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We study deep neural networks and their use in semiparametric inference. We establish novel nonasymptotic high probability bounds for deep feedforward neural nets. These deliver rates of convergence that are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a result also new to the literature. Our nonasymptotic high probability bounds, and the subsequent semiparametric inference, treat the current standard architecture: fully connected feedforward neural networks (multilayer perceptrons), with the now-common rectified linear unit activation function, unbounded weights, and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed-width, very deep networks. We establish the nonasymptotic bounds for these deep nets for a general class of nonparametric regression-type loss functions, which includes as special cases least squares, logistic regression, and other generalized linear models. We then apply our theory to develop semiparametric inference, focusing on causal parameters for concreteness, and demonstrate the effectiveness of deep learning with an empirical application to direct mail marketing.

We study regression discontinuity designs when covariates are included in the estimation. We examine local polynomial estimators that include discrete or continuous covariates in an additive separable way, but without imposing any parametric restrictions on the underlying population regression functions. We recommend a covariate-adjustment approach that retains consistency under intuitive conditions and characterize the potential for estimation and inference improvements. We also present new covariateadjusted mean-squared error expansions and robust bias-corrected inference procedures, with heteroskedasticity-consistent and cluster-robust standard errors. We provide an empirical illustration and an extensive simulation study. All methods are implemented in R and Stata software packages.

Nonparametric methods play a central role in modern empirical work. While they provide inference procedures that are more robust to parametric misspecification bias, they may be quite sensitive to tuning parameter choices. We study the effects of bias correction on confidence interval coverage in the context of kernel density and local polynomial regression estimation, and prove that bias correction can be preferred to undersmoothing for minimizing coverage error and increasing robustness to tuning parameter choice. This is achieved using a novel, yet simple, Studentization, which leads to a new way of constructing kernel-based bias-corrected confidence intervals. In addition, for practical cases, we derive coverage error optimal bandwidths and discuss easy-to-implement bandwidth selectors. For interior points, we show that the mean-squared error (MSE)-optimal bandwidth for the original point estimator (before bias correction) delivers the fastest coverage error decay rate after bias correction when second-order (equivalent) kernels are employed, but is otherwise suboptimal because it is too \"large.\" Finally, for odd-degree local polynomial regression, we show that, as with point estimation, coverage error adapts to boundary points automatically when appropriate Studentization is used; however, the MSE-optimal bandwidth for the original point estimator is suboptimal. All the results are established using valid Edgeworth expansions and illustrated with simulated data. Our findings have important consequences for empirical work as they indicate that bias-corrected confidence intervals, coupled with appropriate standard errors, have smaller coverage error and are less sensitive to tuning parameter choices in practically relevant cases where additional smoothness is available. Supplementary materials for this article are available online.

We present large sample results for partitioning-based least squares non-parametric regression, a popular method for approximating conditional expectation functions in statistics, econometrics and machine learning. First, we obtain a general characterization of their leading asymptotic bias. Second, we establish integrated mean squared error approximations for the point estimator and propose feasible tuning parameter selection. Third, we develop point-wise inference methods based on undersmoothing and robust bias correction. Fourth, employing different coupling approaches, we develop uniform distributional approximations for the undersmoothed and robust bias-corrected t-statistic processes and construct valid confidence bands. In the univariate case, our uniform distributional approximations require seemingly minimal rate restrictions and improve on approximation rates known in the literature. Finally, we apply our general results to three partitioning-based estimators: splines, wavelets and piecewise polynomials. The Supplemental Appendix includes several other general and example-specific technical and methodological results. A companion R package is provided.

