  Chi-squared tests for lack of fit are traditionally employed to find evidence
against a hypothesized model, with the model accepted if the Karl Pearson
statistic comparing observed and expected numbers of observations falling
within cells is not significantly large. However, if one really wants evidence
for goodness of fit, it is better to adopt an equivalence testing approach in
which small values of the chi-squared statistic are evidence for the desired
model. This method requires one to define what is meant by equivalence to the
desired model, and guidelines are proposed. Then a simple extension of the
classical normalizing transformation for the non-central chi-squared
distribution places these values on a simple to interpret calibration scale for
evidence. It is shown that the evidence can distinguish between normal and
nearby models, as well between the Poisson and over-dispersed models.
Applications to evaluation of random number generators and to uniformity of the
digits of pi are included. Sample sizes required to obtain a desired expected
evidence for goodness of fit are also provided.
