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Silber JH, Rosenbaum PR, Reiter JG, Hill AS, Jain S, Wolk DA, Small DS, Hashemi S, Niknam BA, Neuman MD, Fleisher LA, Eckenhoff R. Alzheimer's Dementia After Exposure to Anesthesia and Surgery in the Elderly: A Matched Natural Experiment Using Appendicitis. Ann Surg 2022; 276:e377-e385. [PMID: 33214467 PMCID: PMC8437105 DOI: 10.1097/sla.0000000000004632] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
OBJECTIVE The aim of this study was to determine whether surgery and anesthesia in the elderly may promote Alzheimer disease and related dementias (ADRD). BACKGROUND There is a substantial conflicting literature concerning the hypothesis that surgery and anesthesia promotes ADRD. Much of the literature is confounded by indications for surgery or has small sample size. This study examines elderly patients with appendicitis, a common condition that strikes mostly at random after controlling for some known associations. METHODS A matched natural experiment of patients undergoing appendectomy for appendicitis versus control patients without appendicitis using Medicare data from 2002 to 2017, examining 54,996 patients without previous diagnoses of ADRD, cognitive impairment, or neurological degeneration, who developed appendicitis between ages 68 through 77 years and underwent an appendectomy (the ''Appendectomy'' treated group), matching them 5:1 to 274,980 controls, examining the subsequent hazard for developing ADRD. RESULTS The hazard ratio (HR) for developing ADRD or death was lower in the Appendectomy group than controls: HR = 0.96 [95% confidence interval (CI) 0.94-0.98], P < 0.0001, (28.2% in Appendectomy vs 29.1% in controls, at 7.5 years). The HR for death was 0.97 (95% CI 0.95-0.99), P = 0.002, (22.7% vs 23.1% at 7.5 years). The HR for developing ADRD alone was 0.89 (95% CI 0.86-0.92), P < 0.0001, (7.6% in Appendectomy vs 8.6% in controls, at 7.5 years). No subgroup analyses found significantly elevated rates of ADRD in the Appendectomy group. CONCLUSION In this natural experiment involving 329,976 elderly patients, exposure to appendectomy surgery and anesthesia did not increase the subsequent rate of ADRD.
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Affiliation(s)
- Jeffrey H. Silber
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
- Department of Anesthesiology and Critical Care, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA
- The Departments of Pediatrics, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA
- Department of Health Care Management, The Wharton School, University of Pennsylvania, Philadelphia, PA
| | - Paul R. Rosenbaum
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
| | - Joseph G. Reiter
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
| | - Alexander S. Hill
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
| | - Siddharth Jain
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
| | - David A. Wolk
- Department of Neurology, The Perelman School of Medicine, University of Pennsylvania
| | - Dylan S. Small
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
| | - Sean Hashemi
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
| | - Bijan A. Niknam
- Center for Outcomes Research, Children’s Hospital of Philadelphia, Philadelphia, PA
| | - Mark D. Neuman
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
- Department of Anesthesiology and Critical Care, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA
- Center for Perioperative Outcomes Research and Transformation, University of Pennsylvania, Philadelphia, PA
| | - Lee A. Fleisher
- The Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
- Department of Anesthesiology and Critical Care, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA
- Center for Perioperative Outcomes Research and Transformation, University of Pennsylvania, Philadelphia, PA
| | - Roderic Eckenhoff
- Department of Anesthesiology and Critical Care, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA
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Zhao A, Lee Y, Small DS, Karmakar B. Evidence factors from multiple, possibly invalid, instrumental variables. Ann Stat 2022. [DOI: 10.1214/21-aos2148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Anqi Zhao
- Department of Statistics and Data Science, National University of Singapore
