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confidence.js
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confidence.js
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(function (root, factory) {
if (typeof define === 'function' && define.amd) {
// AMD. Register as an anonymous module.
define([], factory);
} else if (typeof exports === 'object') {
// Node. Does not work with strict CommonJS, but
// only CommonJS-like environments that support module.exports,
// like Node.
module.exports = factory();
} else {
// Browser globals (root is window)
root.Confidence = factory();
}
}(this, function () {
/** Private Constants */
var DEFAULT_Z_SCORE = 1.96;
var DEFAULT_MARGIN_OF_ERROR = 0.01;
var VERSION = '1.1.0';
var Confidence = function(settings) {
settings = settings || { };
this._zScore = settings.hasOwnProperty('zScore') ? settings.zScore : DEFAULT_Z_SCORE;
this._marginOfError = settings.hasOwnProperty('marginOfError') ? settings.marginOfError : DEFAULT_MARGIN_OF_ERROR;
this._variants = { };
this._confidenceIntervals = { };
};
// Updates the API version number
Confidence.version = VERSION;
/*
Given a z-score, this function returns a polynomial approximation of the corresponding confidence-level.
Adapted from:
Ibbetson D, Algorithm 209
Collected Algorithms of the CACM 1963 p. 616
*/
var Z_MAX = 6;
var zScoreProbability = function(z) {
var w, x, y;
if (z === 0.0) {
x = 0.0;
} else {
y = 0.5 * Math.abs(z);
if (y > (Z_MAX * 0.5)) {
x = 1.0;
} else if (y < 1.0) {
w = y * y;
x = ((((((((0.000124818987 * w - 0.001075204047) * w + 0.005198775019) * w - 0.019198292004) * w + 0.059054035642) * w - 0.151968751364) * w + 0.319152932694) * w - 0.531923007300) * w + 0.797884560593) * y * 2.0;
} else {
y -= 2.0;
x = (((((((((((((-0.000045255659 * y + 0.000152529290) * y - 0.000019538132) * y - 0.000676904986) * y + 0.001390604284) * y - 0.000794620820) * y - 0.002034254874) * y + 0.006549791214) * y - 0.010557625006) * y + 0.011630447319) * y - 0.009279453341) * y + 0.005353579108) * y - 0.002141268741) * y + 0.000535310849) * y + 0.999936657524;
}
}
if (z > 0.0) {
return (x + 1) / 2.0;
} else {
return (1 - x) / 2.0;
}
};
var BIGX = 20.0; /* max value to represent exp(x) */
function ex(x) {
return (x < -BIGX) ? 0.0 : Math.exp(x);
}
/* POCHISQ -- probability of chi-square value
Adapted from:
Hill, I. D. and Pike, M. C. Algorithm 299
Collected Algorithms for the CACM 1967 p. 243
Updated for rounding errors based on remark in
ACM TOMS June 1985, page 185
*/
var pochisq = function(x, df) {
var a, y, s;
var e, c, z;
var even; /* True if df is an even number */
var LOG_SQRT_PI = 0.5723649429247000870717135; /* log(sqrt(pi)) */
var I_SQRT_PI = 0.5641895835477562869480795; /* 1 / sqrt(pi) */
if (x <= 0.0 || df < 1) {
return 1.0;
}
a = 0.5 * x;
even = !(df & 1);
if (df > 1) {
y = ex(-a);
}
s = (even ? y : (2.0 * zSquareProbability(-Math.sqrt(x))));
if (df > 2) {
x = 0.5 * (df - 1.0);
z = (even ? 1.0 : 0.5);
if (a > BIGX) {
e = (even ? 0.0 : LOG_SQRT_PI);
c = Math.log(a);
while (z <= x) {
e = Math.log(z) + e;
s += ex(c * z - a - e);
z += 1.0;
}
return s;
} else {
e = (even ? 1.0 : (I_SQRT_PI / Math.sqrt(a)));
c = 0.0;
while (z <= x) {
e = e * (a / z);
c = c + e;
z += 1.0;
}
return c * y + s;
}
} else {
return s;
}
};
/* CRITCHI -- Compute critical chi-square value to
produce given p. We just do a bisection
search for a value within CHI_EPSILON,
relying on the monotonicity of pochisq(). */
var critchi = function(p, df) {
var CHI_EPSILON = 0.000001; /* Accuracy of critchi approximation */
var CHI_MAX = 99999.0; /* Maximum chi-square value */
var minchisq = 0.0;
var maxchisq = CHI_MAX;
