Goodness of Fit: Models

4 Introduction Goodness of fit means how well a statistical model 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 (Kolmogorov-Smirnov test), or whether outcome frequencies follow a specified distribution (Pearson's chi-square test). A statistical test in which the validity of one hypothesis is tested without specification of an alternative hypothesis is called a goodness-of-fit-test. The general procedure consist in defining a test statistic, which is some function of the data measuring the distance between the hypothesis and the data (in fact, the badness-of-fit), and than calculating the probability of obtaining data which have a still larger value of this test statistic than the value observed, assuming that the hypothesis is true. This probability is called the size of the test or confidence level. Small probabilities (say, less than one percent) indicate a poor fit. Especially high probabilities (close to one) correspond to a fit which is too good to happen very often, and may indicate a