Measures of Dependence and Model Selection in Multiple Regression

Kjell A Doksum，DMS-0505651多元回归中的相关性度量和模型选择研究人员通过将重点放在响应和一组协变量之间相关性度量的估计上，来探讨这样一个两难困境：对于高维数据，估计曲线和曲面(如条件平均值)实际上是不可能的。这些相依性度量采用信号除以噪声的形式。它们用于选择要包括在模型中的协变量子集，以及选择调整参数。该信号衡量了Y对一组协变量的依赖程度。噪声是标准误差，也就是估计信号的标准偏差的估计。当模型中包含的变量太多时，以及当调整参数为了较小的偏差而牺牲精度时，它将变得很小。选择使信噪比最小的变量和调谐参数会导致程序以传统的根n速率收敛。研究人员使用了符号方法和蒙特卡罗方法来研究这些过程的性质。他们还将它们与传统的模型和调整参数选择程序联系起来，并将传统程序与信噪比方法进行比较。在过去的几年中，建立了包含大量相关和比较变量的大型数据库。一个很好的例子是人类基因组计划产生的数据，其中大量基因需要被认为是某种疾病的可能贡献者。处理大量的变量是很困难的，因为它们中的许多会产生变异性(噪声)，这可能会淹没潜在的有趣的关系(信号)。研究人员通过使用通用的灵活的模型方程来表示变量之间的重要关系来解决这个问题。选择方程的变量和方面来最小化信噪比。这一过程会自动剔除主要造成噪声的变量，并选择一个强调数据中存在的信号的方程。

Measures of Dependence and Model Selection in Multiple Regression Kjell A Doksum, DMS-0505651 AbstractThe investigators approach the dilemma that estimation of curves andsurfaces such as the conditional mean is virtually impossible with highdimensional data by focusing instead on the estimation of measures ofdependence between a response and a set of covariates. These measures ofdependence take the form of signal divided by noise. They are used toselect the subset of covariates to include in the model, and to choosetuning parameters. The signal measures the strength of the dependence of Yon a set of covariates. The noise is a standard error, that is, anestimate of the standard deviation of the estimated signal. It will besmall when too many variables are included in the model and when thetuning parameter sacrifices precision for smaller bias. Choosing variablesand tuning parameters that minimize signal to noise leads to proceduresthat converge at the traditional root-n rate. The investigators useasymptotic and Monte Carlo methods to investigate the properties of suchprocedures. They also relate them to traditional model and tuningparameter selection procedures and compare traditional procedures with thesignal to noise approach.The last few years have seen the establishment of large databases'containing a large number of variables that are to be related andcompared. A good example is the data produced by the human genome projectwhere a great number of genes need to be considered as possiblecontributers to a certain disease. Dealing with a large number ofvariables is difficult because many of them will contribute variability(noise) that may drown out potential interesting relationships (signals).The investigators approach this problem by using general flexible modelequations to represent important relationships between variables. Thenvariables and aspects of the equations are selected to minimize the ratioof signal to noise. This procedure automatically weeds out the variablesthat contribute mostly noise and selects an equation that emphasizes thesignals present in the data.