# ms_quant_stats Class Reference [Quantitation analysis for Mascot Server and Distiller]

Mathematical and statistical routines for peptide and protein quantitation. More...

#include <ms_quant_stats.hpp>

List of all members.

## Static Public Member Functions

static double arithmeticMean (const std::vector< double > &data)
Calculate the arithmetic mean.
static double arithmeticStandardDeviation (const std::vector< double > &data, double arithmeticMean)
Calculate the arithmetic sample standard deviation.
static double arithmeticStandardErrorOfMean (double arithmeticStdev, unsigned long sampleSize)
Calculate the standard error of the arithmetic sample mean.
static double arithmeticStandardErrorOfMedian (double arithmeticStdev, unsigned long sampleSize)
Calculate the standard error of the sample median.
static double binomialCoefficient (unsigned int n, unsigned int k)
Compute the binomial coefficient.
static double calculateNormalMeanPvalue (double populationMean, double mean, double stdev, unsigned long sampleSize)
Calculate the p-value that the population mean is populationMean, given the sample statistics and assuming the population is normally distributed.
static double calculateNormalMedianPvalue (double populationMedian, double median, double stdev, unsigned long sampleSize)
Calculate the p-value that the population median is populationMedian, given the sample statistics and assuming the population is normally distributed.
static double calculateNormalWeightedPvalue (double populationMean, double weightedMean, const std::vector< double > &weights, double weightedStdev, unsigned long sampleSize)
Calculate the p-value that the population mean is populationMean, given the weighted sample statistics and assuming the population is normally distributed.
static double chisqLowerCriticalValue (unsigned int df, double sigLevel)
Return the lower critical value of the chi-square distribution at the given significance level.
static double chisqUpperCriticalValue (unsigned int df, double sigLevel)
Return the upper critical value of the chi-square distribution at the given significance level.
static void detectOutliers (const std::vector< double > &sample, std::string outlierMethod, double sigLevel, unsigned int *const numLow, unsigned int *const numHigh)
Given a sample and an outlier method, return the number of outliers at low and high extreme at the given significance level.
static void detectOutliers (const std::vector< double > &sample, std::string outlierMethod, unsigned int *const numLow, unsigned int *const numHigh)
Given a sample and an outlier method, return the number of outliers at low and high extreme at significance level 0.05.
static std::vector< double > expTransform (const std::vector< double > &logData)
Antilog (exp) transform of the input data.
static double geometricMean (const std::vector< double > &data)
Calculate the geometric mean.
static double geometricStandardDeviation (const std::vector< double > &data, double geometricMean)
Calculate the geometric sample standard deviation.
static std::vector< double > logTransform (const std::vector< double > &data)
Log transform of the input data.
static std::vector< double > meanCentre (const std::vector< double > &data, double mean)
Centre the data values about the given mean.
static double medianStandardErrorCorrection (unsigned long sampleSize)
Return the correction for the standard error of the median.
static double normalCriticalValue (double p)
Return the critical value of the standard normal distribution at the given cumulative probability.
static double normalCumulativeProbability (double x0)
Return the cumulative probability that the standard normal statistic is at most x0.
static double poissonDensity (double lambda, unsigned int k)
Calculate the probability density of the Poisson distribution.
static double snedecorsCriticalValue (unsigned int df1, unsigned int df2, double sigLevel)
Return the critical value of Snedecor's F distribution at the given significance level.
static double sortedMedian (const std::vector< double > &data)
Find the median of a sorted data vector.
static double sortedMedian (const std::deque< double > &data)
Find the median of a sorted data deque.
static double studentsCriticalValue (unsigned int df, double sigLevel)
Return the critical value of Student's t distribution at the given significance level.
static double sum (const std::vector< double > &data)
Calculate the sum of a sample vector.
static double sumsq (const std::vector< double > &data)
Calculate the sum of squares of a sample vector.
static int testDixonsN9 (unsigned long sampleSize, double min, double x2, double xn_1, double max, double sigLevel)
Apply Dixon's N9 (r11) outlier test to the sample.
static int testGrubbsN2 (unsigned long sampleSize, double min, double mean, double max, double stdev, double sigLevel)
Apply Grubb's N2 outlier test to the sample statistics.
static void testRosners (const std::vector< double > &sample, unsigned int maxOutliers, double sigLevel, unsigned int *const numLow, unsigned int *const numHigh)
Apply the generalised extreme Studentised deviate (ESD) outlier test (aka Rosner's test) to the sample.
static bool testShapiroWilkW (const std::vector< double > &sample, double *const W, double *const pValue)
Apply Shapiro-Wilk W normality test to the sample.
static double unsortedMedian (const std::vector< double > &data)
Find the median of an unsorted data vector.
static double weightedArithmeticMean (const std::vector< double > &data, const std::vector< double > &weights)
Calculate the weighted arithmetic mean.
static double weightedArithmeticStandardDeviation (const std::vector< double > &data, const std::vector< double > &weights, double weightedMean)
Calculate the weighted arithmetic sample standard deviation.
static double weightedArithmeticStandardError (const std::vector< double > &weights, double weightedStdev, unsigned long sampleSize)
Calculate the standard error of the weighted arithmetic sample mean.
static double weightedGeometricMean (const std::vector< double > &data, const std::vector< double > &weights)
Calculate the weighted geometric mean.
static double weightedGeometricStandardDeviation (const std::vector< double > &data, const std::vector< double > &weights, double weightedMean)
Calculate the weighted geometric sample standard deviation.

