背景:降维方法并不总是将其基础指标降低为单个综合得分。此外,这种方法通常基于需要丢弃一些信息的最优性标准。我们建议,在某些条件下,要使用p维随机变量(p指标)的联合概率密度函数(联合pdf或JPD),作为索引或综合分数。事实证明,该指数比任何替代综合得分都更具信息性。在两个例子中,我们将JPD指数与传统方法构建的一些替代方案进行了比较。
方法:我们基于多变量数据的概率密度开发了一种概率无监督降维方法。我们证明了给定JPD的变量的条件分布是均匀的,这意味着JPD是最常见的信息概念下信息最丰富的标量摘要。B.我们在一些广泛合理的条件下表明,JPD可以用作索引。要使用JPD作为索引,除了有一个合理的解释,所有的随机变量应该具有与密度值(共向性)大致相同的方向(单向性)。我们将这些想法应用于两个数据集:首先,在从8,889名患有慢性疼痛的美国退伍军人获得的7个简短疼痛清单干扰量表(BPI-I)项目上,第二,基于912名美国退伍军人的行政数据的一项新措施。要估计这两个例子中的JPD,在可用的JPD估计方法中,我们使用了它的条件规范,为每个因子条件(回归)规范确定了一个拟合良好的参数模型,and,通过最大化相应的可能性,估计他们的参数。由于条件规范的非唯一性,所有估计条件规格的平均值被用作最终估计.由于指数的普遍使用是排名,我们使用单调依赖性的度量[例如,Spearman的等级相关性(rho)]来评估单向性和共方向性的强度。最后,我们将JPD分数与基于方差-协方差的分数(一维模型中的因子分数)进行交叉验证,以及(广义)部分信用和分级响应IRT模型的“人的参数”估计。我们使用PearsonDivergence作为信息和Shannon熵的度量来比较这些替代分数中的不确定性(信息量)。
结果:基于多维数据的联合概率密度(JPD)开发了一种无监督降维方法。JPD,在规律性条件下,可以用作索引。对于完善的简短疼痛干扰清单(BPI-I(7个项目的简短表格)和具有6个指标的新的心理健康严重程度指数(MoPSI),我们估计了JPD的得分.我们比较,假设一维,因子得分,部分信用模型的个人得分,广义部分信用模型,以及具有JPD评分的分级响应模型。不出所料,这两个例子中的所有分数排名都是单调依赖的,具有不同的优势。Shannon熵是JPD得分最小的。对于JPD评分,不同指数的估计密度对均匀分布的Pearson发散最大。
结论:无监督概率降维是可能的。在适当的时候,联合概率密度函数可以作为信息量最大的指标。演示了模型规范和估计以及实施评分的步骤。不出所料,当满足因子分析和IRT模型中所需的假设时,JPD评分与这些既定评分一致。然而,当这些假设被违反时,JPD分数以最小的假设保留指标中的所有信息。
BACKGROUND: Dimension reduction methods do not always reduce their underlying indicators to a single composite score. Furthermore, such methods are usually based on optimality criteria that require discarding some information. We suggest, under some conditions, to use the joint probability density function (joint pdf or JPD) of p-dimensional random variable (the p indicators), as an index or a composite score. It is proved that this index is more informative than any alternative composite score. In two examples, we compare the JPD index with some alternatives constructed from traditional methods.
METHODS: We develop a probabilistic unsupervised dimension reduction method based on the probability density of multivariate data. We show that the conditional distribution of the variables given JPD is uniform, implying that the JPD is the most informative scalar summary under the most common notions of information. B. We show under some widely plausible conditions, JPD can be used as an index. To use JPD as an index, in addition to having a plausible interpretation, all the random variables should have approximately the same direction(unidirectionality) as the density values (codirectionality). We applied these ideas to two data sets: first, on the 7 Brief Pain Inventory Interference scale (BPI-I) items obtained from 8,889 US Veterans with chronic pain and, second, on a novel measure based on administrative data for 912 US Veterans. To estimate the JPD in both examples, among the available JPD estimation methods, we used its conditional specifications, identified a well-fitted parametric model for each factored conditional (regression) specification, and, by maximizing the corresponding likelihoods, estimated their parameters. Due to the non-uniqueness of conditional specification, the average of all estimated conditional specifications was used as the final estimate. Since a prevalent common use of indices is ranking, we used measures of monotone dependence [e.g., Spearman\'s rank correlation (rho)] to assess the strength of unidirectionality and co-directionality. Finally, we cross-validate the JPD score against variance-covariance-based scores (factor scores in unidimensional models), and the \"person\'s parameter\" estimates of (Generalized) Partial Credit and Graded Response IRT models. We used Pearson Divergence as a measure of information and Shannon\'s entropy to compare uncertainties (informativeness) in these alternative scores.
RESULTS: An unsupervised dimension reduction was developed based on the joint probability density (JPD) of the multi-dimensional data. The JPD, under regularity conditions, may be used as an index. For the well-established Brief Pain Interference Inventory (BPI-I (the short form with 7 Items) and for a new mental health severity index (MoPSI) with 6 indicators, we estimated the JPD scoring. We compared, assuming unidimensionality, factor scores, Person\'s scores of the Partial Credit model, the Generalized Partial Credit model, and the Graded Response model with JPD scoring. As expected, all scores\' rankings in both examples were monotonically dependent with various strengths. Shannon entropy was the smallest for JPD scores. Pearson Divergence of the estimated densities of different indices against uniform distribution was maximum for JPD scoring.
CONCLUSIONS: An unsupervised probabilistic dimension reduction is possible. When appropriate, the joint probability density function can be used as the most informative index. Model specification and estimation and steps to implement the scoring were demonstrated. As expected, when the required assumption in factor analysis and IRT models are satisfied, JPD scoring agrees with these established scores. However, when these assumptions are violated, JPD scores preserve all the information in the indicators with minimal assumption.