| | - Youjin Lee
- Department of Biostatistics, Brown University
| | - Dylan S. Small
- Department of Statistics and Data Science, University of Pennsylvania
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Heng S, Kang H, Small DS, Fogarty CB. Increasing power for observational studies of aberrant response: An adaptive approach. J R Stat Soc Series B Stat Methodol 2021. [DOI: 10.1111/rssb.12424] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Affiliation(s)
- Siyu Heng
- University of Pennsylvania Philadelphia PA USA
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Small DS, Firth DW, Keele LJ, Huber M, Passarella M, Lorch SA, Burris HH. Surface mining and low birth weight in central appalachia. ENVIRONMENTAL RESEARCH 2021; 196:110340. [PMID: 33098818 DOI: 10.1016/j.envres.2020.110340] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2020] [Revised: 09/28/2020] [Accepted: 10/06/2020] [Indexed: 06/11/2023]
Abstract
BACKGROUND Surface mining has become a significant method of coal mining in the Central Appalachian region of the eastern United States alongside the traditional underground mining. Concerns have been raised about the health effects of this surface mining, particularly mountaintop removal mining where coal is mined upon steep mountaintops by removing the mountaintop through clearcutting forests and explosives. METHODS We used a control group design with a pretest and a posttest to assess the associations of surface mining in Central Appalachia with low birth weight and other adverse birth outcomes. The pretest period is 1977-1989, a period of low surface mining activity. We consider three posttest periods: 1990-1998, 1999-2011 and 2012-2017, with 1999-2011 as the primary analysis and the other periods as secondary analyses. Surface mining in Central Appalachia increased after 1989, partly resulting from the Clean Air Act Amendments of 1990 which made surface mining in Appalachia more financially attractive. For the primary analysis, we fit a logistic regression model of the primary outcome (low birth weight, <2500 g) on dummy variables for county and year; individual level maternal/infant covariates (maternal race, maternal age, infant sex and whether birth was a multiple birth); and the amount of surface mining during the year of the birth in the maternal county of residence. RESULTS Our analysis sample consisted of 783,328 infants -- 482,284 infants born from 1977 to 2017 to women residing in substantial surface mining activity counties and 301,044 infants born from 1977 to 2017 to women residing in matched control counties. Compared to the pre-period of low surface mining from 1977 to 1989, for the primary analysis posttest period of 1999-2011, there was an estimated relative increase in low birth weight in surface mining counties compared to matched control counties that was not statistically significant (odds ratio for a 5 percentage point increase in area disturbed by surface mining: 1.07, 95% confidence interval (0.96, 1.20), p-value: .22). For the secondary analysis posttest period of 1990-1998, there was no increase (odds ratio: 0.91, 95% confidence interval: (0.74, 1.13), p-value: .41). For the secondary analysis posttest period of 2012-2017, there was a statistically significant relative increase (odds ratio: 1.28, 95% confidence interval: (1.08, 1.50), p-value: .004). Qualitatively similar results were found for the outcomes of very low birth weight, preterm birth and small-for-gestational age. CONCLUSIONS We examined the hypothesis that surface mining activity in Central Appalachia contributes to low birth weight using an observational study. We found evidence in secondary analyses that surface mining was associated with low birth weight in the 2012-2017 time period and potentially beginning in the early to mid 2000's. Evidence for an association was not found prior to 2000. A potential explanation for this pattern of association is that surface mining caused an increase in low birth weight but its onset was delayed. Future research is needed to clarify the findings and if replicated, identify the mechanism necessary to mitigate the impacts of mining on adverse birth outcomes.
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Affiliation(s)
- Dylan S Small
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA, 19104, USA.