## Detailed Description

Mathematical and statistical routines for peptide and protein quantitation.

ms_quant_stats is a collection of mathematical and statistical routines used in peptide and protein quantitation by ms_ms2quantitation and ms_customquantitation. It is possible to use the routines outside this context, as they are not tied to either class.

The class contains only static class methods; it is impossible and meaningless to create instances of ms_quant_stats.

## Member Function Documentation

 double arithmeticMean ( const std::vector< double > & data ) [static]

Calculate the arithmetic mean.

The arithmetic mean is the usual average:

mean(data) = (1/N)sumi xi

where N is the number of elements in data and the summation is over all elements xi.

Parameters:
 data Any vector of numbers.
Returns:
The arithmetic mean of data, or 0 if the vector is empty.
 double arithmeticStandardDeviation ( const std::vector< double > & data, double arithmeticMean ) [static]

Calculate the arithmetic sample standard deviation.

The arithmetic sample deviation is the square root of the arithmetic sample variance. The sample variance is the second sample moment about the sample mean, scaled by N - 1, where N is the sample size. That is,

var(data) = 1/(N - 1)sumi( (xi - arithmeticMean)2 )

where the sum is over all elements xi.

Parameters:
 data A sample of data. arithmeticMean Sample arithmetic mean.
Returns:
Sample standard deviation, or 0 if sample size is 0 or 1.
 double arithmeticStandardErrorOfMean ( double arithmeticStdev, unsigned long sampleSize ) [static]

Calculate the standard error of the arithmetic sample mean.

The standard error of the arithmetic sample mean in a normally distributed sample is simple: standard deviation divided by the square root of the sample size.

Parameters:
 arithmeticStdev Arithmetic standard deviation, for example from arithmeticStandardDeviation(). sampleSize The sample size.
Returns:
Standard error of the mean. If sample size is 0 or 1, this function returns 0.
 double arithmeticStandardErrorOfMedian ( double arithmeticStdev, unsigned long sampleSize ) [static]

Calculate the standard error of the sample median.

The standard error of the sample median in a normally distributed sample is the same as the standard error of the sample mean, times a correction factor. The correction factor is documented in medianStandardErrorCorrection().

Parameters:
 arithmeticStdev Arithmetic standard deviation, for example from arithmeticStandardDeviation(). sampleSize The sample size.
Returns:
Standard error of the mean. If sample size is 0 or 1, this function returns 0.
 double binomialCoefficient ( unsigned int n, unsigned int k ) [static]

Compute the binomial coefficient.

The method returns 1 in the following cases, by definition:

• n = 0
• k = 0
• n < k

Note that although the binomial coefficient is an integer, the return value is double to allow values beyond about 4 billion on 32-bit systems. The value is exact up to about 74C37.

Parameters:
 n Population size. k Sample size.
Returns:
The binomial coefficient nCk or "n choose k".
 double calculateNormalMeanPvalue ( double populationMean, double mean, double stdev, unsigned long sampleSize ) [static]

Calculate the p-value that the population mean is populationMean, given the sample statistics and assuming the population is normally distributed.

Hypothesis: the population you are sampling from is normally distributed with mean populationMean and unknown variance.

Statistic: Student's t = (mean - populationMean)/(stdev/sqrt(sampleSize))

Assume the hypothesis is true. This method returns the probability (the p-value) that the statistic t could have a value as extreme (or more extreme) as the one observed. Low p-value means that either the hypothesis is wrong or the sample is exceptionally rare.

(Note that there is no alternative hypothesis. The p-value with this interpretation is part of Fisher's significance test, which does not decide between hypotheses. Recommended reading: Hubbard, R. et al. (2003): Confusion over Measures of Evidence (p's) versus Errors (alphas) in Classical Statistical Testing. The American Statistician 57(3):171-182.)

