Simultaneous Sparse Estimation of Canonical Vectors in the <i>p</i> ≫ <i>N</i> Setting

<p>This article considers the problem of sparse estimation of canonical vectors in linear discriminant analysis when <i>p</i> ≫ <i>N</i>. Several methods have been proposed in the literature that estimate one canonical vector in the two-group case. However, <i>G</i> − 1 canonical vectors can be considered if the number of groups is <i>G</i>. In the multi-group context, it is common to estimate canonical vectors in a sequential fashion. Moreover, separate prior estimation of the covariance structure is often required. We propose a novel methodology for direct estimation of canonical vectors. In contrast to existing techniques, the proposed method estimates all canonical vectors at once, performs variable selection across all the vectors and comes with theoretical guarantees on the variable selection and classification consistency. First, we highlight the fact that in the <i>N</i> > <i>p</i> setting the canonical vectors can be expressed in a closed form up to an orthogonal transformation. Secondly, we propose an extension of this form to the <i>p</i> ≫ <i>N</i> setting and achieve feature selection by using a group penalty. The resulting optimization problem is convex and can be solved using a block-coordinate descent algorithm. The practical performance of the method is evaluated through simulation studies as well as real data applications. Supplementary materials for this article are available online.</p

The Charlson Comorbidity Index Can Be Used Prospectively to Identify Patients Who Will Incur High Future Costs

<div><p>Background</p><p>Reducing health care costs requires the ability to identify patients most likely to incur high costs. Our objective was to evaluate the ability of the Charlson comorbidity score to predict the individuals who would incur high costs in the subsequent year and to contrast its predictive ability with other commonly used predictors.</p><p>Methods</p><p>We contrasted the prior year Charlson comorbidity index, costs, Diagnostic Cost Group (DCG) and hospitalization as predictors of subsequent year costs from claims data of fund that provides comprehensive health benefits to a large union of health care workers. Total costs in the subsequent year was the principal outcome.</p><p>Results</p><p>Of the 181,764 predominantly Black and Latino beneficiaries, 70% were adults (mean age 45.7 years; 62% women). As the comorbidity index increased, total yearly costs increased significantly (P<.001). At lower comorbidity, the costs were similar across different chronic diseases. Using regression to predict total costs, top 5^th and 10^th percentile of costs, the comorbidity index, prior costs and DCG achieved almost identical explained variance in both adults and children.</p><p>Conclusions and Relevance</p><p>The comorbidity index predicted health costs in the subsequent year, performing as well as prior cost and DCG in identifying those in the top 5% or 10%. The comorbidity index can be used prospectively to identify patients who are likely to incur high costs.</p><p>Trial Registration</p><p>ClinicalTrials.gov <a href="http://clinicaltrials.gov/show/NCT01761253" target="_blank">NCT01761253</a></p></div

Percent of beneficiaries according to 2010 hospitalizations and average 2010 yearly costs according to prior year comorbidity index.

<p><b>The numbers of patients in each comorbidity group are shown on </b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0112479#pone-0112479-t001" target="_blank"><b>Table 1</b></a><b>.</b></p><p>Costs adjusted for age, gender, major mental health diagnoses, and zip code of residence.</p><p>Percent of beneficiaries according to 2010 hospitalizations and average 2010 yearly costs according to prior year comorbidity index.</p

Regression models evaluating prior year (2009) predictors of subsequent year (2010) costs.

<p>Each model had only the single independent variable in the first column, controlling for age, gender and mental health diagnosis.</p><p>t indicates the strength of the association and the p value the statistical significance. R² is the explained variance, that is, the extent to which the prior year variables predict subsequent year costs; the higher the R², the greater the explanatory or predictive power.</p><p>Regression models evaluating prior year (2009) predictors of subsequent year (2010) costs.</p

The y-axis is total costs, that is, the total yearly costs for patients with that disease according to the comorbidity level.

<p>The x-axis is the adjusted comorbidity index found by subtracting the weight of each disease from the patient's comorbidity index <i>for those patients with the stated disease</i>. Thus, a patient with an adjusted comorbidity index of 0 has only that chronic disease.</p

Predictors of the adults and children who would have the top 5% and 10% of subsequent (2010) costs using quantile regression.

<p>Each model had only a single independent variable controlling for age, gender and mental health diagnosis.</p><p>t indicates the strength of the association and the p value the statistical significance. R² is the explained variance, that is, the extent to which the prior year variables predict subsequent year costs; the higher the R², the greater the explanatory or predictive power.</p><p>Predictors of the adults and children who would have the top 5% and 10% of subsequent (2010) costs using quantile regression.</p

The y axis is the proportion of patients with a given chronic disease according to the adjusted comorbidity index.

<p>The x-axis is the adjusted comorbidity index found by subtracting the weight of each disease from the patient's comorbidity index <i>for those patients with the stated disease</i>. Thus, a patient with an adjusted comorbidity index of 0 has only that chronic disease.</p

Different weights assigned for specific conditions in the comorbidity index.

<p>Different weights assigned for specific conditions in the comorbidity index.</p

Beneficiaries and 2010 health care costs according to the prior year Charlson comorbidity index.

<p>Yearly cost per person adjusted for age, gender, major mental health diagnoses, and zip code of residence.</p><p>Beneficiaries and 2010 health care costs according to the prior year Charlson comorbidity index.</p