| | - Daniel W Firth
- Daniel W Firth Dba 4E Analytics, Kingsport, TN, 37664, USA
| | - Luke J Keele
- Department of Surgery, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Matthew Huber
- Division of Neonatology, Children's Hospital of Philadelphia, Philadelphia, PA, 19104, USA
| | - Molly Passarella
- Division of Neonatology, Children's Hospital of Philadelphia, Philadelphia, PA, 19104, USA
| | - Scott A Lorch
- Division of Neonatology, Children's Hospital of Philadelphia, Philadelphia, PA, 19104, USA; Department of Pediatrics, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Heather H Burris
- Division of Neonatology, Children's Hospital of Philadelphia, Philadelphia, PA, 19104, USA; Department of Pediatrics, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, 19104, USA
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Karmakar B, Small DS, Rosenbaum PR. Reinforced Designs: Multiple Instruments Plus Control Groups as Evidence Factors in an Observational Study of the Effectiveness of Catholic Schools. J Am Stat Assoc 2021. [DOI: 10.1080/01621459.2020.1745811] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- Bikram Karmakar
- Department of Statistics, University of Florida, Gainesville, FL
| | - Dylan S. Small
- Statistics Department, University of Pennsylvania, Philadelphia, PA
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Nattino G, Lu B, Shi J, Lemeshow S, Xiang H. Triplet Matching for Estimating Causal Effects With Three Treatment Arms: A Comparative Study of Mortality by Trauma Center Level. J Am Stat Assoc 2021. [DOI: 10.1080/01621459.2020.1737078] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- Giovanni Nattino
- Division of Biostatistics, College of Public Health, The Ohio State University, Columbus, OH
| | - Bo Lu
- Division of Biostatistics, College of Public Health, The Ohio State University, Columbus, OH
| | - Junxin Shi
- Center for Pediatric Trauma Research, Center for Injury Research and Policy, Abigail Wexner Research Institute at Nationwide Children’s Hospital, Columbus, OH
| | - Stanley Lemeshow
- Division of Biostatistics, College of Public Health, The Ohio State University, Columbus, OH
| | - Henry Xiang
- Center for Pediatric Trauma Research, Center for Injury Research and Policy, Abigail Wexner Research Institute at Nationwide Children’s Hospital, Columbus, OH
- College of Medicine, The Ohio State University, Columbus, OH
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Yu R, Silber JH, Rosenbaum PR. Rejoinder: Matching Methods for Observational Studies Derived from Large Administrative Databases. Stat Sci 2020. [DOI: 10.1214/20-sts790] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Karmakar B, Doubeni CA, Small DS. EVIDENCE FACTORS IN A CASE-CONTROL STUDY WITH APPLICATION TO THE EFFECT OF FLEXIBLE SIGMOIDOSCOPY SCREENING ON COLORECTAL CANCER. Ann Appl Stat 2020; 14:829-849. [PMID: 38465229 PMCID: PMC10924422 DOI: 10.1214/20-aoas1329] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/12/2024]
Abstract
As in any observational study, in a case-control study a primary concern is potential unmeasured confounders. Bias, due to unmeasured confounders, can result in a false discovery of an apparent treatment effect when there is none. Replication of an observational study, which tries to provide multiple analyses of the data where the biases affecting each analysis are thought to be different, is one way to strengthen the evidence from an observational study. Evidence factors allow for internal replication by testing a hypothesis using multiple comparisons in a way that the comparisons yield independent evidence and differ in the sources of potential bias. We construct evidence factors in a case-control study in which there are two types of cases, "narrow" cases which are thought to be potentially more affected by the exposure and "marginal" cases which are thought to have more heterogeneous causes. We develop and study an inference procedure for using such evidence factors and apply it to a study of the effect of sigmoidoscopy screening on colorectal cancer.
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Affiliation(s)
- Bikram Karmakar
- Department of Statistics, College of Liberal Arts and Sciences, University of Florida
| | - Chyke A Doubeni
- Center for Health Equity and Community Engagement Research, Mayo Clinic
| | - Dylan S Small
- Department of Statistics, The Wharton School, University of Pennsylvania
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Karmakar B, Small DS, Rosenbaum PR. Using Evidence Factors to Clarify Exposure Biomarkers. Am J Epidemiol 2020; 189:243-249. [PMID: 31912138 DOI: 10.1093/aje/kwz263] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2019] [Revised: 08/15/2019] [Accepted: 08/26/2019] [Indexed: 11/14/2022] Open
Abstract
A study has 2 evidence factors if it permits 2 statistically independent inferences about 1 treatment effect such that each factor is immune to some bias that would invalidate the other factor. Because the 2 factors are statistically independent, the evidence they provide can be combined using methods associated with meta-analysis for independent studies, despite using the same data twice in different ways. We illustrate evidence factors, applying them in a new way in investigations that have both an exposure biomarker and a coarse external measure of exposure to a treatment. To illustrate, we consider the possible effects of cigarette smoking on homocysteine levels, with self-reported smoking and a cotinine biomarker. We examine joint sensitivity of 2 factors to bias from confounding, a central aspect of any observational study.