Parameters:
 populationMean Assumed population mean. (Population variance is assumed unknown.) mean Sample arithmetic mean. stdev Sample standard deviation. This must be positive. sampleSize Sample size. This must be at least 1.
Returns:
The p-value of the statistic under the hypothesis of normality.
 double calculateNormalMedianPvalue ( double populationMedian, double median, double stdev, unsigned long sampleSize ) [static]

Calculate the p-value that the population median is populationMedian, given the sample statistics and assuming the population is normally distributed.

Hypothesis: the population you are sampling from is normally distributed with median populationMedian and unknown variance. In particular, the population median is equal to the population mean.

Statistic: Student's t for sample median, t = 1/C*(median - populationMedian)/(stdev/sqrt(sampleSize))

Assume the hypothesis is true. This method returns the probability (the p-value) that the statistic t could have a value as extreme (or more extreme) as the one observed. Low p-value means that either the hypothesis is wrong or the sample is exceptionally rare.

(Note that there is no alternative hypothesis. The p-value with this interpretation is part of Fisher's significance test, which does not decide between hypotheses. Recommended reading: Hubbard, R. et al. (2003): Confusion over Measures of Evidence (p's) versus Errors (alphas) in Classical Statistical Testing. The American Statistician 57(3):171-182.)

The constant C is a correction to the standard error of the median, retrieved from medianStandardErrorCorrection().

Parameters:
 populationMedian Assumed population median. (Population variance is assumed unknown.) median Sample median. stdev Sample standard deviation. This must be positive. sampleSize Sample size. This must be at least 1.
Returns:
The p-value of the statistic under the hypothesis of normality.
 double calculateNormalWeightedPvalue ( double populationMean, double weightedMean, const std::vector< double > & weights, double weightedStdev, unsigned long sampleSize ) [static]

Calculate the p-value that the population mean is populationMean, given the weighted sample statistics and assuming the population is normally distributed.

Hypothesis: the population you are sampling from is normally distributed with mean populationMean and unknown variance.

Statistic: Student's t = (weightedMean - populationMean)/(stdev sqrt(W)/b)),

where stdev is the sample standard deviation, W is the sum of weights and b is the effective base, as defined in weightedArithmeticStandardError(). Note that this function takes as argument the weighted standard deviation and the weights, not stdev.

Assume the hypothesis is true. This method returns the probability (the p-value) that the statistic t could have a value as extreme (or more extreme) as the one observed. Low p-value means that either the hypothesis is wrong or the sample is exceptionally rare.

(Note that there is no alternative hypothesis. The p-value with this interpretation is part of Fisher's significance test, which does not decide between hypotheses. Recommended reading: Hubbard, R. et al. (2003): Confusion over Measures of Evidence (p's) versus Errors (alphas) in Classical Statistical Testing. The American Statistician 57(3):171-182.)

Parameters:
 populationMean Assumed population mean. (Population variance is assumed unknown.) weightedMean Weighted sample arithmetic mean. weights Positive weight for each datum. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than sampleSize, the weights of those elements are taken to be 1.0. weightedStdev Weighted sample standard deviation. This must be positive. sampleSize Sample size. This must be at least 1.
Returns:
The p-value of the statistic under the hypothesis of normality.
 double chisqLowerCriticalValue ( unsigned int df, double sigLevel ) [static]

Return the lower critical value of the chi-square distribution at the given significance level.

Suppose the random variable X has the chi-square distribution with df degrees of freedom. This routine returns the lower critical value x0 for which

P(X <= x0) = p

where p is the desired significance level. Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.050 < s <= 0.1000.1
0.025 < s <= 0.0500.05
0.010 < s <= 0.0250.025
0.001 < s <= 0.0100.01
s <= 0.0010.001

If sigLevel exceeds 0.100, it is truncated to 0.100.

Parameters:
 df The degrees of freedom between 1 and 100 inclusive. sigLevel Significance level. See the above table for supported values.
Returns:
The lower critical value x0.
 double chisqUpperCriticalValue ( unsigned int df, double sigLevel ) [static]

Return the upper critical value of the chi-square distribution at the given significance level.

Suppose the random variable X has the chi-square distribution with df degrees of freedom. This routine returns the upper critical value x0 for which

P(X <= x0) = 1 - p

where p is the desired significance level. Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.050 < s <= 0.1000.1
0.025 < s <= 0.0500.05
0.010 < s <= 0.0250.025
0.005 < s <= 0.0100.01
0.001 < s <= 0.0050.005
s <= 0.0010.001

If sigLevel exceeds 0.100, it is truncated to 0.100.

Parameters:
 df The degrees of freedom between 1 and 100 inclusive. sigLevel Significance level. See the above table for supported values.
Returns:
The upper critical value x0.
 void detectOutliers ( const std::vector< double > & sample, std::string outlierMethod, double sigLevel, unsigned int *const numLow, unsigned int *const numHigh ) [static]

Given a sample and an outlier method, return the number of outliers at low and high extreme at the given significance level.

