将数据分组到可变宽度箱中导致的潜在拟合偏倚
Potential fitting biases resulting from grouping data into variable width bins
摘要：
当阅读描述任何经验数据分析的同行评审科学文献时，自然和正确的是继续进行基本假设，即实验已经做出了真诚的努力，以确保其分析产生无偏见的结果。然而，粒子物理学实验是昂贵和耗时的，因此，如果分析有固有的偏见（即使是无意的），许多金钱和精力可以浪费在试图复制或理解结果，特别是如果分析是我们对宇宙的理解的基础。在本说明中，我们讨论了数据分箱方案可能导致的重大偏差。正如我们将要展示的，如果数据被分组，使得它们提供了与特定（但不正确）模型的最佳比较，那么当拟合分组数据时，得到的模型参数估计值可能会显着偏差，导致我们经常接受模型假设，而实际上它并不真实。当使用分箱似然或最小二乘法，当然没有先验的要求，数据箱的大小需要是恒定的，但我们表明，拟合数据分组到可变宽度箱是特别容易产生偏差的结果，如果箱的边界被选择来优化比较的分箱数据到一个错误的模型。简单地使用可变分箱可以实现的偏差程度可以令人惊讶地大。在可能的情况下，用未分组的似然方法拟合数据，是研究人员证明他们的分析不会受到分组效应影响的最佳方法。如果不能做到这一点，则应尽可能使用相等的箱宽作为拟合分析的交叉检查。
英文摘要：
When reading peer-reviewed scientific literature describing any analysis of empirical data, it is natural and correct to proceed with the underlying assumption that experiments have made good faith efforts to ensure that their analyses yield unbiased results. However, particle physics experiments are expensive and time consuming to carry out, thus if an analysis has inherent bias (even if unintentional), much money and effort can be wasted trying to replicate or understand the results, particularly if the analysis is fundamental to our understanding of the universe. In this note we discuss the significant biases that can result from data binning schemes. As we will show, if data are binned such that they provide the best comparison to a particular (but incorrect) model, the resulting model parameter estimates when fitting to the binned data can be significantly biased, leading us to too often accept the model hypothesis when it is not in fact true. When using binned likelihood or least squares methods there is of course no a priori requirement that data bin sizes need to be constant, but we show that fitting to data grouped into variable width bins is particularly prone to produce biased results if the bin boundaries are chosen to optimize the comparison of the binned data to a wrong model. The degree of bias that can be achieved simply with variable binning can be surprisingly large. Fitting the data with an unbinned likelihood method, when possible to do so, is the best way for researchers to show their analyses are not biased by binning effects. Failing that, equal bin widths should be employed as a cross-check of the fitting analysis whenever possible.
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