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Reduced chi-squared statistic. In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation ( MSWD) in isotopic dating [1] and variance of unit weight in the context of weighted least squares. [2] [3]
A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables ( two dimensions of the contingency table) are independent in influencing the test statistic ...
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.
From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable J has a Poisson distribution with mean , and the conditional distribution of Z given J = i is chi-squared with k + 2 i degrees of freedom.
Pearson's chi-square test. Pearson's chi-square test uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies (that is, counts of observations), each squared and divided by the expectation: where: Oi = an observed count for bin i. Ei = an expected count for bin i, asserted by the null ...
The Pearson's chi-squared test statistic is defined as . The p-value of the test statistic is computed either numerically or by looking it up in a table. If the p-value is small enough (usually p < 0.05 by convention), then the null hypothesis is rejected, and we conclude that the observed data does not follow the multinomial distribution.
Relation to the chi-squared test. The commonly used chi-squared tests for goodness of fit to a distribution and for independence in contingency tables are in fact approximations of the log-likelihood ratio on which the G-tests are based. The general formula for Pearson's chi-squared test statistic is
CF. In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution [1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It arises in Bayesian inference, where it can be used as the prior and posterior distribution for ...