If outlierMethod is "none" or the empty string, or if outlierMethod is not listed in Type mqm:outlierType, then the method simply returns with 0 outliers. Otherwise the corresponding outlier test will be used to count the number of low and high outliers at the given significance level.

If outlierMethod is "auto", then the method uses Dixon's N9 if sample size is less than 25 and Rosner's test otherwise.

Note that not all tests support the same set of significance levels. The common values between all three tests are as follows:

sigLevel (s)p
0.100 < s <= 0.2000.2
0.050 < s <= 0.1000.1
0.020 < s <= 0.0500.05
0.010 < s <= 0.0200.02
0.005 < s <= 0.0100.01

If you specify a significance level that is outside these ranges, then the effective significance levels depends on which test is being used. Please refer to testDixonsN9(), testGrubbsN2() and testRosners() for more details.

Note that this method does not modify sample. You will need to remove the indicated outliers yourself if desired.

Parameters:
 [in] sample A sample of data sorted in increasing numerical order. If the sample size is not large enough for outlierMethod, both numLow and numHigh will be 0 and no testing will be done. [in] outlierMethod Outlier method to use. Outlier methods are listed in Type mqm:outlierType. Note that the value is case sensitive. [in] sigLevel Significance level to use in tests. [out] numLow Number of outliers at the low end of sample; values at indices from 0 to numLow - 1 are outliers based on the chosen test. If numLow is 0, the sample has no outliers at the low end. [out] numHigh Number of outliers at the high end of sample; values at indices from N - numHigh to N - 1 are outliers based on the chosen test (N is the size of sample). If numHigh is 0, the sample has no outliers at the high end.
 void detectOutliers ( const std::vector< double > & sample, std::string outlierMethod, unsigned int *const numLow, unsigned int *const numHigh ) [static]

Given a sample and an outlier method, return the number of outliers at low and high extreme at significance level 0.05.

If outlierMethod is "none" or the empty string, or if outlierMethod is not listed in Type mqm:outlierType, then the method simply returns with 0 outliers. Otherwise the corresponding outlier test will be used to count the number of low and high outliers at significance level 0.05. If you wish to specify a different significance level, use detectOutliers(const std::vector<double>&, std::string, double, unsigned int*const, unsigned int*const).

If outlierMethod is "auto", then the method uses Dixon's N9 if sample size is less than 25 and Rosner's test otherwise.

Note that this method does not modify sample. You will need to remove the indicated outliers yourself if desired.

Parameters:
 [in] sample A sample of data sorted in increasing numerical order. If the sample size is not large enough for outlierMethod, both numLow and numHigh will be 0 and no testing will be done. [in] outlierMethod Outlier method to use. Outlier methods are listed in Type mqm:outlierType. Note that the value is case sensitive. [out] numLow Number of outliers at the low end of sample; values at indices from 0 to numLow - 1 are outliers based on the chosen test. If numLow is 0, the sample has no outliers at the low end. [out] numHigh Number of outliers at the high end of sample; values at indices from N - numHigh to N - 1 are outliers based on the chosen test (N is the size of sample). If numHigh is 0, the sample has no outliers at the high end.
 std::vector< double > expTransform ( const std::vector< double > & logData ) [static]

Antilog (exp) transform of the input data.

Parameters:
 logData Any vector of logarithms.
Returns:
The elementwise exp transform of the input vector.
 double geometricMean ( const std::vector< double > & data ) [static]

Calculate the geometric mean.

The geometric mean is

geom(data) = (prodi xi)1/N

where N is the number of elements in data and the product is over all the elements xi. That is, the geometric mean is the Nth root of the product of all input values.

Equivalently, the logarithm of the geometric mean is equal to the arithmetic mean of the data in log space. This is actually how the mean is calculated by the method.

Parameters:
 data Any vector of numbers.
Returns:
The geometric mean of data, or 1 if the vector is empty. If any of the data elements are 0 or negative, the mean is 0.
 double geometricStandardDeviation ( const std::vector< double > & data, double geometricMean ) [static]

Calculate the geometric sample standard deviation.

The geometric sample deviation is the square root of the geometric sample variance. The geometric sample variance is the second sample moment about the sample mean, scaled by N - 1, where N is the sample size. That is,

var(data) = e1/(N - 1)sumi( (log(xi) - log(geometricMean))2 )

where the sum is over all elements xi.

Equivalently, the logarithm of the geometric standard deviation is equal to the arithmetic standard deviation of the data in log space.

Parameters:
 data A sample of data. geometricMean Sample geometric mean.
Returns:
Sample standard deviation, or 0 if sample size is 0 or 1. If any data element is 0 or negative, or if the geometric mean is 0 or negative, then the standard deviation is 0.
 std::vector< double > logTransform ( const std::vector< double > & data ) [static]

Log transform of the input data.

