Chi-squared distribution

Definitions

If Z1, ..., Z**k are independent, standard normal random variables, then the sum of their squares, X\ = \sum_{i=1}^k Z_i^2, is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as X\ \sim\ \chi^2(k)\ \ \text{or}\ \ X\ \sim\ \chi^2_k.

The chi-squared distribution has one parameter: a positive integer k that specifies the number of degrees of freedom (the number of random variables Z**i being summed).

Introduction

The chi-squared distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables
• Wald test
• Score test

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason for which the chi-squared distribution is extensively used in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Suppose that Z is a random variable sampled from the standard normal distribution, where the mean is 0 and the variance is 1: Z \sim N(0,1). Now, consider the random variable X = Z^2. The distribution of the random variable X is an example of a chi-squared distribution: \ X\ \sim\ \chi^2_1. The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT). LRTs have several desirable properties; in particular, simple LRTs commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for a small sample size, and it is preferable to use Fisher's exact test. Ramsey shows that the exact binomial test is always more powerful than the normal approximation.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

\chi = \frac{m - Np}{\sqrt{Npq}}

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 - p.

Squaring both sides of the equation gives

\chi^2 = \frac{\left(m - Np\right)^2}{Npq}

Using N = Np + N(1 - p), N = m + (N - m), and q = 1 - p, this equation can be rewritten as

\chi^2 = \frac{\left(m - Np\right)^2}{Np} + \frac{\left(N - m - Nq\right)^2}{Nq}

The expression on the right is of the form that Karl Pearson would generalize to the form

\chi^2 = \sum_{i=1}^n \frac{\left(O_i - E_i\right)^2}{E_i}

where

• \chi^2 = Pearson's cumulative test statistic, which asymptotically approaches a \chi^2 distribution;
• O_i = the number of observations of type i;
• E_i = N p_i = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is p_i; and
• n = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a degenerate multivariate normal approximation to the multinomial distribution (the numbers in each category add up to the total sample size, which is considered fixed). Pearson showed that the chi-squared distribution arose from such a multivariate normal approximation to the multinomial distribution, taking careful account of the statistical dependence (negative correlations) between numbers of observations in different categories.

Probability density function

The probability density function (pdf) of the chi-squared distribution is f(x;,k) = \begin{cases} \dfrac{x^{k/2 -1} e^{-x/2}}{2^{k/2} , \Gamma{\left(\frac k 2 \right)}}, & x 0; \ 0, & \text{otherwise}. \end{cases} where \Gamma(k/2) denotes the gamma function, which has closed-form values for integer k.

For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chi-squared distribution.

Cumulative distribution function

Its cumulative distribution function is: F(x;,k) = \frac{\gamma{\left(\frac{k}{2},,\frac{x}{2}\right)}}{\Gamma{\left(\frac{k}{2}\right)}} = P{\left(\frac{k}{2},,\frac{x}{2}\right)}, where \gamma(s,t) is the lower incomplete gamma function and P(s,t) is the regularized gamma function.

In a special case of k = 2 this function has the simple form: F(x;,2) = 1 - e^{-x/2} which can be easily derived by integrating f(x;,2) = \frac{1}{2}e^{-x/2} directly. The integer recurrence of the gamma function makes it easy to compute F(x;,k) for other small, even k.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.

Letting z \equiv x/k, Chernoff bounds on the lower and upper tails of the CDF may be obtained. For the cases when 0 (which include all of the cases when this CDF is less than half): F(z k;,k) \leq (z e^{1-z})^{k/2}.

The tail bound for the cases when z 1, similarly, is 1-F(z k;,k) \leq (z e^{1-z})^{k/2}.

For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-squared distribution.

Properties

Cochran's theorem

Main article: Cochran's theorem

The following is a special case of Cochran's theorem.

Theorem. If Z_1,...,Z_n are independent identically distributed (i.i.d.), standard normal random variables, then \sum_{t=1}^n \left(Z_t - \bar Z\right)^2 \sim \chi^2_{n-1} where \bar Z = \frac{1}{n} \sum_{t=1}^n Z_t.

