.apache.commons.math.MathException;
import org.apache.commons.math.distribution.NormalDistribution;
import org.apache.commons.math.distribution.NormalDistributionImpl;
import org.broadinstitute.gatk.engine.GenomeAnalysisEngine;
import org.broadinstitute.gatk.utils.collections.Pair;
import org.broadinstitute.gatk.utils.exceptions.GATKException;
import java.io.Serializable;
import java.util.Comparator;
import java.util.TreeSet;
/**
* Created by IntelliJ IDEA.
* User: chartl
*/
public class MannWhitneyU {
private static Normal STANDARD_NORMAL = new Normal(0.0,1.0,null);
private static NormalDistribution APACHE_NORMAL = new NormalDistributionImpl(0.0,1.0,1e-2);
private static double LNSQRT2PI = Math.log(Math.sqrt(2.0*Math.PI));
private TreeSet<Pair<Number,USet>> observations;
private int sizeSet1;
private int sizeSet2;
private ExactMode exactMode;
public MannWhitneyU(ExactMode mode, boolean dither) {
if ( dither )
observations = new TreeSet<Pair<Number,USet>>(new DitheringComparator());
else
observations = new TreeSet<Pair<Number,USet>>(new NumberedPairComparator());
sizeSet1 = 0;
sizeSet2 = 0;
exactMode = mode;
}
public MannWhitneyU() {
this(ExactMode.POINT,true);
}
public MannWhitneyU(boolean dither) {
this(ExactMode.POINT,dither);
}
public MannWhitneyU(ExactMode mode) {
this(mode,true);
}
/**
* Add an observation into the observation tree
* @param n: the observation (a number)
* @param set: whether the observation comes from set 1 or set 2
*/
public void add(Number n, USet set) {
observations.add(new Pair<Number,USet>(n,set));
if ( set == USet.SET1 ) {
++sizeSet1;
} else {
++sizeSet2;
}
}
public Pair<Long,Long> getR1R2() {
long u1 = calculateOneSidedU(observations,MannWhitneyU.USet.SET1);
long n1 = sizeSet1*(sizeSet1+1)/2;
long r1 = u1 + n1;
long n2 = sizeSet2*(sizeSet2+1)/2;
long u2 = n1*n2-u1;
long r2 = u2 + n2;
return new Pair<Long,Long>(r1,r2);
}
/**
* Runs the one-sided test under the hypothesis that the data in set "lessThanOther" stochastically
* dominates the other set
* @param lessThanOther - either Set1 or Set2
* @return - u-based z-approximation, and p-value associated with the test (p-value is exact for small n,m)
*/
@Requires({"lessThanOther != null"})
@Ensures({"validateObservations(observations) || Double.isNaN(result.getFirst())","result != null", "! Double.isInfinite(result.getFirst())", "! Double.isInfinite(result.getSecond())"})
public Pair<Double,Double> runOneSidedTest(USet lessThanOther) {
long u = calculateOneSidedU(observations, lessThanOther);
int n = lessThanOther == USet.SET1 ? sizeSet1 : sizeSet2;
int m = lessThanOther == USet.SET1 ? sizeSet2 : sizeSet1;
if ( n == 0 || m == 0 ) {
// test is uninformative as one or both sets have no observations
return new Pair<Double,Double>(Double.NaN,Double.NaN);
}
// the null hypothesis is that {N} is stochastically less than {M}, so U has counted
// occurrences of {M}s before {N}s. We would expect that this should be less than (n*m+1)/2 under
// the null hypothesis, so we want to integrate from K=0 to K=U for cumulative cases. Always.
return calculateP(n, m, u, false, exactMode);
}
/**
* Runs the standard two-sided test,
* returns the u-based z-approximate and p values.
* @return a pair holding the u and p-value.