Parameters:
 data Any vector of positive numbers.
Returns:
The elementwise log transform of the input vector. If any element is 0 or negative, its logarithm will be substituted with a large negative number (but not negative infinity).
 std::vector< double > meanCentre ( const std::vector< double > & data, double mean ) [static]

Centre the data values about the given mean.

Parameters:
 data Any vector of numbers. mean Sample mean.
Returns:
A mean centred vector of the same size as data.
 double medianStandardErrorCorrection ( unsigned long sampleSize ) [static]

Return the correction for the standard error of the median.

The standard error of the median is the same as the standard error of the mean, times a correction factor. That is, the standard error of the mean is s/sqrt(n), where s is the estimated standard deviation of the sample and n is the sample size. The standard error of the median is C * s/sqrt(n), where the constant C is the correction factor returned by this function.

The correction factor C is a function of sample size and is always between 1 and about 1.25. The sample median is an unbiased estimator of the population mean, but it has larger sampling variance than the sample mean (it is less efficient). For a normal population, the ratio between the variance of the sample mean and sample median tends to pi/2 as the sample size tends to infinity, which means C = sqrt(pi/2).

If the sample size is small (below about 100), then the constant is less than sqrt(pi/2) and depends on whether the sample size is even or odd. This method uses exact values for sample size between 1 and 12, and an accurate polynomial approximation for sample sizes between 13 and 100. These numbers and the polynomial are described in the following two publications, respectively:

• Hojo, T. and K. Pearson (1931): Distribution of the Median, Quartiles and Interquartile Distance in Samples From a Normal Population. Biometrika 23(3/4):315-363.
• Cadwell, J. H. (1952): The Distribution of Quantiles of Small Samples. Biometrika 39(1/2):207-211.

(Lower efficiency means that although the median is a robust estimator, this statistic always has a less extreme value than if the sample mean had been used. In other words, the p-value will be higher than for calculateNormalMeanPvalue() for the same sample.)

Parameters:
 sampleSize The sample size.
Returns:
Correction constant as described above.
 double normalCriticalValue ( double p ) [static]

Return the critical value of the standard normal distribution at the given cumulative probability.

Suppose the random variable X has standard normal distribution. This routine returns the critical value x0 in the right tail such that

P(X <= abs(x0)) = p

where p is the desired cumulative probability.

Note that due to implementation reasons, the precision of the calculated critical value is limited to 3 decimal places. That is, the return value ranges from 0.000 to 4.090.

Parameters:
 p The cumulative probability between 0.5 and 1.0. If you need critical values for the left tail, give 1 - p as the argument and negate the result.
Returns:
The critical value x0.
 double normalCumulativeProbability ( double x0 ) [static]

Return the cumulative probability that the standard normal statistic is at most x0.

Suppose the random variable X has standard normal distribution. This routine returns the cumulative probability p such that

p = P(0 <= X <= abs(x0))

where abs is the absolute value function. That is, the routine calculates the cumulative probability in the right tail. Since the normal distribution is symmetric, this can be used to calculate any cumulative probability of the distribution. Refer to a standard text on statistics for details.

Note that due to implementation reasons, the precision of the calculated critical value is limited to 6 decimal places. That is, the return value ranges from 0.000000 to 0.500000.

Parameters:
 x0 The critical value between 0.000 and 4.090, inclusive. Values beyond 4.090 are taken to be infinite (so cumulative probability is 0.50000), while the sign of negative value is changed to positive.
Returns:
The cumulative probability that 0 <= X <= abs(x0).
 double poissonDensity ( double lambda, unsigned int k ) [static]

Calculate the probability density of the Poisson distribution.

Poisson probability density is the probability of observing exactly k events, when the events occur at mean rate per unit lambda.

Parameters:
 lambda The rate parameter, which must be positive. If lambda is negative or zero, the function returns 0. k The number of events.
Returns:
The probability density, or 0 if lambda is not positive.
 double snedecorsCriticalValue ( unsigned int df1, unsigned int df2, double sigLevel ) [static]

Return the critical value of Snedecor's F distribution at the given significance level.

Suppose the random variable F has Snedecor's F distribution with df1 and df2 degrees of freedom. This routine returns the critical value f0 for which

P(F <= f0) = 1 - p

where p is the desired significance level. Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.050 < s <= 0.1000.1
0.010 < s <= 0.0500.05
s <= 0.0100.01

If sigLevel exceeds 0.100, it is truncated to 0.100.

Parameters:
 df1 The degrees of freedom for the "numerator" between 1 and 20, inclusive. df2 The degrees of freedom for the "denominator" between 1 and 100 inclusive. sigLevel Significance level. See the above table for supported values.
Returns:
The critical value f0.
 double sortedMedian ( const std::vector< double > & data ) [static]

Find the median of a sorted data vector.