Background: Concern regarding wide variations in spending and intensive care unit use for patients at the end of life hinges on the assumption that such treatment offers little or no survival benefit. Objective: To explore the relationship between hospital \"end-of-life\" (EOL) treatment intensity and postadmission survival. Research Design: Retrospective cohort analysis of Pennsylvania Health Care Cost Containment Council discharge data April 2001 to March 2005 linked to vital statistics data through September 2005 using hospital-level correlation, admission-level marginal structural logistic regression, and pooled logistic regression to approximate a Cox survival model. Subjects: A total of 1,021,909 patients ≥65 years old, incurring 2,216,815 admissions in 169 Pennsylvania acute care hospitals. Measures: EOL treatment intensity (a summed index of standardized intensive care unit and life-sustaining treatment use among patients with a high predicted probability of dying [PPD] at admission) and 30- and 180-day postadmission mortality. Results: There was a nonlinear negative relationship between hospital EOL treatment intensity and 30-day mortality among all admissions, although patients with higher PPD derived the greatest benefit. Compared with admission at an average intensity hospital, admission to a hospital 1 standard deviation below versus 1 standard deviation above average intensity resulted in an adjusted odds ratio of mortality for admissions at low PPD of 1.06 (1.04–1.08) versus 0.97 (0.96–0.99); average PPD: 1.06 (1.04–1.09) versus 0.97 (0.96–0.99); and high PPD: 1.09 (1.07–1.11) versus 0.97 (0.95–0.99), respectively. By 180 days, the benefits to intensity attenuated (low PPD: 1.03 [1.01–1.04] vs. 1.00 [0.98–1.01]; average PPD: 1.03 [1.02–1.05] vs. 1.00 [0.98–1.01]; and high PPD: 1.06 [1.04–1.09] vs. 1.00 [0.98–1.02]), respectively. Conclusions: Admission to higher EOL treatment intensity hospitals is associated with small gains in postadmission survival. The marginal returns to intensity diminish for admission to hospitals above average EOL treatment intensity and wane with time.

Background: Health care utilization among decedents is increasingly used as a measure of health care efficiency, but decedent-based measures may be biased estimates of care received by \"dying\" patients. Objective: To develop and validate new measures of hospital \"end-of-life\" treatment intensity. Research Design: Retrospective cohort study using Pennsylvania Health Care Cost Containment Council (PHC4) discharge data (April 2001–March 2005) and Centers for Medicare and Medicaid Services (CMS) data (January 1999–December 2003). Subjects: Patients 65 and older admitted to 174 Pennsylvania acute care hospitals. Measures: Hospital-specific standardized ratios of intensive care unit (ICU) and life-sustaining treatment (LST) use among terminal admissions (decedents) and admissions with a high probability of dying, and spending and use of hospitals, ICUs, and physicians among patients in their last 6 months of life. Results: There was marked between-hospital variation in the use of the ICU and LSTs among decedents and admissions with high probability of dying. All hospital decedent and high probability of dying measures were highly correlated (P < 00001). In principal components factor analysis, all 4 of the last-6-months cohort-based measures, the decedent and high-risk admission-based ICU measures, and 8 of the 12 decedent and high probability of dying LST measures loaded onto a single factor, explaining 42% of the variation in the data. Conclusions: Hospitals' end-of-life intensity varies in the use of specific life-sustaining treatments that are somewhat emblematic of aggressive end-of-life care. End-of-life intensity is a relatively stable hospital attribute that is robust to multiple measurement approaches.

Background: Chronic obstructive pulmonary disease (COPD) is a leading cause of death and 70% of the cost of COPD is due to hospitalizations. Self-reported daily physical activity and health status have been reported as predictors of a hospitalization in COPD but are not routinely assessed. Objectives: We tested the hypothesis that self-reported daily physical activity and health status assessed by a simple question were predictors of a hospitalization in a well-characterized cohort of patients with severe emphysema. Methods: Investigators gathered daily physical activity and health status data assessed by a simple question in 597 patients with severe emphysema and tested the association of those patient-reported outcomes to the occurrence of a hospitalization in the following year. Multiple logistic regression analyses were used to determine predictors of hospitalization during the first 12 months after randomization. Results: The two variables tested in the hypothesis were significant predictors of a hospitalization after adjusting for all univariable significant predictors: >2 h of physical activity per week had a protective effect [odds ratio (OR) 0.60; 95% confidence interval (95% CI) 0.41–0.88] and self-reported health status as fair or poor had a deleterious effect (OR 1.57; 95% CI 1.10–2.23). In addition, two other variables became significant in the multivariate model: total lung capacity (every 10% increase) had a protective effect (OR 0.88; 95% CI 0.78–0.99) and self-reported anxiety had a deleterious effect (OR 1.75; 95% CI 1.13–2.70). Conclusion: Self-reported daily physical activity and health status are independently associated with COPD hospitalizations. Our findings, assessed by simple questions, suggest the value of patient-reported outcomes in developing risk assessment tools that are easy to use.

