[section:chi_squared_dist Chi Squared Distribution] ``#include `` namespace boost{ namespace math{ template class chi_squared_distribution; typedef chi_squared_distribution<> chi_squared; template class chi_squared_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor: chi_squared_distribution(RealType i); // Accessor to parameter: RealType degrees_of_freedom()const; // Parameter estimation: static RealType find_degrees_of_freedom( RealType difference_from_mean, RealType alpha, RealType beta, RealType sd, RealType hint = 100); }; }} // namespaces The Chi-Squared distribution is one of the most widely used distributions in statistical tests. If [chi][sub i][space] are [nu][space] independent, normally distributed random variables with means [mu][sub i][space] and variances [sigma][sub i][super 2], then the random variable: [equation chi_squ_ref1] is distributed according to the Chi-Squared distribution. The Chi-Squared distribution is a special case of the gamma distribution and has a single parameter [nu][space] that specifies the number of degrees of freedom. The following graph illustrates how the distribution changes for different values of [nu]: [graph chi_squared_pdf] [h4 Member Functions] chi_squared_distribution(RealType v); Constructs a Chi-Squared distribution with /v/ degrees of freedom. Requires v > 0, otherwise calls __domain_error. RealType degrees_of_freedom()const; Returns the parameter /v/ from which this object was constructed. static RealType find_degrees_of_freedom( RealType difference_from_variance, RealType alpha, RealType beta, RealType variance, RealType hint = 100); Estimates the sample size required to detect a difference from a nominal variance in a Chi-Squared test for equal standard deviations. [variablelist [[difference_from_variance][The difference from the assumed nominal variance that is to be detected: Note that the sign of this value is critical, see below.]] [[alpha][The maximum acceptable risk of rejecting the null hypothesis when it is in fact true.]] [[beta][The maximum acceptable risk of falsely failing to reject the null hypothesis.]] [[variance][The nominal variance being tested against.]] [[hint][An optional hint on where to start looking for a result: the current sample size would be a good choice.]] ] Note that this calculation works with /variances/ and not /standard deviations/. The sign of the parameter /difference_from_variance/ is important: the Chi Squared distribution is asymmetric, and the caller must decide in advance whether they are testing for a variance greater than a nominal value (positive /difference_from_variance/) or testing for a variance less than a nominal value (negative /difference_from_variance/). If the latter, then obviously it is a requirement that `variance + difference_from_variance > 0`, since no sample can have a negative variance! This procedure uses the method in Diamond, W. J. (1989). Practical Experiment Designs, Van-Nostrand Reinhold, New York. See also section on Sample sizes required in [@http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm the NIST Engineering Statistics Handbook, Section 7.2.3.2]. [h4 Non-member Accessors] All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. (We have followed the usual restriction of the mode to degrees of freedom >= 2, but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0, and provide an extended definition that would avoid a discontinuity in the mode as alternative code in a comment). The domain of the random variable is \[0, +[infin]\]. [h4 Examples] Various [link math_toolkit.stat_tut.weg.cs_eg worked examples] are available illustrating the use of the Chi Squared Distribution. [h4 Accuracy] The Chi-Squared distribution is implemented in terms of the [link math_toolkit.sf_gamma.igamma incomplete gamma functions]: please refer to the accuracy data for those functions. [h4 Implementation] In the following table /v/ is the number of degrees of freedom of the distribution, /x/ is the random variate, /p/ is the probability, and /q = 1-p/. [table [[Function][Implementation Notes]] [[pdf][Using the relation: pdf = __gamma_p_derivative(v / 2, x / 2) / 2 ]] [[cdf][Using the relation: p = __gamma_p(v / 2, x / 2) ]] [[cdf complement][Using the relation: q = __gamma_q(v / 2, x / 2) ]] [[quantile][Using the relation: x = 2 * __gamma_p_inv(v / 2, p) ]] [[quantile from the complement][Using the relation: x = 2 * __gamma_q_inv(v / 2, p) ]] [[mean][v]] [[variance][2v]] [[mode][v - 2 (if v >= 2)]] [[skewness][2 * sqrt(2 / v) == sqrt(8 / v)]] [[kurtosis][3 + 12 / v]] [[kurtosis excess][12 / v]] ] [h4 References] * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm NIST Exploratory Data Analysis] * [@http://en.wikipedia.org/wiki/Chi-square_distribution Chi-square distribution] * [@http://mathworld.wolfram.com/Chi-SquaredDistribution.html Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.] [endsect][/section:chi_squared_dist Chi Squared] [/ chi_squared.qbk Copyright 2006 John Maddock and Paul A. Bristow. Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt). ]