The median of a sorted input vector can be found very easily:

• If the vector has an odd number of elements N, then the median is simply the element at index m = (N-1)/2.
• If the vector has an even number of elements N, then the median is the mean between the two central elements m1 = floor((N-1)/2) = N/2 - 1 and m2 = ceil((N-1)/2) = N/2.

If data is empty the method returns zero.

Parameters:
 data Any sorted vector of numbers.
Returns:
The median value of the input vector.
 double sortedMedian ( const std::deque< double > & data ) [static]

Find the median of a sorted data deque.

The median of a sorted input deque can be found very easily:

• If the deque has an odd number of elements N, then the median is simply the element at index m = (N-1)/2.
• If the deque has an even number of elements N, then the median is the mean between the two central elements m1 = floor((N-1)/2) = N/2 - 1 and m2 = ceil((N-1)/2) = N/2.

If data is empty the method returns zero.

Parameters:
 data Any sorted deque of numbers.
Returns:
The median value of the input deque.
 double studentsCriticalValue ( unsigned int df, double sigLevel ) [static]

Return the critical value of Student's t distribution at the given significance level.

Suppose the random variable t has Student's t distribution with df degrees of freedom. This routine returns the critical value t0 for which

P(t <= t0) = 1 - p

where p is the desired significance level. Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.050 < s <= 0.1000.1
0.025 < s <= 0.0500.05
0.010 < s <= 0.0250.025
0.005 < s <= 0.0100.01
0.001 < s <= 0.0050.005
s <= 0.0010.001

If sigLevel exceeds 0.100, it is truncated to 0.100.

If you need critical values for df exceeding 100, you can use the standard normal critical values as a good approximation; see normalCriticalValue().

Parameters:
 df The degrees of freedom. If df exceeds 100, it is taken to be infinite. sigLevel Significance level. See the above table for supported values.
Returns:
The critical value t0.
 double sum ( const std::vector< double > & data ) [static]

Calculate the sum of a sample vector.

Parameters:
 data Any vector of numbers.
Returns:
The sum of the elements of data.
 double sumsq ( const std::vector< double > & data ) [static]

Calculate the sum of squares of a sample vector.

Parameters:
 data Any vector of numbers.
Returns:
The sum of the squares of elements of data.
 int testDixonsN9 ( unsigned long sampleSize, double min, double x2, double xn_1, double max, double sigLevel ) [static]

Apply Dixon's N9 (r11) outlier test to the sample.

Dixon's N9 outlier test assumes that the data is normally distributed with unknown mean and variance, and that it could be contaminated by value(s) from another normal distribution with a different mean or variance. (For example, such contamination could result from gross measurement errors.) A related but different test is Grubbs N2; see testGrubbsN2().

Dixon's N9 is based on sample order statistics, which are used to estimate sample standard deviation. The test tries to remove one of two extremes: sample maximum or sample minimum. The extreme with the higher statistic will be chosen for possible removal. The method returns -1 or +1 if the chosen extreme is an outlier (statistic exceeds a critical value) and 0 otherwise. If you suspect the sample contains multiple outliers, use the test repeatedly until it returns 0, removing the corresponding outlier at each step. (Note that in sequential use, sigLevel is a significance level strictly speaking only for the first test.)

If your data is multiplicative, for example log-normal (such as peptide ratios), then you need to do a log transformation first before calling this method. See logTransform().

Description of the test procedure and critical values are presented in the following references:

• Dixon, W. J. (1950): Analysis of Extreme Values. The Annals of Mathematical Statistics 21(4):488-506.
• Dixon, W. J. (1951): Ratios Involving Extreme Values. The Annals of Mathematical Statistics 22(1):68-78.
• Verma, S. P. and A. Quiroz-Ruiz (2006): Critical values for six Dixon tests for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geológicas 23(2):133-161.

Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.200 < s <= 0.3000.3
0.100 < s <= 0.2000.2
0.050 < s <= 0.1000.1
0.020 < s <= 0.0500.05
0.010 < s <= 0.0200.02
0.005 < s <= 0.0100.01
s <= 0.0050.005

If sigLevel exceeds 0.300, it is truncated to 0.300.

Parameters:
 sampleSize Sample size. If this is not between 4 and 100, the method returns 0. min Minimum value in the sample. x2 Second smallest value in the sample. xn_1 Second largest value in the sample. max Maximum value in the sample. sigLevel Significance level. See the above table for supported values.
Returns:
One of three values: 0 if neither extreme is an outlier; -1 if min is an outlier; and +1 if max is an outlier.
 int testGrubbsN2 ( unsigned long sampleSize, double min, double mean, double max, double stdev, double sigLevel ) [static]

Apply Grubb's N2 outlier test to the sample statistics.