*/
@Ensures({"result != null", "! Double.isInfinite(result.getFirst())", "! Double.isInfinite(result.getSecond())"})
//@Requires({"validateObservations(observations)"})
public Pair<Double,Double> runTwoSidedTest() {
Pair<Long,USet> uPair = calculateTwoSidedU(observations);
long u = uPair.first;
int n = uPair.second == USet.SET1 ? sizeSet1 : sizeSet2;
int m = uPair.second == USet.SET1 ? sizeSet2 : sizeSet1;
if ( n == 0 || m == 0 ) {
// test is uninformative as one or both sets have no observations
return new Pair<Double,Double>(Double.NaN,Double.NaN);
}
return calculateP(n, m, u, true, exactMode);
}
/**
* Given a u statistic, calculate the p-value associated with it, dispatching to approximations where appropriate
* @param n - The number of entries in the stochastically smaller (dominant) set
* @param m - The number of entries in the stochastically larger (dominated) set
* @param u - the Mann-Whitney U value
* @param twoSided - is the test twosided
* @return the (possibly approximate) p-value associated with the MWU test, and the (possibly approximate) z-value associated with it
* todo -- there must be an approximation for small m and large n
*/
@Requires({"m > 0","n > 0"})
@Ensures({"result != null", "! Double.isInfinite(result.getFirst())", "! Double.isInfinite(result.getSecond())"})
protected static Pair<Double,Double> calculateP(int n, int m, long u, boolean twoSided, ExactMode exactMode) {
Pair<Double,Double> zandP;
if ( n > 8 && m > 8 ) {
// large m and n - normal approx
zandP = calculatePNormalApproximation(n,m,u, twoSided);
} else if ( n > 5 && m > 7 ) {
// large m, small n - sum uniform approx
// todo -- find the appropriate regimes where this approximation is actually better enough to merit slowness
// pval = calculatePUniformApproximation(n,m,u);
zandP = calculatePNormalApproximation(n, m, u, twoSided);
} else if ( n > 8 || m > 8 ) {
zandP = calculatePFromTable(n, m, u, twoSided);
} else {
// small m and n - full approx
zandP = calculatePRecursively(n,m,u,twoSided,exactMode);
}
return zandP;
}
public static Pair<Double,Double> calculatePFromTable(int n, int m, long u, boolean twoSided) {
// todo -- actually use a table for:
// todo - n large, m small
return calculatePNormalApproximation(n,m,u, twoSided);
}
/**
* Uses a normal approximation to the U statistic in order to return a cdf p-value. See Mann, Whitney [1947]
* @param n - The number of entries in the stochastically smaller (dominant) set
* @param m - The number of entries in the stochastically larger (dominated) set
* @param u - the Mann-Whitney U value
* @param twoSided - whether the test should be two sided
* @return p-value associated with the normal approximation
*/
@Requires({"m > 0","n > 0"})
@Ensures({"result != null", "! Double.isInfinite(result.getFirst())", "! Double.isInfinite(result.getSecond())"})
public static Pair<Double,Double> calculatePNormalApproximation(int n,int m,long u, boolean twoSided) {
double z = getZApprox(n,m,u);
if ( twoSided ) {
return new Pair<Double,Double>(z,2.0*(z < 0 ? STANDARD_NORMAL.cdf(z) : 1.0-STANDARD_NORMAL.cdf(z)));
} else {
return new Pair<Double,Double>(z,STANDARD_NORMAL.cdf(z));
}
}
/**
* Calculates the Z-score approximation of the u-statistic
* @param n - The number of entries in the stochastically smaller (dominant) set
* @param m - The number of entries in the stochastically larger (dominated) set
* @param u - the Mann-Whitney U value
* @return the asymptotic z-approximation corresponding to the MWU p-value for n < m
*/
@Requires({"m > 0","n > 0"})