var chisqval;
if (p <= 0.0) {
return maxchisq;
} else {
if (p >= 1.0) {
return 0.0;
}
}
chisqval = df / Math.sqrt(p); /* fair first value */
while ((maxchisq - minchisq) > CHI_EPSILON) {
if (pochisq(chisqval, df) < p) {
maxchisq = chisqval;
} else {
minchisq = chisqval;
}
chisqval = (maxchisq + minchisq) * 0.5;
}
return chisqval;
};
/** Public Constants **/
Confidence.prototype.addVariant = function(variant) {
// check if variant ID already exists
if (this.variantExists(variant.id)) {
var message = 'A variant with ID \'' + variant.id + '\' already exists.';
throw new Error (message);
} else {
// add the variant!
// variant must have properties conversionCount, eventCount
if (variant.hasOwnProperty('id') &&
variant.hasOwnProperty('conversionCount') &&
variant.hasOwnProperty('eventCount')) {
if(!variant.hasOwnProperty('name')) {
variant['name'] = 'Variant ' + variant.id;
}
this._variants[variant.id] = variant;
} else {
throw new Error('variant object needs conversionCount and eventCount properties');
}
}
};
Confidence.prototype.getVariant = function(variantID) {
if (this.variantExists(variantID)) {
return this._variants[variantID];
} else {
throw new Error('The variant you requested does not exist.');
}
};
Confidence.prototype.variantExists = function(variantID) {
if (this._variants.hasOwnProperty(variantID)) {
return true;
} else {
return false;
}
};
Confidence.prototype.hasVariants = function(variants) {
for(var variant in variants) {
if(variants.hasOwnProperty(variant)) {
return true;
}
}
return false;
};
//**************************************************************************//
// Z-TEST
//**************************************************************************//
Confidence.prototype.getResult = function() {
if (this.hasVariants(this._variants)) {
confidenceIntervals = {};
var result;
// for each variant in variants
for (var variantID in this._variants) {
// get sample size required for statistically significant answer
var requiredSampleSize = this.getRequiredSampleSize(variantID);
// do we have a winner to pass into the result
var hasWinner = false;
// verify whether we have enough data
var hasEnoughData = this.hasEnoughData(variantID);
// If we don't have enough data yet, we cannot yet declare a result.
if (!hasEnoughData) {
result = {
hasWinner: hasWinner,
hasEnoughData: hasEnoughData,
winnerID: null,
winnerName: null,
confidencePercent: null,
confidenceInterval: null,
readable: "There is not enough data to determine a conclusive result."
};
return result;
}
//calculate confidence interval
var confidenceInterval = this.getConfidenceInterval(variantID);
confidenceIntervals[variantID] = confidenceInterval;
}
// At this point, we have enough data to determine whether there is a winner or not
result = this.analyzeConfidenceIntervals(confidenceIntervals);
return result;
} else {
throw new Error('There are no variants available.');
}
};
Confidence.prototype.analyzeConfidenceIntervals = function(confidenceIntervals) {
var minimums = [];
var maximums = [];
// split confidence intervals into a list of mins and maxes.
for (var id in confidenceIntervals) {
// instantiate object
var min = confidenceIntervals[id].min;
minimums.push({ id: id, val: min });
var max = confidenceIntervals[id].max;
maximums.push({ id: id, val: max });
}
// sort list of minimums greatest to least
minimums = this.sortList(minimums);
// identify ID with the largest min
var idWithLargestMin = minimums[0].id;
var largestMin = minimums[0].val;
// sort list of maximums greatest to least
maximums = this.sortList(maximums);
// remove the ID with the largest min from the list of maximums.