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Affiliation(s)
- Bikram Karmakar
- Department of Statistics, College of Liberal Arts and Sciences, University of Florida, Gainesville, Florida
| | - Dylan S Small
- Department of Statistics, the Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania
| | - Paul R Rosenbaum
- Department of Statistics, the Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania
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Zhao Q, Small DS, Rosenbaum PR. Cross-Screening in Observational Studies That Test Many Hypotheses. J Am Stat Assoc 2018. [DOI: 10.1080/01621459.2017.1407770] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- Qingyuan Zhao
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
| | - Dylan S. Small
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
| | - Paul R. Rosenbaum
- Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
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de Los Angeles Resa M, Zubizarreta JR. Evaluation of subset matching methods and forms of covariate balance. Stat Med 2016; 35:4961-4979. [PMID: 27442072 DOI: 10.1002/sim.7036] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2015] [Revised: 02/10/2016] [Accepted: 06/05/2016] [Indexed: 01/25/2023]
Abstract
This paper conducts a Monte Carlo simulation study to evaluate the performance of multivariate matching methods that select a subset of treatment and control observations. The matching methods studied are the widely used nearest neighbor matching with propensity score calipers and the more recently proposed methods, optimal matching of an optimally chosen subset and optimal cardinality matching. The main findings are: (i) covariate balance, as measured by differences in means, variance ratios, Kolmogorov-Smirnov distances, and cross-match test statistics, is better with cardinality matching because by construction it satisfies balance requirements; (ii) for given levels of covariate balance, the matched samples are larger with cardinality matching than with the other methods; (iii) in terms of covariate distances, optimal subset matching performs best; (iv) treatment effect estimates from cardinality matching have lower root-mean-square errors, provided strong requirements for balance, specifically, fine balance, or strength-k balance, plus close mean balance. In standard practice, a matched sample is considered to be balanced if the absolute differences in means of the covariates across treatment groups are smaller than 0.1 standard deviations. However, the simulation results suggest that stronger forms of balance should be pursued in order to remove systematic biases due to observed covariates when a difference in means treatment effect estimator is used. In particular, if the true outcome model is additive, then marginal distributions should be balanced, and if the true outcome model is additive with interactions, then low-dimensional joints should be balanced. Copyright © 2016 John Wiley & Sons, Ltd.
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Affiliation(s)
- María de Los Angeles Resa
- Department of Statistics, Columbia University, 1255 Amsterdam Avenue, 901 SSW, New York, 10027, NY, U.S.A..
| | - José R Zubizarreta
- Division of Decision, Risk and Operations, and Department of Statistics, Columbia University, 3022 Broadway, 417 Uris Hall, New York, 10027, NY, U.S.A
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Rosenbaum PR. Weighted M-statistics With Superior Design Sensitivity in Matched Observational Studies With Multiple Controls. J Am Stat Assoc 2014. [DOI: 10.1080/01621459.2013.879261] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Silber JH, Rosenbaum PR, Ross RN, Ludwig JM, Wang W, Niknam BA, Mukherjee N, Saynisch PA, Even-Shoshan O, Kelz RR, Fleisher LA. Template matching for auditing hospital cost and quality. Health Serv Res 2014; 49:1446-74. [PMID: 24588413 DOI: 10.1111/1475-6773.12156] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022] Open
Abstract
OBJECTIVE Develop an improved method for auditing hospital cost and quality. DATA SOURCES/SETTING Medicare claims in general, gynecologic and urologic surgery, and orthopedics from Illinois, Texas, and New York between 2004 and 2006. STUDY DESIGN A template of 300 representative patients was constructed and then used to match 300 patients at hospitals that had a minimum of 500 patients over a 3-year study period. DATA COLLECTION/EXTRACTION METHODS From each of 217 hospitals we chose 300 patients most resembling the template using multivariate matching. PRINCIPAL FINDINGS The matching algorithm found close matches on procedures and patient characteristics, far more balanced than measured covariates would be in a randomized clinical trial. These matched samples displayed little to no differences across hospitals in common patient characteristics yet found large and statistically significant hospital variation in mortality, complications, failure-to-rescue, readmissions, length of stay, ICU days, cost, and surgical procedure length. Similar patients at different hospitals had substantially different outcomes. CONCLUSION The template-matched sample can produce fair, directly standardized audits that evaluate hospitals on patients with similar characteristics, thereby making benchmarking more believable. Through examining matched samples of individual patients, administrators can better detect poor performance at their hospitals and better understand why these problems are occurring.
