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A chi-squared test (also chi-square or χ 2 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 statistic can then be used to calculate a p-value by comparing the value of the statistic to a chi-squared distribution. The number of degrees of freedom is equal to the number of cells , minus the reduction in degrees of freedom, . The chi-squared statistic can be also calculated as
These values can be calculated evaluating the quantile function (also known as "inverse CDF" or "ICDF") of the chi-squared distribution; [23] e. g., the χ 2 ICDF for p = 0.05 and df = 7 yields 2.1673 ≈ 2.17 as in the table above, noticing that 1 – p is the p-value from the table.
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]
In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables .
The resulting value can be compared with a chi-square distribution to determine the goodness of fit. The chi-square distribution has ( k − c ) degrees of freedom , where k is the number of non-empty bins and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one.
Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit as the fitted model deviates less from the saturated model. When assessed upon a chi-square distribution, non-significant chi-square values indicate very little unexplained variance and thus, good model fit.
Using the poisson-weighted mixture representation for , and the fact that the sum of chi-squared random variables is also a chi-square, completes the result. The indices in the series are (1 + 2 i ) + ( k − 1) = k + 2 i as required.