Goodness of fit statistics

See also: Fitting distributions to data, Fitting in ModelRisk, Analyzing and using data, Goodness of Fit Plots, Kolmogorov-Smirnoff (K-S) Statistic, Anderson-Darling (A-D) Statistic

Many goodness-of-fit statistics have been developed but two are in most common use. These are the Chi-Squared (c2) and Kolmogorov-Smirnoff (K-S) statistics, generally used for discrete and continuous distributions respectively. The Anderson-Darling (A-D) statistic is also explained as a sophistication of the K-S statistic. The lower the value of these statistics, the more likely the data could have come from the hypothesised distribution.

Data to be analyzed may come in one of three forms: the raw data points; relative frequencies from a histogram (xi, pi) pairs; or points from a cumulative plot (xi, Pi) pairs. The c2 and K-S statistics can be calculated from any data form, but the raw data points must be available in order to calculate the A-D statistic. Data can also be censored, for which use of goodness-of-fit statistics is more complex.

Goodness-of-fit statistics are not intuitively easy to understand or interpret. They do not provide a true measure of the probability that the data actually comes from the fitted distribution. Instead, they provide a probability that random data generated from the fitted distribution would have produced a goodness-of-fit statistic value as low as that calculated for the observed data. By far the most intuitive measure of goodness-of-fit is a visual comparison of probability distributions. We recommend you produce these plots to assure yourself of the validity of the fit before labouring over goodness-of-fit statistics.

Information criteria

Although still popular today, the Chi-Squared, Kolmogorov-Smirnoff and Anderson-Darling goodness of fit statistics are technically all inappropriate as a method of comparing fits of distributions to data. They are also limited to having precise observations and cannot incorporate censored, truncated or binned data.

Realistically, most of the time we are fitting a continuous distribution to a set of precise observations and then the Anderson-Darling does a reasonable job.

For important work you should instead consider using statistical measures of fit called information criteria.

 

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