Proof. Let Z\sim\mathcal{N}(\bar 0,1!!1) be a vector of n independent normally distributed random variables, and \bar Z their average. Then \sum_{t=1}^n(Z_t-\bar Z)^2 = \sum_{t=1}^n Z_t^2 -n\bar Z^2 = Z^\top[1!!1 -{\textstyle\frac1n}\bar 1\bar 1^\top]Z =: Z^\top!M Z where 1!!1 is the identity matrix and \bar 1 the all ones vector. M has one eigenvector b_1:={\textstyle\frac{1}{\sqrt{n}}} \bar 1 with eigenvalue 0, and n-1 eigenvectors b_2,...,b_n (all orthogonal to b_1) with eigenvalue 1, which can be chosen so that Q:=(b_1,...,b_n) is an orthogonal matrix. Since also X:=Q^\top!Z\sim\mathcal{N}(\bar 0,Q^\top!1!!1 Q) =\mathcal{N}(\bar 0,1!!1), we have \sum_{t=1}^n(Z_t-\bar Z)^2 = Z^\top!M Z = X^\top!Q^\top!M Q X = X_2^2+...+X_n^2 \sim \chi^2_{n-1}, which proves the claim.

Additivity

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if X_i,i=\overline{1,n} are independent chi-squared variables with k_i, i=\overline{1,n} degrees of freedom, respectively, then Y = X_1 + \cdots + X_n is chi-squared distributed with k_1 + \cdots + k_n degrees of freedom.

Sample mean

The sample mean of n i.i.d. chi-squared variables of degree k is distributed according to a gamma distribution with shape \alpha and scale \theta parameters: \overline X = \frac{1}{n} \sum_{i=1}^n X_i \sim \operatorname{Gamma}\left(\alpha{=}\tfrac{n k}{2}, ,\theta{=}\tfrac{2}{n} \right) \qquad \text{where } X_i \sim \chi^2(k)

Asymptotically, given that for a shape parameter \alpha going to infinity, a Gamma distribution converges towards a normal distribution with expectation \mu = \alpha \theta and variance the sample mean converges towards:

\overline X \xrightarrow{n \to \infty} N{\left(\mu{=}k, , \sigma^2{=}\tfrac{2k}{n} \right)}

Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-squared variable of degree k the expectation is k , and its variance 2k (and hence the variance of the sample mean \overline{X} being \sigma^2 = \tfrac{2k}{n} ).

Entropy

The differential entropy is given by \begin{align} h &= \int_0^\infty f(x;,k) \ln f(x;,k) , dx \ &= \frac k 2 + \ln \left[2,\Gamma{\left(\frac k 2 \right)}\right] + \left(1-\frac k 2 \right) \psi!\left(\frac k 2 \right), \end{align} where \psi(x) is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate X for which \operatorname{E}(X)=k and \operatorname{E}(\ln(X))=\psi(k/2)+\ln(2) are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the log moment of gamma. For derivation from more basic principles, see the derivation in moment-generating function of the sufficient statistic.

Noncentral moments

The noncentral moments (raw moments) of a chi-squared distribution with k degrees of freedom are given by \begin{align} \operatorname{E}(X^m) &= k (k+2) (k+4) \cdots (k+2m-2) \[1ex] &= 2^m \frac{\Gamma{\left(m+\frac{k}{2}\right)}}{\Gamma{\left(\frac{k}{2}\right)}}. \end{align}

Cumulants

The cumulants are readily obtained by a power series expansion of the logarithm of the characteristic function: \kappa_n = 2^{n-1}(n-1)!,k with cumulant generating function \ln \operatorname{E}[e^{tX}] = - \frac{k}{2} \ln(1-2t) .

Concentration

The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart bounds are: \Pr(X - k \ge 2 \sqrt{k x} + 2x) \le e^{-x} \Pr(k - X \ge 2 \sqrt{k x}) \le e^{-x} One consequence is that, if Z \sim N(0, 1)^k is a Gaussian random vector in \R^k, then as the dimension k grows, the squared length of the vector is concentrated tightly around k with a width k^{1/2 + \alpha}:\Pr\left(\left|Z\right|^2 \in \left[k - 2k^{1/2+\alpha}, ; k + 2k^{1/2+\alpha} + 2k^{\alpha}\right]\right) \geq 1-e^{-k^\alpha}where the exponent \alpha can be chosen as any value in \R.

