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Supplemental Appendices\1. Instrumental variable analysisDescription of instrumental variable analysesTwo-stage residual inclusion (2SRI) is a method for instrumental variable analysis appropriate for the setting of exposure and outcome variables which are not normally distributed continuous measures (and therefore do not meet the assumptions required for valid two stage least squares estimation(2SLS)ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"0017-9124","PMID":"18546544","abstract":"OBJECTIVE To investigate potential bias in the use of the conventional linear instrumental variables (IV) method for the estimation of causal effects in inherently nonlinear regression settings. DATA SOURCES Smoking Supplement to the 1979 National Health Interview Survey, National Longitudinal Alcohol Epidemiologic Survey, and simulated data. STUDY DESIGN Potential bias from the use of the linear IV method in nonlinear models is assessed via simulation studies and real world data analyses in two commonly encountered regression setting: (1) models with a nonnegative outcome (e.g., a count) and a continuous endogenous regressor; and (2) models with a binary outcome and a binary endogenous regressor. PRINCIPAL FINDINGS The simulation analyses show that substantial bias in the estimation of causal effects can result from applying the conventional IV method in inherently nonlinear regression settings. Moreover, the bias is not attenuated as the sample size increases. This point is further illustrated in the survey data analyses in which IV-based estimates of the relevant causal effects diverge substantially from those obtained with appropriate nonlinear estimation methods. CONCLUSIONS We offer this research as a cautionary note to those who would opt for the use of linear specifications in inherently nonlinear settings involving endogeneity.","author":[{"dropping-particle":"V","family":"Terza","given":"Joseph","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Bradford","given":"W David","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Dismuke","given":"Clara E","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Health services research","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2008","6"]]},"page":"1102--20","title":"The use of linear instrumental variables methods in health services research and health economics: a cautionary note.","type":"article-journal","volume":"43"},"uris":[""]}],"mendeley":{"formattedCitation":"[27]","plainTextFormattedCitation":"[27]","previouslyFormattedCitation":"[26]"},"properties":{"noteIndex":0},"schema":""}[27]ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jhealeco.2007.09.009","ISSN":"01676296","author":[{"dropping-particle":"V.","family":"Terza","given":"Joseph","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Basu","given":"Anirban","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Rathouz","given":"Paul J.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Health Economics","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2008","5"]]},"page":"531-543","title":"Two-stage residual inclusion estimation: Addressing endogeneity in health econometric modeling","type":"article-journal","volume":"27"},"uris":[""]},{"id":"ITEM-2","itemData":{"DOI":"10.1111/1475-6773.12714","ISSN":"1475-6773","PMID":"28568477","abstract":"OBJECTIVES Empirical analyses in health services research and health economics often require implementation of nonlinear models whose regressors include one or more endogenous variables-regressors that are correlated with the unobserved random component of the model. In such cases, implementation of conventional regression methods that ignore endogeneity will likely produce results that are biased and not causally interpretable. Terza et al. (2008) discuss a relatively simple estimation method that avoids endogeneity bias and is applicable in a wide variety of nonlinear regression contexts. They call this method two-stage residual inclusion (2SRI). In the present paper, I offer a 2SRI how-to guide for practitioners and a step-by-step protocol that can be implemented with any of the popular statistical or econometric software packages. STUDY DESIGN We introduce the protocol and its Stata implementation in the context of a real data example. Implementation of 2SRI for a very broad class of nonlinear models is then discussed. Additional