@Ensures({"! Double.isNaN(result)", "! Double.isInfinite(result)"})
private static double getZApprox(int n, int m, long u) {
double mean = ( ((long)m)*n+1.0)/2;
double var = (((long) n)*m*(n+m+1.0))/12;
double z = ( u - mean )/Math.sqrt(var);
return z;
}
/**
* Uses a sum-of-uniform-0-1 random variable approximation to the U statistic in order to return an approximate
* p-value. See Buckle, Kraft, van Eeden [1969] (approx) and Billingsly [1995] or Stephens, MA [1966, biometrika] (sum of uniform CDF)
* @param n - The number of entries in the stochastically smaller (dominant) set
* @param m - The number of entries in the stochastically larger (dominated) set
* @param u - mann-whitney u value
* @return p-value according to sum of uniform approx
* todo -- this is currently not called due to not having a good characterization of where it is significantly more accurate than the
* todo -- normal approxmation (e.g. enough to merit the runtime hit)
*/
public static double calculatePUniformApproximation(int n, int m, long u) {
long R = u + (n*(n+1))/2;
double a = Math.sqrt(m*(n+m+1));
double b = (n/2.0)*(1-Math.sqrt((n+m+1)/m));
double z = b + ((double)R)/a;
if ( z < 0 ) { return 1.0; }
else if ( z > n ) { return 0.0; }
else {
if ( z > ((double) n) /2 ) {
return 1.0-1/(Arithmetic.factorial(n))*uniformSumHelper(z, (int) Math.floor(z), n, 0);
} else {
return 1/(Arithmetic.factorial(n))*uniformSumHelper(z, (int) Math.floor(z), n, 0);
}
}
}
/**
* Helper function for the sum of n uniform random variables
* @param z - value at which to compute the (un-normalized) cdf
* @param m - a cutoff integer (defined by m <= z < m + 1)
* @param n - the number of uniform random variables
* @param k - holder variable for the recursion (alternatively, the index of the term in the sequence)
* @return the (un-normalized) cdf for the sum of n random variables
*/
private static double uniformSumHelper(double z, int m, int n, int k) {
if ( k > m ) { return 0; }
int coef = (k % 2 == 0) ? 1 : -1;
return coef*Arithmetic.binomial(n,k)*Math.pow(z-k,n) + uniformSumHelper(z,m,n,k+1);
}
/**
* Calculates the U-statistic associated with a two-sided test (e.g. the RV from which one set is drawn
* stochastically dominates the RV from which the other set is drawn); two-sidedness is accounted for
* later on simply by multiplying the p-value by 2.
*
* Recall: If X stochastically dominates Y, the test is for occurrences of Y before X, so the lower value of u is chosen
* @param observed - the observed data
* @return the minimum of the U counts (set1 dominates 2, set 2 dominates 1)
*/
@Requires({"observed != null", "observed.size() > 0"})
@Ensures({"result != null","result.first > 0"})
public static Pair<Long,USet> calculateTwoSidedU(TreeSet<Pair<Number,USet>> observed) {
int set1SeenSoFar = 0;
int set2SeenSoFar = 0;
long uSet1DomSet2 = 0;
long uSet2DomSet1 = 0;
USet previous = null;
for ( Pair<Number,USet> dataPoint : observed ) {
if ( dataPoint.second == USet.SET1 ) {
++set1SeenSoFar;
} else {
++set2SeenSoFar;
}
if ( previous != null ) {
if ( dataPoint.second == USet.SET1 ) {
uSet2DomSet1 += set2SeenSoFar;
} else {
uSet1DomSet2 += set1SeenSoFar;
}
}
previous = dataPoint.second;
}
return uSet1DomSet2 < uSet2DomSet1 ? new Pair<Long,USet>(uSet1DomSet2,USet.SET1) : new Pair<Long,USet>(uSet2DomSet1,USet.SET2);
}
/**
* Calculates the U-statistic associated with the one-sided hypothesis that "dominator" stochastically dominates
* the other U-set. Note that if S1 dominates S2, we want to count the occurrences of points in S2 coming before points in S1.