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Affiliation(s)
- Jeffrey H Silber
- The Department of Pediatrics, The University of Pennsylvania School of Medicine, Philadelphia, PA; Department of Anesthesiology and Critical Care, The University of Pennsylvania School of Medicine, Philadelphia, PA; Department of Health Care Management, The Wharton School, The University of Pennsylvania, Philadelphia, PA; The Leonard Davis Institute of Health Economics, The University of Pennsylvania, Philadelphia, PA
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Hsu JY, Small DS, Rosenbaum PR. Effect Modification and Design Sensitivity in Observational Studies. J Am Stat Assoc 2013. [DOI: 10.1080/01621459.2012.742018] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
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Zubizarreta JR, Neuman M, Silber JH, Rosenbaum PR. Contrasting Evidence Within and Between Institutions That Provide Treatment in an Observational Study of Alternate Forms of Anesthesia. J Am Stat Assoc 2012; 107:901-915. [PMID: 26664027 PMCID: PMC4673003 DOI: 10.1080/01621459.2012.682533] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Abstract
In a randomized trial, subjects are assigned to treatment or control by the flip of a fair coin. In many nonrandomized or observational studies, subjects find their way to treatment or control in two steps, either or both of which may lead to biased comparisons. By a vague process perhaps affected by proximity or sociodemographic issues, subjects find their way to institutions that provide treatment. Once at such an institution, a second process, perhaps thoughtful and deliberate, assigns individuals to treatment or control. In the current paper, the institutions are hospitals, and the treatment under study is the use of general anesthesia alone versus some use of regional anesthesia during surgery. For a specific operation, the use of regional anesthesia may be typical in one hospital and atypical in another. A new matched design is proposed for studies of this sort, one that creates two types of nonoverlapping matched pairs. Using a new extension of optimal matching with fine balance, pairs of the first type exactly balance treatment assignment across institutions, so each institution appears in the treated group with the same frequency that it appears in the control group; hence, differences between institutions that affect everyone in the same way cannot bias this comparison. Pairs of the second type compare institutions that assign most subjects to treatment and other institutions that assign most subjects to control, so each institution is represented in the treated group if it typically assigns subjects to treatment or alternatively in the control group if it typically assigns subjects to control, and no institution appears in both groups. By and large, in the second type of matched pair, subjects became treated subjects or controls by choosing an institution, not by a thoughtful and deliberate process of selecting subjects for treatment within institutions. The design provides two evidence factors, that is, two tests of the null hypothesis of no treatment effect that are independent when the null hypothesis is true, where each factor is largely unaffected by certain unmeasured biases that could readily invalidate the other factor. The two factors permit separate and combined sensitivity analyses, where the magnitude of bias affecting the two factors may differ. The case of knee surgery in the study of regional versus general anesthesia is considered in detail.
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Affiliation(s)
- José R Zubizarreta
- Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340 USA
| | - Mark Neuman
- Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340 USA
| | - Jeffrey H Silber
- Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340 USA
| | - Paul R Rosenbaum
- Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340 USA
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