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What is the Goodness of Fit Test
The goodness of fit of a statistical model describes how well it fits a set of observations and the measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures are mainly used in statistical hypothesis testing.
To test whether two samples are drawn from identical distributions and whether outcome frequencies follow a specified distribution and in the analysis of variance one of the components into which the variance is partitioned may be a Lack-of-fit sum of squares.
One way in which a measure of goodness of this can be constructed is in the case where the variance of the measurement error is known. This is to construct a weighted sum of squared errors and it is denoted as
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where is the known variance of the observation and O is the observed data, as well as E, is the theoretical data this definition is only useful when one has estimates for the error in the measurements. It leads to a situation where a Chi-squared distribution can be used to test the goodness of fit and this is used when the errors are assumed to have a normal distribution.
The chi-squared statistic that is reduced is simply the chi-square divided by the number of degrees of freedom and it is denoted as follows
where is the number of degrees of freedom usually given by, where is the number of observations and is the number of fitted parameters. It assumes that the mean value is an additional fitted parameter and the advantage of the reduced chi-squared is that it already normalizes for the number of data points and model complexity.
As a rule of thumb, a large indicates a poor model fit and indicates that the model is 'over-fitting' the data either the model is improperly fitting noise or there has been an overestimation of the error variance.
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A indicates that the fit has not fully captured the data or that the error variance has been underestimated and the principle a value of indicates the extent of the match between observations. It is used to estimate is in accord with the error variance.
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