Since the cumulant generating function for \chi^2(k) is K(t) = -\frac k2 \ln(1-2t) , and its convex dual is K^*(q) = \frac{1}{2} \left(q - k + k\ln\frac{k}{q}\right) , the standard Chernoff bound yields\begin{aligned} \ln \Pr(X \geq (1 + \varepsilon) k) &\leq -\frac{k}{2} \left( \varepsilon - \ln(1+\varepsilon)\right) \ \ln \Pr(X \leq (1 - \varepsilon) k) &\leq -\frac{k}{2} \left(-\varepsilon - \ln(1-\varepsilon)\right) \end{aligned}where 0 . By the union bound,Pr(X \in (1\pm \varepsilon ) k ) \geq 1 - 2 e^{-\frac k2 (\frac{1}{2} \varepsilon^2 - \frac 13 \varepsilon^3)} This result is used in proving the Johnson–Lindenstrauss lemma.

Asymptotic properties

By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k50 the distribution is sufficiently close to a normal distribution, so the difference is ignorable. Specifically, if X \sim \chi^2(k), then as k tends to infinity, the distribution of (X-k)/\sqrt{2k} tends to a standard normal distribution. However, convergence is slow as the skewness is \sqrt{8/k} and the excess kurtosis is 12/k.

The sampling distribution of \ln(\chi^2) converges to normality much faster than the sampling distribution of \chi^2, as the logarithmic transform removes much of the asymmetry.

Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

• If X \sim \chi^2(k) then \sqrt{2X} is approximately normally distributed with mean \sqrt{2k-1} and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of Johnson).
• If X \sim \chi^2(k) then \sqrt[3]{X/k} is approximately normally distributed with mean 1-\frac{2}{9k} and variance \frac{2}{9k} . This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of Johnson.
  • This normalizing transformation leads directly to the commonly used median approximation k\bigg(1-\frac{2}{9k}\bigg)^3; by back-transforming from the mean, which is also the median, of the normal distribution.

Related distributions

• As k\to\infty, (\chi^2_k-k)/\sqrt{2k} ~ \xrightarrow{d}\ N(0,1) , (normal distribution)
• \chi_k^2 \sim {\chi'}^2_k(0) (noncentral chi-squared distribution with non-centrality parameter \lambda = 0 )
• If Y \sim \mathrm{F}(\nu_1, \nu_2) then X = \lim_{\nu_2 \to \infty} \nu_1 Y has the chi-squared distribution \chi^2_{\nu_{1}}
  • As a special case, if Y \sim \mathrm{F}(1, \nu_2), then X = \lim_{\nu_2 \to \infty} Y, has the chi-squared distribution \chi^2_{1}
• \left|\boldsymbol{N}_{i=1,\ldots,k} (0,1) \right|^2 \sim \chi^2_k (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
• If X \sim \chi^2_\nu, and c0 ,, then cX \sim \Gamma(k=\nu/2, \theta=2c),. (gamma distribution)
• If X \sim \chi^2_k then \sqrt{X} \sim \chi_k (chi distribution)
• If X \sim \chi^2_2, then X \sim \operatorname{exp}(1/2) is an exponential distribution. (See gamma distribution for more.)
• If X \sim \chi^2_{2k}, then X \sim \operatorname{Erlang}(k, 1/2) is an Erlang distribution.
• If X \sim \operatorname{Erlang}(k,\lambda), then 2\lambda X\sim \chi^2_{2k}
• If X \sim \operatorname{Rayleigh}(1), (Rayleigh distribution) then X^2 \sim \chi^2_2,
• If X \sim \operatorname{Maxwell}(1), (Maxwell distribution) then X^2 \sim \chi^2_3,
• If X \sim \chi^2_\nu then \tfrac{1}{X} \sim \operatorname{Inv-}\chi^2_\nu, (Inverse-chi-squared distribution)
• The chi-squared distribution is a special case of type III Pearson distribution
• If X \sim \chi^2_{\nu_1}, and Y \sim \chi^2_{\nu_2}, are independent then \tfrac{X}{X+Y} \sim \operatorname{Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2}), (beta distribution)
• If X \sim \operatorname{U}(0,1), (uniform distribution) then -2\log(X) \sim \chi^2_2,
• If X_i \sim \operatorname{Laplace}(\mu,\beta), then \sum_{i=1}^n \frac{2 |X_i-\mu|}{\beta} \sim \chi^2_{2n},
• If X_i follows the generalized normal distribution (version 1) with parameters \mu,\alpha,\beta then \sum_{i=1}^n \frac{2 |X_i-\mu|^\beta}{\alpha} \sim \chi^2_{2n/\beta},
• The chi-squared distribution is a transformation of Pareto distribution
• Student's t-distribution is a transformation of chi-squared distribution
• Student's t-distribution can be obtained from chi-squared distribution and normal distribution
• The noncentral beta distribution can be obtained as a transformation of chi-squared distribution and noncentral chi-squared distribution
• The noncentral t-distribution can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

If Y is a k-dimensional Gaussian random vector with mean vector \mu and rank k covariance matrix C, then X = (Y-\mu )^\mathsf{T}C^{-1}(Y-\mu) is chi-squared distributed with k degrees of freedom.