* @param observed - the observed data points, tagged by each set
* @param dominator - the set that is hypothesized to be stochastically dominating
* @return the u-statistic associated with the hypothesis that dominator stochastically dominates the other set
*/
@Requires({"observed != null","dominator != null","observed.size() > 0"})
@Ensures({"result >= 0"})
public static long calculateOneSidedU(TreeSet<Pair<Number,USet>> observed,USet dominator) {
long otherBeforeDominator = 0l;
int otherSeenSoFar = 0;
for ( Pair<Number,USet> dataPoint : observed ) {
if ( dataPoint.second != dominator ) {
++otherSeenSoFar;
} else {
otherBeforeDominator += otherSeenSoFar;
}
}
return otherBeforeDominator;
}
/**
* The Mann-Whitney U statistic follows a recursive equation (that enumerates the proportion of possible
* binary strings of "n" zeros, and "m" ones, where a one precedes a zero "u" times). This accessor
* calls into that recursive calculation.
* @param n: number of set-one entries (hypothesis: set one is stochastically less than set two)
* @param m: number of set-two entries
* @param u: number of set-two entries that precede set-one entries (e.g. 0,1,0,1,0 -> 3 )
* @param twoSided: whether the test is two sided or not. The recursive formula is symmetric, multiply by two for two-sidedness.
* @param mode: whether the mode is a point probability, or a cumulative distribution
* @return the probability under the hypothesis that all sequences are equally likely of finding a set-two entry preceding a set-one entry "u" times.
*/
@Requires({"m > 0","n > 0","u >= 0"})
@Ensures({"result != null","! Double.isInfinite(result.getFirst())", "! Double.isInfinite(result.getSecond())"})
public static Pair<Double,Double> calculatePRecursively(int n, int m, long u, boolean twoSided, ExactMode mode) {
if ( m > 8 && n > 5 ) { throw new GATKException(String.format("Please use the appropriate (normal or sum of uniform) approximation. Values n: %d, m: %d",n,m)); }
double p = mode == ExactMode.POINT ? cpr(n,m,u) : cumulativeCPR(n,m,u);
//p *= twoSided ? 2.0 : 1.0;
double z;
try {
if ( mode == ExactMode.CUMULATIVE ) {
z = APACHE_NORMAL.inverseCumulativeProbability(p);
} else {
double sd = Math.sqrt((1.0+1.0/(1+n+m))*(n*m)*(1.0+n+m)/12); // biased variance empirically better fit to distribution then asymptotic variance
//System.out.printf("SD is %f and Max is %f and prob is %f%n",sd,1.0/Math.sqrt(sd*sd*2.0*Math.PI),p);
if ( p > 1.0/Math.sqrt(sd*sd*2.0*Math.PI) ) { // possible for p-value to be outside the range of the normal. Happens at the mean, so z is 0.
z = 0.0;
} else {
if ( u >= n*m/2 ) {
z = Math.sqrt(-2.0*(Math.log(sd)+Math.log(p)+LNSQRT2PI));
} else {
z = -Math.sqrt(-2.0*(Math.log(sd)+Math.log(p)+LNSQRT2PI));
}
}
}
} catch (MathException me) {
throw new GATKException("A math exception occurred in inverting the probability",me);
}
return new Pair<Double,Double>(z,(twoSided ? 2.0*p : p));
}
/**
* Hook into CPR with sufficient warning (for testing purposes)
* calls into that recursive calculation.
* @param n: number of set-one entries (hypothesis: set one is stochastically less than set two)
* @param m: number of set-two entries
* @param u: number of set-two entries that precede set-one entries (e.g. 0,1,0,1,0 -> 3 )
* @return same as cpr
*/
protected static double calculatePRecursivelyDoNotCheckValuesEvenThoughItIsSlow(int n, int m, long u) {
return cpr(n,m,u);
}
/**
* For testing
*
* @param n: number of set-one entries (hypothesis: set one is stochastically less than set two)
* @param m: number of set-two entries
* @param u: number of set-two entries that precede set-one entries (e.g. 0,1,0,1,0 -> 3 )
*/
protected static long countSequences(int n, int m, long u) {
if ( u < 0 ) { return 0; }
if ( m == 0 || n == 0 ) { return u == 0 ? 1 : 0; }
return countSequences(n-1,m,u-m) + countSequences(n,m-1,u);
}
/**
* : just a shorter name for calculatePRecursively. See Mann, Whitney, [1947]
* @param n: number of set-1 entries
* @param m: number of set-2 entries
* @param u: number of times a set-2 entry as preceded a set-1 entry
* @return recursive p-value
*/
private static double cpr(int n, int m, long u) {
if ( u < 0 ) {
return 0.0;
}
if ( m == 0 || n == 0 ) {
// there are entries in set 1 or set 2, so no set-2 entry can precede a set-1 entry; thus u must be zero.