for (var index = 0; index < maximums.length; index++) {
var obj = maximums[index];
if (obj.id == idWithLargestMin) {
maximums.splice(index, 1);
}
}
// identify ID with the largest max and its value
var idWithLargestMax = maximums[0].id;
var largestMax = maximums[0].val;
var confidencePercent = this.getConfidencePercent();
var result;
var hasWinner;
var hasEnoughData;
// if highest min > highest max, declare the ID of the winner to be the min
if (largestMin > largestMax) {
winningVariantName = this._variants[idWithLargestMin].name;
var roundedMin = (Math.round(10000 * confidenceIntervals[idWithLargestMin].min)/100);
var roundedMax = (Math.round(10000 * confidenceIntervals[idWithLargestMin].max)/100);
var message = "With " + confidencePercent + "% confidence, the true population parameter of the \"";
message += winningVariantName + "\" variant will fall between ";
message += roundedMin + "% and ";
message += roundedMax + "%.";
result = {
hasWinner: true,
hasEnoughData: true,
winnerID: idWithLargestMin,
winnerName: winningVariantName,
confidencePercent: confidencePercent,
confidenceInterval: { min: roundedMin, max: roundedMax },
readable: message
};
return result;
} else {
// otherwise, there is no winner
var messageNoWinner = "There is no winner, the results are too close.";
result = {
hasWinner: false,
hasEnoughData: true,
winnerID: null,
winnerName: null,
confidencePercent: confidencePercent,
confidenceInterval: null,
readable: messageNoWinner
};
}
return result;
};
// Sorts list from greatest to least
Confidence.prototype.sortList = function(list) {
list.sort(function(a, b) {
return b.val - a.val;
});
return list;
};
// Gets Confidence percentage from the configured zscore
Confidence.prototype.getConfidencePercent = function(zScore) {
zScore = typeof zScore === 'number' ? zScore : this._zScore;
var normalProbability = zScoreProbability(zScore);
return (100 * (2 * normalProbability - 1)).toFixed(2);
};
// Are these result statistically significant?
// (zscore^2 * stdErr * (1 - stdErr)) / marginErr^2
Confidence.prototype.getRequiredSampleSize = function(variantID) {
var standardError = this.getStandardError(variantID);
var numerator = (Math.pow(this._zScore, 2) * standardError * (1 - standardError));
var denominator = Math.pow(this._marginOfError, 2);
var requiredSampleSize = Math.max((numerator/denominator), 100);
return requiredSampleSize;
};
Confidence.prototype.hasEnoughData = function(variantID) {
var variant = this.getVariant(variantID);
var requiredSampleSize = this.getRequiredSampleSize(variantID);
if (variant.eventCount >= requiredSampleSize) {
return true;
} else {
return false;
}
};
Confidence.prototype.getRate = function(variantID) {
var variant = this.getVariant(variantID);
if (variant.eventCount === 0) {
throw new Error('Total is zero: cannot divide by zero to produce rate.');
} else if (variant.eventCount < 0) {
throw new Error('Total is negative: cannot use a negative number to produce rate.');
} else {
var rate = variant.conversionCount / variant.eventCount;
return rate;
}
};
// Calculate the interval for which we are <zscore> confident
// rate +- (zscore * standard error)
Confidence.prototype.getConfidenceInterval = function(variantID) {
var confidenceInterval = {};
var rate = this.getRate(variantID);
var standardError = this.getStandardError(variantID);
//lower limit
var min = rate - (this._zScore * standardError);
if (min < 0) {
min = 0.00;
}
//upper limit
var max = rate + (this._zScore * standardError);
if (max > 1) {
max = 1.00;
}
confidenceInterval = { min: min, max: max };
return confidenceInterval;
};
// Calculate standard error:
// SE = sqrt(rate*(1-rate)/total)
Confidence.prototype.getStandardError = function(variantID) {
var variant = this.getVariant(variantID);
//Throw error if rate is not okay.
var rate = this.getRate(variantID);
//Check total Count - if event count is 0
var standardError = Math.sqrt(rate * (1 - rate) / variant.eventCount);
return standardError;
};
//**************************************************************************//
// CHI-SQUARED AND MARASCUILLO'S PROCEDURE
//**************************************************************************//
Confidence.prototype.getMarascuilloResult = function() {
var result;
// Calculate observed and expected values.