Grubbs N2 outlier test tries to remove one of two extremes: sample minimum or sample maximum. The data is assumed normally distributed with unknown mean and variance, and a statistic is computed for both extremes. The extreme with the higher statistic will be chosen for possible removal. The method returns -1 or +1 if the chosen extreme is an outlier (statistic exceeds a critical value) and 0 otherwise.

It is usually best to use a normal probability plot to assess whether outlier removal is needed and how many outliers there seem to be. The Grubbs N2 test assumes the data has at most one outlier, and may fail to remove the outlier if there are actually two or more outliers in the data (this is called masking). If you suspect more than one outlier, the generalised ESD test is more appropriate; see testRosners().

If your data is multiplicative, for example log-normal (such as peptide ratios), then you need to do a log transformation first before calling this method. See logTransform().

If you suspect the data is contaminated by values from another normal distribution with a different mean or variance, then Dixon's N9 test could be more appropriate. See testDixonsN9().

Details of the test procedure and critical values are presented in the following references:

• Grubbs, F. E. (1969): Procedures for Detecting Outlying Observations in Samples. Technometrics 11:1-21.
• Verma, S. P. and A. Quiroz-Ruiz (2006): Critical values for 22 discordancy test variants for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geológicas 23(3):302-319.
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.17.1

Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.200 < s <= 0.3000.3
0.100 < s <= 0.2000.2
0.050 < s <= 0.1000.1
0.020 < s <= 0.0500.05
0.010 < s <= 0.0200.02
0.005 < s <= 0.0100.01
s <= 0.0050.005

If sigLevel exceeds 0.300, it is truncated to 0.300.

Parameters:
 sampleSize Sample size. If this is not between 3 and 100, the method returns 0. min Minimum value in the sample. mean Sample arithmetic mean. max Maximum value in the sample. stdev Sample standard deviation. If stdev is not positive, the method returns 0. sigLevel Significance level. See the above table for supported values.
Returns:
One of three values: 0 if neither extreme is an outlier; -1 if min is an outlier; and +1 if max is an outlier.
 void testRosners ( const std::vector< double > & sample, unsigned int maxOutliers, double sigLevel, unsigned int *const numLow, unsigned int *const numHigh ) [static]

Apply the generalised extreme Studentised deviate (ESD) outlier test (aka Rosner's test) to the sample.

The generalised ESD test tries to remove at least 1 and up to maxOutliers outliers from the sample. If you suspect there is at most one outlier in the data, the Grubbs N2 test may be more appropriate; see testGrubbsN2().

The data is assumed normally distributed with unknown mean and variance. If your data is multiplicative, for example log-normal (such as peptide ratios), then you need to do a log transformation first before calling this method. See logTransform().

Description of the test procedure can be found in the following references:

Significance level is determined by finding the smallest standard significance level that exceeds sigLevel:

sigLevel (s)p
0.100 < s <= 0.2000.2
0.050 < s <= 0.1000.1
0.020 < s <= 0.0500.05
0.010 < s <= 0.0200.02
0.002 < s <= 0.0100.01
s <= 0.0020.002

If sigLevel exceeds 0.200, it is truncated to 0.200.

Parameters:
 [in] sample A sample of data sorted in increasing numerical order. Sample size must be at least 25 for meaningful results. [in] maxOutliers Number between 1 and 10 (inclusive) indicating the maximum number of suspected outliers in the sample. [in] sigLevel Significance level. See the above table for supported values. [out] numLow Number of outliers at the low end of sample; values at indices from 0 to numLow - 1 are outliers based on the test. If numLow is 0, the sample has no outliers at the low end. [out] numHigh Number of outliers at the high end of sample; values at indices from N - numHigh to the end of the vector are outliers based on the test (N is the size of sample). If numHigh is 0, the sample has no outliers at the high end.
 bool testShapiroWilkW ( const std::vector< double > & sample, double *const W, double *const pValue ) [static]

Apply Shapiro-Wilk W normality test to the sample.

See ms_shapiro_wilk for the test details. This method returns false if any of the following conditions apply:

• Sample size is less than 3.
• Sample range (max - min) is not positive.

In these cases the test is not conducted, and the values of W and pValue are not defined.

The number of right-censored values is assumed to be 0. If your sample contains censored values, use ms_shapiro_wilk directly.

Parameters:
 [in] sample A sample of data sorted in increasing numerical order. The number of elements should be between 3 and 5000, inclusive. If sample size exceeds 5000, the test statistic and p-value will be calculated successfully, but the p-value is not guaranteed to be accurate. [out] W The Shapiro-Wilk W statistic calculated from the data. [out] pValue The p-value of W under the null hypothesis.
Returns:
True if the test was conducted; false otherwise.
 double unsortedMedian ( const std::vector< double > & data ) [static]

Find the median of an unsorted data vector.

