Goodness-of-Fit Measures

Additional Examples

Goodness-of-fit measures are not limited to regression models but may apply to any hypothetical statement about the study POPULATION. The following are three examples of models and corresponding goodness-of-fit measures.

The HYPOTHESIS that a variable X has a NORMAL DISTRIBUTION in the population is an example of a model. With a sample of observations on X, the goodness-of-fit statistic D can be calculated as a measure of how well the sample data conform to a normal distribution. In addition, the D statistic can be used in the KOLMOGOROV-SMIRNOV TEST to formally test the hypothesis that X is normally distributed in the population.

Another example of a model is the hypothesis that the mean, denoted μ, of the distribution of X is equal to 100, or μX = 100. The T-STATISTIC can be calculated from a sample of data to measure the goodness of fit between the sample mean X¯ and the hypothesized population mean μX. The t-statistic can then be used in a T-TEST to decide if the hypothesis μX = 100 should be rejected based on the sample data.

One more example is the INDEPENDENCE model, which states that two categorical variables are unrelated in the population. The numbers expected in each category of a CONTINGENCY TABLE of the sample data, if the independence model is true, are called the “EXPECTED FREQUENCIES,” and the numbers actually observed in the sample are called the “OBSERVED FREQUENCIES.” The goodness of fit between the hypothesized independence model and the sample data is measured with the χ2 (chi-square) statistic, which can then be used with the χ2 test (CHI-SQUARE TEST) to decide if the independence model should be rejected.

Hypothesis Tests

The examples above illustrate two distinct classes of goodness-of-fit measures that serve different purposes. Each summarizes the consistency between sample data and a model, but R2 is a descriptive statistic with a clear intuitive interpretation, whereas D,t, and χ2[Page 436]are INFERENTIAL STATISTICS that are less meaningful intrinsically. The inferential measures are valuable for conducting HYPOTHESIS TESTS about goodness of fit, principally because each has a known SAMPLING DISTRIBUTION that often has the same name as the statistic (e.g., t and χ2). Even if a model accurately represents the population under study, a goodness-of-fit statistic calculated from a sample of data will rarely indicate a perfect fit with the model, and the extent of departure from perfect fit will vary from one sample to another. This variation is captured in the sampling distribution of a goodness-of-fit measure, which is known for D,t, and χ2. These statistics become interpretable as measures of fit when compared to their sampling distribution, so that if the sample statistic lies sufficiently far out in the tail of its sampling distribution, it suggests that the sample data are improbable if the model is true (i.e., a poor fit between data and model). Depending on the rejection criteria for the test and the SIGNIFICANCE LEVEL chosen, the sample statistic may warrant rejection of the proposed model.

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