// If any of the expected values < 5 there is not enough data.
var observedValues = this.getObservedValues();
var pooledProportion = this.getPooledProportion(observedValues);
var expectedValues = this.getExpectedValues(observedValues, pooledProportion);
// If any expected value < 5, we do not have enough data
if (expectedValues.hasEnoughData === false) {
result = {
hasWinner: false,
hasEnoughData: false,
winnerID: null,
winnerName: null
};
return result;
}
// Calculate "Chi Parts".
var chiPartValues = this.getChiParts(observedValues, expectedValues);
var chiPartSum = this.sumChiParts(chiPartValues);
// Calculate critical value.
var degreesOfFreedom = this.getDegreesOfFreedom();
var probability = (1 - zScoreProbability(this._zScore));
var critChi = critchi(probability, 2);
if (chiPartSum > critChi) {
// there is a difference, proceed to marascuillo
var bestVariant = this.getBestVariant();
result = this.marascuillo(bestVariant, critChi);
} else {
// Enough data, no winner
result = {
hasWinner: false,
hasEnoughData: true,
winnerID: null,
winnerName: null
};
return result;
}
return result;
};
// Compute and store observed successes (ie. clickthroughs),
// failures(total - clickthroughs), and totals for each variant.
Confidence.prototype.getObservedValues = function() {
var observedValues = {};
var variants = this._variants;
for (var variant in variants) {
var observedStats = {};
var success = variants[variant].conversionCount;
var fail = variants[variant].eventCount - variants[variant].conversionCount;
var total = variants[variant].eventCount;
observedStats['success'] = success;
observedStats['fail'] = fail;
observedStats['total'] = total;
observedValues[variant] = observedStats;
}
return observedValues;
};
// Compute pooled proportion p
// p = sum of successes / sum of totals
Confidence.prototype.getPooledProportion = function(observedValues) {
var summedSuccesses = 0;
var summedTotals = 0;
// sum the successes across all variants, and
// sum the totals across all variants
for (var variant in observedValues){
summedSuccesses += observedValues[variant].success;
summedTotals += observedValues[variant].total;
}
if (summedTotals === 0) {
throw new Error('Summed total is zero: cannot divide by zero to produce rate.');
} else {
var result = summedSuccesses / summedTotals;
return result;
}
};
// Compute and store expected successes and failures.
// For each expected success:
// E@i = p*total@i
// For each expected failure:
// E@i = (1-p)*total@i
Confidence.prototype.getExpectedValues = function(observedValues, pooledProportion) {
var expectedValues = {};
for (var variant in observedValues) {
var expectedStats = {};
var success;
var fail;
// If any expected count < 5, we don't have enough data.
success = pooledProportion * observedValues[variant].total;
fail = (1 - pooledProportion) * observedValues[variant].total;
if (success < 5 || fail < 5) {
expectedValues['hasEnoughData'] = false;
}
expectedStats['success'] = success;
expectedStats['fail'] = fail;
expectedValues[variant] = expectedStats;
}
if (expectedValues['hasEnoughData'] !== false) {
expectedValues['hasEnoughData'] = true;
}
return expectedValues;
};
// Calculate "Chi Parts".