The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi-squared distribution.

If Y is a vector of k i.i.d. standard normal random variables and A is a k\times k symmetric, idempotent matrix with rank k-n, then the quadratic form Y^\mathsf{T}!AY is chi-square distributed with k-n degrees of freedom.

If \Sigma is a p\times p positive-semidefinite covariance matrix with strictly positive diagonal entries, then for X\sim N(0,\Sigma) and w a random p-vector independent of X such that w_1+\cdots+w_p=1 and w_i\geq 0, i=1,\ldots,p, then

\frac{1}{\tilde{w}^\mathsf{T} \Sigma \tilde{w}} \sim \chi_1^2.where \tilde{w} = (w_1/X_1, \dots, w_p/X_p).

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

• Y is F-distributed, Y \sim F(k_1, k_2) if Y = \frac{ {X_1}/{k_1} }{ {X_2}/{k_2} }, where X_1 \sim \chi^2_{k_1} and X_2 \sim \chi^2_{k_2} are statistically independent.
• If X_1 \sim \chi^2_{k_1} and X_2 \sim \chi^2_{k_2} are statistically independent, then X_1 + X_2\sim \chi^2_{k_1+k_2}. If X_1 and X_2 are not independent, then X_1+X_2 is not chi-square distributed.

Generalizations

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

Linear combination

If X_1,\ldots,X_n are chi square random variables and a_1,\ldots,a_n\in\mathbb{R}{0}, then the distribution of X = \sum{i=1}^n a_i X_i is a special case of the generalized chi-squared distribution. A closed expression for this distribution is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-square random variables.{{cite journal |article-number=505202

Chi-squared distributions

Noncentral chi-squared distribution

Main article: Noncentral chi-squared distribution

The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-squared distribution

Main article: Generalized chi-squared distribution

The generalized chi-squared distribution is obtained from the quadratic form z'Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions

The chi-squared distribution X \sim \chi_k^2 is a special case of the gamma distribution, in that X \sim \Gamma{\left(\tfrac{k}{2},\tfrac{1}{2}\right)} using the rate parameterization of the gamma distribution (or X \sim \Gamma {\left(\tfrac{k}{2},2 \right)} using the scale parameterization of the gamma distribution) where k is an integer.

Because the exponential distribution is also a special case of the gamma distribution, we also have that if X \sim \chi_2^2, then X\sim \operatorname{exp}\left(\tfrac 1 2\right) is an exponential distribution.

The Erlang distribution is also a special case of the gamma distribution and thus we also have that if X \sim\chi_k^2 with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

Occurrence and applications{{anchor|Applications}}

The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.

• if X_1, ..., X_n are i.i.d. N(\mu, \sigma^2) random variables, then \sum_{i=1}^n \left(X_i - \bar{X}\right)^2 \sim \sigma^2 \chi^2_{n-1} where \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i.
• The box below shows some statistics based on X_i \sim N(\mu_i, \sigma^2_i), i= 1, \ldots, k independent random variables that have probability distributions related to the chi-squared distribution:

The chi-squared distribution is also often encountered in magnetic resonance imaging.

Computational methods

Table of {{math|''χ''²}} values vs {{math|''p''}}-values

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the p-value. A low p-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results.

The table below gives a number of p-values matching to \chi^2 for the first 10 degrees of freedom.

These values can be calculated evaluating the quantile function (also known as "inverse CDF" or "ICDF") of the chi-squared distribution; e. g., the χ2 ICDF for and yields 2.1673 ≈ 2.17 as in the table above, noticing that 1 – p is the p-value from the table.

History

This distribution was first described by the German geodesist and statistician Friedrich Robert Helmert in papers of 1875–6, where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmert'sche ("Helmertian") or "Helmert distribution".

The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-squared test, published in 1900, with computed table of values published in , collected in . The name "chi-square" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing −χ2 for what would appear in modern notation as −xTΣ−1x (Σ being the covariance matrix).R. L. Plackett, Karl Pearson and the Chi-Squared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics. The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.

References

Sources

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