ABSTRACT 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 indicate evidence for the desired model. This method requires one to define what is meant by equivalence to the desired model, and guidelines are proposed. It is shown that the evidence for equivalence can distinguish between normal and nearby models, as well between the Poisson and over-dispersed models. Applications to the 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.

中文翻译：

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Karl Pearson 卡方统计中拟合优度的证据

摘要 卡方缺乏拟合的检验传统上用于寻找反对假设模型的证据，如果 Karl Pearson 统计比较观察到的和预期的落在单元格内的观察数不是很大，则该模型被接受。但是，如果真的需要拟合优度的证据，最好采用等价检验方法，其中卡方统计量的小值表明所需模型的证据。这种方法需要定义与所需模型等效的含义，并提出了指导方针。结果表明，等效性证据可以区分正常模型和附近模型，以及泊松模型和过度分散模型。包括评估随机数生成器和 pi 数字均匀性的应用。