// For each cell (where i is the cell):
// ChiPart@i = ((ObservedVal@i - ExpectedVal@i) ^ 2) / ExpectedVal@i
// SHORTCUT THIS? calculate critChi here and check at every step
Confidence.prototype.getChiParts = function(observedValues, expectedValues) {
var chiPartValues = {};
var success;
var fail;
for (var variant in observedValues) {
var chiPartStats = {};
// Compute Chi-Part for success
var observedSuccess = observedValues[variant].success;
var expectedSuccess = expectedValues[variant].success;
if (expectedSuccess === 0) {
throw new Error('Cannot divide by zero to produce chi parts.');
} else {
success = Math.pow((observedSuccess - expectedSuccess), 2) / expectedSuccess;
chiPartStats['success'] = success;
}
// Compute Chi-Part for fail
var observedFail = observedValues[variant].fail;
var expectedFail = expectedValues[variant].fail;
if (expectedFail === 0) {
throw new Error('Cannot divide by zero to produce chi parts.');
} else {
fail = Math.pow((observedFail - expectedFail), 2) / expectedFail;
chiPartStats['fail'] = fail;
}
chiPartValues[variant] = chiPartStats;
}
return chiPartValues;
};
// Sum the Chi Parts to get the Chi-Square value.
Confidence.prototype.sumChiParts = function(chiPartValues) {
var chiPartSum = 0;
for (var variant in chiPartValues) {
chiPartSum += chiPartValues[variant].success;
chiPartSum += chiPartValues[variant].fail;
}
return chiPartSum;
};
// Calculate degrees of freedom.
// k - 1 where k is the number of variants
Confidence.prototype.getDegreesOfFreedom = function() {
var len = (Object.keys(this._variants).length) - 1;
return len;
};
// Find the variant with the highest rate
Confidence.prototype.getBestVariant = function() {
var bestRate = 0;
var bestVariantID;
for (var variantID in this._variants) {
var rate = this.getRate(variantID);
if (rate > bestRate){
bestRate = rate;
bestVariantID = variantID;
}
}
return bestVariantID;
};
// Compare the best variant to each other variant.
// if each test stat is greater than the corresponding critical value,
// then the best variant is the winner
// otherwise, there is no winner?
Confidence.prototype.marascuillo = function(bestVariantID, critChi) {
var result;
// TODO if there is only one variant there needs to be a check somewhere because this will probably mess up.
// This is where the z-test messed up... right?
for (var variantID in this._variants) {
if (variantID === bestVariantID) {
continue;
} else {
// Compare the successes of the best variantID
var testStatistic = this.computeTestStatistic(bestVariantID, variantID);
var criticalValue = this.computeCriticalValue(bestVariantID, variantID, critChi);
if (testStatistic > criticalValue) {
// keep goin', doin' fine, calculate the next one
continue;
} else {
// There is enough data, but no winner.
result = {
hasWinner: false,
hasEnoughData: true,
winnerID: null,
winnerName: null
};
return result;
}
}
}
// There is enough data and there is a winner.
result = {
hasWinner: true,
hasEnoughData: true,
winnerID: bestVariantID,
winnerName: this._variants[bestVariantID]['name']
};
return result;
};
// test stat:
// | pi - pj |
Confidence.prototype.computeTestStatistic = function(bestVariantID, challengerVariantID) {
var bestVariantRate = this.getRate(bestVariantID);
var challengerVariantRate = this.getRate(challengerVariantID);
var testStatistic = Math.abs(bestVariantRate - challengerVariantRate);
return testStatistic;
};
// critical value:
// critChi * (sqrt((pi(1 - pi) / ni) + (pj(1 - pj) / nj))
Confidence.prototype.computeCriticalValue = function(bestVariantID, challengerVariantID, critChi) {
// rates
var bestVariantRate = this.getRate(bestVariantID);
var challengerVariantRate = this.getRate(challengerVariantID);
// totals
var bestVariantTotal = this._variants[bestVariantID].eventCount;
var challengerVariantTotal = this._variants[challengerVariantID].eventCount;
var bestVariantPart = (bestVariantRate * (1 - bestVariantRate)) / bestVariantTotal;
var challengerVariantPart = (challengerVariantRate * (1 - challengerVariantRate)) / challengerVariantTotal;
var criticalValue = Math.sqrt(critChi) * Math.sqrt(bestVariantPart + challengerVariantPart);
return criticalValue;
};
return Confidence;
}));