Background: There is substantial hospital-level variation in end-of-life (EOL) treatment intensity. Objective: To explore the association between organizational factors and EOL treatment intensity in Pennsylvania (PA) hospitals. Research Design: Cross-sectional mixed-mode survey of Chief Nursing Officers of PA hospitals linked to hospital-level measures of EOL treatment intensity calculated from PA Health Care Cost Containment Council (PHC4) hospital discharge data. Hospitals: One hundred sixty-four hospitals, of which 124 (76%) responded to the survey. Measures: The dependent variable was an index of hospital EOL treatment intensity; the independent variables included administrative data-derived structural and market characteristics and 29 survey-derived hospital or ICU programs, policies, or practices. Results: In models restricted to independent variables drawn from administrative sources (available for all 164 hospitals), bed size (P < 0.001), proportion of admissions among black patients (P < 0.001), and county- wide hospital market competitiveness (Herflndahl-Hirschman index) (P = 0.001) were independently associated with greater EOL treatment intensity (adjusted R² = 0.5136). In models that additionally included hospital programs, policies, and practices (available for 124 hospitals), only an ICU long length of stay review committee (P = 0.03) was independently associated with greater EOL treatment intensity (adjusted R² = 0.5357). Conclusions: Information about hospital and ICU programs, policies, and practices believed relevant to the treatment of patients near the end of life offers little additional explanatory power in understanding hospital-level variation in EOL treatment intensity than administratively-derived variables alone. Future studies should explore the contribution of more difficult to measure social norms in shaping hospital practice patterns.

Abstract Rationale In studies that address health-related quality of life (QoL) and survival, subjects who die are usually censored from QoL assessments. This practice tends to inflate the apparent benefits of interventions with a high risk of mortality. Assessing a composite QoL-death outcome is a potential solution to this problem. Objectives To determine the effect of lung volume reduction surgery (LVRS) on a composite endpoint consisting of the occurrence of death or a clinically meaningful decline in QoL defined as an increase of at least eight points in the St. George's Respiratory Questionnaire total score from the National Emphysema Treatment Trial. Methods In patients with chronic obstructive pulmonary disease and emphysema randomized to receive medical treatment (n = 610) or LVRS (n = 608), we analyzed the survival to the composite endpoint, the hazard functions and constructed prediction models of the slope of QoL decline. Measurements and Main Results The time to the composite endpoint was longer in the LVRS group (2 years) than the medical treatment group (1 year) (P < 0.0001). It was even longer in the subsets of patients undergoing LVRS without a high risk for perioperative death and with upper-lobe-predominant emphysema. The hazard for the composite event significantly favored the LVRS group, although it was most significant in patients with predominantly upper-lobe emphysema. The beneficial impact of LVRS on QoL decline was most significant during the 2 years after LVRS. Conclusions LVRS has a significant effect on the composite QoL-survival endpoint tested, indicating its meaningful palliative role, particularly in patients with upper-lobe–predominant emphysema.

This paper concerns robust inference on average treatment effects following model selection. In the selection on observables framework, we show how to construct confidence intervals based on a doubly-robust estimator that are robust to model selection errors and prove that they are valid uniformly over a large class of treatment effect models. The class allows for multivalued treatments with heterogeneous effects (in observables), general heteroskedasticity, and selection amongst (possibly) more covariates than observations. Our estimator attains the semiparametric efficiency bound under appropriate conditions. Precise conditions are given for any model selector to yield these results, and we show how to combine data-driven selection with economic theory. For implementation, we give a specific proposal for selection based on the group lasso, which is particularly well-suited to treatment effects data, and derive new results for high-dimensional, sparse multinomial logistic regression. A simulation study shows our estimator performs very well in finite samples over a wide range of models. Revisiting the National Supported Work demonstration data, our method yields accurate estimates and tight confidence intervals.