examples are given. EMPIRICAL APPLICATION We analyze cigarette smoking as a determinant of infant birthweight using data from Mullahy (1997). CONCLUSION It is hoped that the discussion will serve as a practical guide to implementation of the 2SRI protocol for applied researchers.","author":[{"dropping-particle":"V","family":"Terza","given":"Joseph","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Health services research","id":"ITEM-2","issue":"3","issued":{"date-parts":[["2018","6"]]},"page":"1890-1899","title":"Two-Stage Residual Inclusion Estimation in Health Services Research and Health Economics.","type":"article-journal","volume":"53"},"uris":[""]},{"id":"ITEM-3","itemData":{"DOI":"10.1186/s12874-018-0513-y","ISSN":"1471-2288","author":[{"dropping-particle":"","family":"Koladjo","given":"Babagnidé Fran?ois","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Escolano","given":"Sylvie","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Tubert-Bitter","given":"Pascale","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"BMC Medical Research Methodology","id":"ITEM-3","issue":"1","issued":{"date-parts":[["2018","12","22"]]},"page":"61","title":"Instrumental variable analysis in the context of dichotomous outcome and exposure with a numerical experiment in pharmacoepidemiology","type":"article-journal","volume":"18"},"uris":[""]}],"mendeley":{"formattedCitation":"[24–26]","plainTextFormattedCitation":"[24–26]","previouslyFormattedCitation":"[23–25]"},"properties":{"noteIndex":0},"schema":""}[24–26]). The first stage of this process employed a logistic regression model to predict the probability of receipt of a PNB conditional on age (restricted cubic spline with 5 knots), biological sex (binary), neighborhood income quintile (5-level categorical variable), rurality (binary), procedure (categorical), HOMR score (continuous linear), each Elixhauser comorbidity (binary), each specified drug class (binary), year of surgery (categorical), resource utilization band (categorical), frailty (binary) and preoperative long-term care residence (binary) plus the instrumental variable (proportion of patients at the same hospital who received a PNB in the previous year; continuous).The second stage model was a generalized linear model with a log link and gamma response distribution. Postoperative LoS was the dependent variable, receipt of a PNB was the exposure of interest, the raw residual from the first stage model was included, and covariate adjustment was made for age (restricted cubic spline with 5 knots), biological sex (binary), neighborhood income quintile (5-level categorical variable), rurality (binary), procedure (categorical), HOMR score (continuous linear), each Elixhauser comorbidity (binary), each specified drug class (binary), year of surgery (categorical), resource utilization band (categorical), frailty (binary) and preoperative long-term care residence (binary).We calculated both the regression coefficient and the attributable effect size estimate (i.e., difference in days between exposure groups) using bootstrap techniques. All 2SRI effect estimate and 95% confidence intervals were calculated using 1000 bootstrap samples that were created with a 1:1 sampling rate with replacement. Confidence intervals were based on the 2.5th and 97.5th percentiles. To estimate the regression coefficient, our model was run in each of the 1000 bootstrap samples and the regression coefficient was based on the median value for the exposure of interest. For the attributable effect size estimate, we predicted the LoS for all individuals using the primary adjusted regression model, and the difference in days was based on the median difference between the predicted LoS values for comparing those who did and did not receive a PNB in each bootstrap sample.We tested the robustness of our IV estimation in two additional analyses. First, because we lack consensus on the type of residual (e.g., raw vs. a variety of available scaled residuals) most appropriate for 2SRI analyses, we re-calculated the attributable days using deviance scaled residuals (as opposed to the raw residuals used in the first 2SRI analysis). We also completed a 2SLS analysis (using the same covariate adjustment above) using PROC SYSLIN in SAS. Results of additional instrumental variable analyses2SRI with deviance residuals: Adjusted decrease in LOS 1.04, 95%CI 0.89-1.182SLS: 2.25 day decrease in LOS with PNB, 95%CI 1.47-3.03 2. Propensity score analysisDescription of propensity score analysisThe second additional approach was a propensity score matched analysis where each individual who received a PNB was matched 1:1 to an individual without a PNB. The propensity score was generated from a logistic regression model that predicted receipt of a PNB with the following predictor variables: age (restricted cubic spline with 5 knots), biological sex (binary), neighborhood income quintile (5-level categorical variable), rurality (binary), procedure (categorical), HOMR score (continuous linear), each Elixhauser comorbidity (binary), each specified drug class (binary), year of surgery (categorical), resource utilization band (categorical), frailty (binary) and preoperative long-term care residence (binary).The match was made exactly on hospital (to account for clustering) and then using a caliper equivalent to 0.2 standard deviations of the logit of the propensity score and employing a greedy matching algorithm. This process resulted in successful matching of 8 261 (82.4%) of patients who received a PNB to a similar patient from the same hospital who did not receive a PNB. Absolute standardized differences (ASD) for all measured covariates were <0.10 between matched groups, and overall standardized differences across variables decreased from 0.104 to 0.01 after matching. Comparison between the matched PNB and no PNB groups were calculated using a generalized linear model with a log link and gamma distributed error that accounted for the paired nature of the data. Hospital fixed-effects analysisDescription of hospital fixed-effects analysisThe third additional approach, post-hoc as it was reviewer-requested, was a regression analysis that adjusted for all of the measured covariates plus a categorical variable which represented each hospital as a fixed effect. All covariates included in the primary generalized linear model with a log link and gamma response distributions were added, with length of stay as the dependent variable and, instead of clustering at the hospital-level, we included the hospital indicator variable as a fixed effect. SDC Table - Characteristics of Population by High (>8%) vs Low (<8%) Peripheral Nerve Block Use??High PNB n=10 030Low PNB n=55 241ASD?Demographics????Age at surgery (years; mean (SD))79 (14)78 (14)0.07Female68.867.80.04Income Quintile 1 (lowest)21.822.10.01?219.621.10.05?319.520.00.00?419.519.00.01?5 (highest)19.517.90.04Year of surgery 201121.316.00.07201222.017.40.04201320.721.10.01201417.922.90.03201518.122.50.14Comorbidities????Alcohol abuse 3.73.40.05Atrial arrhythmia 8.78.50.01Blood loss anemia ?18.118.40.05Cardiac valve disease ?3.23.40.01Coagulopathy 2.72.60.03Chronic obstructive pulmonary disease 11.811.70.03Cerebrovascular disease 4.94.90.01Disease of pulmonary circulation 2.52.40.02Dementia 17.618.30.01Depression 4.75.00.01Deficiency anemia 0.60.60.00Diabetes mellitus without complications 13.413.80.00Diabetes mellitus with complications 14.113.90.02Dialysis 1.51.50.02Drug abuse 0.80.90.02Heart failure 21.221.00.01Hemiplegia 0.90.80.02Hypertension without complications 39.843.50.04Hypertension with complications 0.90.90.01Liver disease 1.41.50.00Malignancy ?6.76.50.02Metastases ?1.92.10.00Obesity 1.41.70.01Peptic ulcer disease 1.51.50.00Peripheral vascular disease 2.52.30.01Psychoses 1.01.00.01Renal disease 4.14.10.00Rheumatic disease 1.31.10.03Venous thromboembolism 0.80.70.01Weight loss 3.53.20.00Frail 60.261.20.01One-year mortality risk ????HOMR score (mean (SD))37 (7)37 (7)0.01Medications????Anticoagulant 13.013.00.02Antiplatetlet agent 7.37.60.02Antipsychotic 11.410.20.03Benzodiazepine17.516.50.02Opioid 21.822.50.00Dementia medication 7.87.60.01Healthcare resource use????Long term care facility 14.915.50.01Resource utilization band 2 (lowest)2.42.00.01314.013.40.00423.524.20.02?5 (highest)60.160.40.01Procedure????Implantation of internal device, pelvis 0.30.20.02Implantation of internal device, hip joint 3935.60.07Fixation, hip joint 17.121.60.09Fixation, femur 50.849.20.01*all+A1:E68 column values indicate n (%) unless otherwise indicated; ?values grater than 0.10 indicate a substantial difference; ASD: absolute standardized difference; PNB: Peripheral Nerve Block SD standard deviation; HOMR: Hospital One-Year Mortality Risk ................
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