IGNOU MS 8 2011 - Test of goodness of fit

Question

Test of goodness of fit

02 Nov 2017

Answer :

Word Count : 982

he goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions or whether outcome frequencies follow a specified distribution In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.

Fit of distributions

In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used:

Regression analysis

In regression analysis, the following topics relate to goodness of fit:

Coefficient of determination (The R squared measure of goodness of fit);

Lack-of-fit sum of squares.

Example

One way in which a measure of goodness of fit statistic can be constructed, in the case where the variance of the measurement error is known, is to construct a weighted sum of squared errors:

Image ignouacademy-com--p-ignou-45920

where σ2 is the known variance of the observation.

This definition is only useful when one has estimates for the error on the measurements, but it leads to a situation where a chi-square distribution can be used to test goodness of fit, provided that the errors can be assumed to have a normal distribution.

The reduced chi-squared statistic is simply the chi-squared divided by the number of degrees of freedom:

where ν is the number of degrees of freedom, usually given by N − n − 1, where N is the number of observations, and n is the number of fitted parameters, assuming that the mean value is an additional fitted parameter. 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. However indicates that the model is 'over-fitting' the data (either the model is improperly fitting noise, or the error variance has been over-estimated). A indicates that the

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