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In addition, model selection is a difficult problem that has puzzled statisticians for a long time in dose finding research. For example, due to the uncertainty of the shape of the dose-response curve, Pinheiro et al. developed a new method for multiple comparison and modeling using the general-parameter model, and pointed out that it is difficult to conduct modeling of any kind if relatively small doses are used[17]. It is worth mentioning that this is also one of the problems that this study tries to solve. The data source of this study is the tumor chemosensitive assay in vitro of HB. However, as a rare disease, HB can provide a small amount of patient sample data, and it is not a random double-blind controlled experiment, which leads to a certain difficulty in carrying out mainstream research.
For example, Lifeng Lin pointed out that when the sample size is small, sample errors would lead to significant deviations in the results of meta-analysis[18]. Similarly, Kenny et al. concluded through comparative study that RMSEA(an absolute fit index) should not be used as an indicator to evaluate the fitting effect of the model in the case of small df or small samples, because it would lead to a wrong evaluation result[19]. In addition, Der-Chiang Li and I-Hsiang Wen indicated that the normality test for small sample data had low efficacy in rejecting the null hypothesis, in other words, it was unnecessary to conduct the normality test for small sample data[20]. Then, in 2013, Ma Lin Song et al. indicated that even the superior DEA method would have the problem of efficiency deviation in the case of small samples, so they introduced Bootstrap method to correct it and took Bootstrap-DEA as a new method suitable for small sample analysis[10].
In addition, for the Bootstrap method, J Martin Bland and Douglas G Altman called it as a resampling process of the original sample data set, and assumed that a single sample could be used to analyze the changes of the whole population[21]. At the same time, Shaowu Dai et al. stated that the traditional Bootstrap method has the defect of limited sample value range in generating random samples, which is easy to cause the calculation result to deviate greatly from the real distribution, therefore, they proposed the Bayes Bootstrap method to improve the traditional method and obtained the result with better precision[22]. In addition to the Bootstrap method, Bayes analysis is also a statistical inference method applied for small samples. For example, as early as 2004, Jeroen and Wicher proposed that under the condition of small samples, maximum likelihood estimates and standard error of logit parameters do not exist or have deviations, and the solution to this problem is to use Bayesian method to assume that parameters have certain prior distribution[23]. Therefore, how to set the prior distribution has become a key problem in rational use of Bayesian analysis. For example, in 2019, Sanne et al. conducted a literature review on a large number of simulation studies to summarize the performance of Bayesian methods in the construction of structural equation models with small samples[24]. They concluded that the use of Bayesian estimation with diffuse default priors would cause serious biases, therefore advised against the naive use of Bayesian estimates and provided suggestions on how to construct thoughtful prior distributions[24]. For solving this problem, Nomura et al. set a good example: they used successfully Akaike Bayesian Information Criterion method to objectively select prior distribution[25].