// note that this exists only for edification, as when we reach this point, the coefficient on this term is zero anyway
return ( u == 0 ) ? 1.0 : 0.0;
}
return (((double)n)/(n+m))*cpr(n-1,m,u-m) + (((double)m)/(n+m))*cpr(n,m-1,u);
}
private static double cumulativeCPR(int n, int m, long u ) {
// from above:
// the null hypothesis is that {N} is stochastically less than {M}, so U has counted
// occurrences of {M}s before {N}s. We would expect that this should be less than (n*m+1)/2 under
// the null hypothesis, so we want to integrate from K=0 to K=U for cumulative cases. Always.
double p = 0.0;
// optimization using symmetry, use the least amount of sums possible
long uSym = ( u <= n*m/2 ) ? u : ((long)n)*m-u;
for ( long uu = 0; uu < uSym; uu++ ) {
p += cpr(n,m,uu);
}
// correct by 1.0-p if the optimization above was used (e.g. 1-right tail = left tail)
return (u <= n*m/2) ? p : 1.0-p;
}
/**
* hook into the data tree, for testing purposes only
* @return observations
*/
protected TreeSet<Pair<Number,USet>> getObservations() {
return observations;
}
/**
* hook into the set sizes, for testing purposes only
* @return size set 1, size set 2
*/
protected Pair<Integer,Integer> getSetSizes() {
return new Pair<Integer,Integer>(sizeSet1,sizeSet2);
}
/**
* Validates that observations are in the correct format for a MWU test -- this is only called by the contracts API during testing
* @param tree - the collection of labeled observations
* @return true iff the tree set is valid (no INFs or NaNs, at least one data point in each set)
*/
protected static boolean validateObservations(TreeSet<Pair<Number,USet>> tree) {
boolean seen1 = false;
boolean seen2 = false;
boolean seenInvalid = false;
for ( Pair<Number,USet> p : tree) {
if ( ! seen1 && p.getSecond() == USet.SET1 ) {
seen1 = true;
}
if ( ! seen2 && p.getSecond() == USet.SET2 ) {
seen2 = true;
}
if ( Double.isNaN(p.getFirst().doubleValue()) || Double.isInfinite(p.getFirst().doubleValue())) {
seenInvalid = true;
}
}
return ! seenInvalid && seen1 && seen2;
}
/**
* A comparator class which uses dithering on tie-breaking to ensure that the internal treeset drops no values
* and to ensure that rank ties are broken at random.
*/
private static class DitheringComparator implements Comparator<Pair<Number,USet>>, Serializable {
public DitheringComparator() {}
@Override
public boolean equals(Object other) { return false; }
@Override
public int compare(Pair<Number,USet> left, Pair<Number,USet> right) {
double comp = Double.compare(left.first.doubleValue(),right.first.doubleValue());
if ( comp > 0 ) { return 1; }
if ( comp < 0 ) { return -1; }
return GenomeAnalysisEngine.getRandomGenerator().nextBoolean() ? -1 : 1;
}
}
/**
* A comparator that reaches into the pair and compares numbers without tie-braking.
*/
private static class NumberedPairComparator implements Comparator<Pair<Number,USet>>, Serializable {
public NumberedPairComparator() {}
@Override
public boolean equals(Object other) { return false; }
@Override
public int compare(Pair<Number,USet> left, Pair<Number,USet> right ) {
return Double.compare(left.first.doubleValue(),right.first.doubleValue());
}
}
public enum USet { SET1, SET2 }
public enum ExactMode { POINT, CUMULATIVE }
}