单次重复试验的三水平析因设计的分析

On the Analysis of Three-Level Factorial Designs with only One Replicate

由于试验材料、费用和时间等条件的限制,仅有单次重复试验的三水平析因设计经常要应用在农业、工业和医学临床试验等领域。例如,在医学临床试验中,为找到影响治疗关节炎效果的重要因子和最佳治疗方案需要考虑2个三水平的因子:A(药物治疗)和B(运动治疗),由于只能找到9位病情相似的病人进行试验,故只能实施仅有单次重复试验的三水平析因设计3~2。不幸的是,交互作用A×B也可能存在,这样就没有剩余自由度用于估计误差的方差,从而通常的方差分析方法不再能用于数据分析。针对上述问题,本文提出了三个基于均方误差的检验统计量用于分析单次重复试验的三水平析因设计。通过实例表明用这些方法不仅能检验所考虑因子的主效应,而且还能同时检验交互效应。相应检验所用的一些常用临界值提供在附录中。并且,还通过大量的模拟研究对所提出的三个检验方法进行了比较。结果显示,T_^((3))检验在三个检验方法中具有最大的功效。

In clinical trials,three-level factorial designs with only one replicate are sometimes needed due to the lack of homogenous trial objects,trial materials,trial time,and so on.For example,to find out the best therapy of improving the ability of activity of knee osteoarthritis（KOA） patients,two factors A（medication treatment） and B（exercise treatment）,each at three levels,should be considered in a clinical trial.The interaction,A×B,between factors A and B might exist.Unfortunately,only a few of patients,whose situations are similar enough fairly to compare the treatments,could be found amongst a lot of KOA patients,so that an unreplicated 3~2 design could just be carried out.Unfortunately,the ordinary ANOVA is inapplicable because of no degrees of freedom left to estimate the error variance in this case.In this paper,three mean-square-based test procedures are proposed to identify possibly significant factors or interactions in three-level factorial designs with only one replicate.Critical values used in the proposed procedures for some three-level factorial designs are tabulated.It is illustrated that not only main effects but also interactions can be identified by the proposed tests at the same time in an unreplicated three-level factorial design.An extensive simulation study shows that the power of three proposed tests varies for different magnitudes and the number of significant factors or interactions.The test procedure T_k^（（3）） is the most powerful over a wide range of the number of significant factors among three compared tests.