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.

摘要

传统上采用卡方检验来检验缺乏拟合的情况，以寻找针对假设模型的证据，如果比较观察到的和期望观察到的观察次数的卡尔·皮尔森（Karl Pearson）统计量不是很大，则接受该模型。但是，如果真的想要证据的拟合优度较好，最好采用等效检验方法，在该方法中，卡方统计量的较小值表示所需模型的证据。此方法需要定义与期望模型等效的含义，并提出了准则。结果表明，等价证据可以区分正常模型和附近模型，也可以区分泊松模型和过度分散模型。包括对随机数生成器的评估以及pi位数均匀性的应用。还提供了获得所需的拟合优度预期证据所需的样本量。