This method works for both sorted and unsorted input, although in the former case it is faster to call sortedMedian() instead.

The trivial implementation is to create a copy of the input data, sort and then call sortedMedian(). This method is slightly faster: If the input vector has an odd number of elements N, then the median would be at index m = (N-1)/2; otherwise the median is between elements m1 = floor((N-1)/2) and m2 = ceil((N-1)/2). The method uses partial sorting to find the mth element (or the m1st and m2nd elements) which means time complexity is slightly less than O(N log(N)) (which is the average time complexity of quicksort).

Parameters:
 data Any vector of numbers.
Returns:
The median of data, or 0 if the vector is empty.
 double weightedArithmeticMean ( const std::vector< double > & data, const std::vector< double > & weights ) [static]

Calculate the weighted arithmetic mean.

The weighted arithmetic mean is

mean(data, weights) = (1/W)sumi wixi

where W is the sum of weights, wi is an individual weight, and the summation is over all elements xi.

Parameters:
 data Any vector of numbers. weights Positive weight for each datum in data. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than data, the missing weights are taken to be 1.0.
Returns:
The weighted arithmetic mean of data, or 0 if the vector is empty.
 double weightedArithmeticStandardDeviation ( const std::vector< double > & data, const std::vector< double > & weights, double weightedMean ) [static]

Calculate the weighted arithmetic sample standard deviation.

The weighted arithmetic sample deviation is the square root of the weighted arithmetic sample variance. The weighted sample variance is

var(data,weights) = var(data)/b(weights)

where the effective base b is

b(weights) = (sumi wi)2/sumi(wi2)

Parameters:
 data A sample of data. weights Positive weight for each datum in data. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than data, the weights of those elements are taken to be 1.0. weightedMean Sample weighted arithmetic mean.
Returns:
Sample standard deviation, or 0 if sample size is 0 or 1.
 double weightedArithmeticStandardError ( const std::vector< double > & weights, double weightedStdev, unsigned long sampleSize ) [static]

Calculate the standard error of the weighted arithmetic sample mean.

The standard error of the weighted arithmetic mean is slightly different from the standard error of the mean. Like the weighted standard deviation, the weights must be taken into account. The effective base b is

b(weights) = (sumi wi)2/sumi(wi2) = W2/S,

where W2 is the squared sum of the weights, and S is the sum of squared weights. As the weighted mean is

mean(data, weights) = (1/W)sumi wixi,

we can use the usual sample properties (independent, identically distributed random variables) to easily derive

var(mean) = sigma2/(b/W)

where sigma2 is the population variance. The sample estimate of the standard error is then

sqrt(W s2/b2) = sqrt(W) s/b,

which is the value returned by this function.

Parameters:
 weights Positive weight for each datum. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than sampleSize, the weights of those elements are taken to be 1.0. weightedStdev Sample weighted arithmetic standard deviation. It is recommended to use weightedArithmeticStandardDeviation() with the same weights. sampleSize The sample size.
Returns:
Weighted standard error, or 0 if sample size is 0 or 1.
 double weightedGeometricMean ( const std::vector< double > & data, const std::vector< double > & weights ) [static]

Calculate the weighted geometric mean.

The weighted geometric mean is

geom(data,weights) = (prodi xiwi)1/W

where W is the sum of weights, wi is an individual weight and the product is over all the elements xi. That is, the geometric mean is the Wth root of the product of the with roots of the input values.

Equivalently, the logarithm of the weighted geometric mean is equal to the weighted arithmetic mean of the data in log space. This is actually how the mean is calculated by the method.

Parameters:
 data Any vector of numbers. weights Positive weight for each datum in data. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than data, the missing weights are taken to be 1.0.
Returns:
The weighted geometric mean of data, or 1 if the vector is empty. If any of the data elements are 0, the mean is 0.
 double weightedGeometricStandardDeviation ( const std::vector< double > & data, const std::vector< double > & weights, double weightedMean ) [static]

Calculate the weighted geometric sample standard deviation.

The weighted geometric sample deviation is the square root of the weighted geometric sample variance. The geometric sample variance is

var(logdata,weights) = evar(logdata)/b

where the effective base b is

b(weights) = (sumi wi)2/sumi(wi2)

and logdata is the log-transformed data.

Equivalently, the logarithm of the weighted geometric standard deviation is equal to the weighted arithmetic standard deviation of the data in log space.

Parameters:
 data A sample of data. weights Positive weight for each datum in data. Negative and zero weights are coerced to 1.0, and if weights has fewer elements than data, the weights of those elements are taken to be 1.0. weightedMean Sample weighted geometric mean.
Returns:
Sample standard deviation, or 0 if sample size is 0 or 1.

The documentation for this class was generated from the following files: