Joseph Hilbe

Logistic regression models

Preface Introduction The Normal Model Foundation of the Binomial Model Historical and Software Considerations Chapter Profiles Concepts Related to the Logistic Model 2 x 2 Table Logistic Model 2 x k Table Logistic Model Modeling a Quantitative Predictor Logistic Modeling Designs Estimation Methods Derivation of the IRLS Algorithm IRLS Estimation Maximum Likelihood Estimation Derivation of the Binary Logistic Algorithm Terms of the Algorithm Logistic GLM and ML Algorithms Other Bernoulli Models Model Development Building a Logistic Model Assessing Model Fit: Link Specification Standardized Coefficients Standard Errors Odds Ratios as Approximations of Risk Ratios Scaling of Standard Errors Robust Variance Estimators Bootstrapped and Jackknifed Standard Errors Stepwise Methods Handling Missing Values Modeling an Uncertain Response Constraining Coefficients Interactions Introduction Binary X Binary Interactions Binary X Categorical Interactions Binary X Continuous Interactions Categorical X Continuous Interaction Thoughts about Interactions Analysis of Model Fit Traditional Fit Tests for Logistic Regression Hosmer-Lemeshow GOF Test Information Criteria Tests Residual Analysis Validation Models Binomial Logistic Regression Overdispersion Introduction The Nature and Scope of Overdispersion Binomial Overdispersion Binary Overdispersion Real Overdispersion Concluding Remarks Ordered Logistic Regression Introduction The Proportional Odds Model Generalized Ordinal Logistic Regression Partial Proportional Odds Multinomial Logistic Regression Unordered Logistic Regression Independence of Irrelevant Alternatives Comparison to Multinomial Probit Alternative Categorical Response Models Introduction Continuation Ratio Models Stereotype Logistic Model Heterogeneous Choice Logistic Model Adjacent Category Logistic Model Proportional Slopes Models Panel Models Introduction Generalized Estimating Equations Unconditional Fixed Effects Logistic Model Conditional Logistic Models Random Effects and Mixed Models Logistic Regression Other Types of Logistic-Based Models Survey Logistic Models Scobit-Skewed Logistic Regression Discriminant Analysis Exact Logistic Regression Exact Methods Alternative Modeling Methods Conclusion Appendix A: Brief Guide to Using Stata Commands Appendix B: Stata and R Logistic Models Appendix C: Greek Letters and Major Functions Appendix D: Stata Binary Logistic Command Appendix E: Derivation of the Beta-Binomial Appendix F: Likelihood Function of the Adaptive Gauss-Hermite Quadrature Method of Estimation Appendix G: Data Sets Appendix H: Marginal Effects and Discrete Change References Author Index Subject Index Exercises and R Code appear at the end of most chapters.

Failure-free survival following brachytherapy alone or external beam irradiation alone for T1–2 prostate tumors in 2222 patients: results from a single practice

PURPOSE To evaluate failure-free survival (FFS) for brachytherapy (BT) alone compared to external beam radiotherapy (EBRT) alone for Stage T1-2 Nx-No Mo patients over the same time period by a single community-based practice in the prostate-specific antigen (PSA) era. MATERIALS AND METHODS The database of Arizona Oncology Services (a multiphysician radiation oncology practice in the Phoenix metropolitan area) was reviewed for patients meeting the following criteria: (1) T1 or T2 Nx-No Mo prostate cancer; (2) no prior or concurrent therapy including hormones; (3) treatment period 12/88-12/95; and (4) treatment with either EBRT alone or BT alone ((125)I or (103)Pd). This yielded 1527 EBRT and 695 BT patients; no patients meeting the above criteria were excluded from analysis. Median follow-up for EBRT patients was 41.3 months and, for BT patients, 51.3 months. Patients were not randomized to either therapy but rather received EBRT or BT based upon patient, treating, and/or referring physician preference. PSA failure was defined according to the ASTRO consensus guidelines. The median patient age was 74 years for both groups. RESULTS Failure-free survival at 5 years for EBRT and BT are 69% and 71%, respectively (p = 0.91). For T stage, no significant difference in FFS at 5 years is observed between EBRT and BT for either T1 (78% vs. 83%, p = 0.47) or T2 (67% vs. 67%, p = 0.89) tumors. Analysis by Gleason score shows superior outcomes for Gleason 8-10 lesions treated with EBRT vs. BT (5-year FFS 52% vs. 28%, p = 0.04); outcomes for lower grade lesions (Gleason 4-6) when analyzed by Gleason score alone do not significantly differ according to treatment received. Patients with initial PSA values of 10-20 ng/dL have an improved FFS with EBRT vs. BT at 5 years (70% vs. 53%, p = 0.001); outcomes for patients with initial PSA ranges of 0-4 ng/dL, of > 4-10 ng/dL, and > 20 ng/dL did not differ significantly by treatment received. FFS was also determined for presenting Gleason score/PSA combinations; all Gleason combinations in the initial PSA range >10-20 ng/dL had superior outcomes with EBRT compared to BT, and this reached statistical significance for Gleason scores of 2-4 (72% vs. 58%, p = 0.026), Gleason 7 (67% vs. 28%, p = 0.002), and Gleason 8-10 (63% vs. 23%, p = 0.05). CONCLUSION In our patient population, either EBRT or BT appear equally efficacious for patients with T1/T2 disease with Gleason scores </= 6 or PSA </= 10 ng/dL. Patients with presenting Gleason scores of 8-10 or PSA > 10 ng/dL (but </= 20 ng/dL) appear to fare significantly worse with BT alone compared to EBRT alone. Neither EBRT nor BT alone was particularly effective for patients with a presenting PSA > 20 ng/dL, as would be anticipated from the significant risks of occult distant metastasis in this group. To our knowledge, this is the first report comparing the outcome of EBRT and BT treatment in patients treated concurrently by a single group, and these results, achieved in a community-based practice, compare favorably to data from academic centers regarding external beam, brachytherapy, or surgical outcomes and should be generalizable to the community at large.

Management and outcomes for black patients with acute myocardial infarction in the reperfusion era

Data from a national registry of myocardial infarction patients from June 1994 to April 1996 were analyzed to compare the presenting characteristics, acute reperfusion strategies, treatment patterns, and clinical outcomes among black and white patients. Blacks presented much later to the hospital after the onset of symptoms (median 145 vs 122 minutes, p <0.001), were more likely to have atypical cardiac symptoms (28% vs 24%, p <0.001), and nondiagnostic electrocardiograms during the initial evaluation period compared with whites (37% vs 31%, p <0.001). Also, blacks were less likely to receive intravenous thrombolytic therapy (adjusted odds ratio [OR] 0.76, 95% confidence intervals [CI] 0.71 to 0.80), coronary arteriography (adjusted OR 0.85, 95% CI 0.77 to 0.95), other elective catheter-based procedures (adjusted OR 0.87, 95% CI 0.78 to 0.96), and coronary artery bypass surgery (adjusted OR 0.66, 95% CI 0.58 to 0.75) than their white counterparts. Despite these differences in treatment, there were no significant differences in hospital mortality between blacks and whites.

Presenting characteristics, treatment patterns, and clinical outcomes of non-black minorities in the national registry of myocardial infarction 2

Data from a national registry (cohort) of myocardial infarction, which has enrolled 275,046 patients from June 1994 to April 1996, were analyzed to compare the baseline demographic and clinical characteristics, treatment patterns, and clinical outcomes among Hispanics, Asian-Pacific islanders, and native Americans with those of white Americans presenting to the hospital with acute myocardial infarction. Non-black minorities were younger, had a higher proportion of men, used the emergency medical services less frequently, and presented later to the hospital after the onset of symptoms (135 vs 122 minutes, p <0.001) than whites. Also, non-black minorities were less likely to receive beta-blocker therapy at discharge (crude odds ratio 0.86, confidence interval 0.82 to 0.90) than whites, but they were generally as likely to receive intravenous thrombolytic therapy (with the exception of Asian-Pacific islanders) and undergo both coronary arteriography and revascularization procedures as their white counterparts. There were no significant differences in hospital mortality for non-black minorities compared with whites.

A comparison of several approaches for choosing between working correlation structures in generalized estimating equation analysis of longitudinal binary data.

The method of generalized estimating equations (GEE) models the association between the repeated observations on a subject with a patterned correlation matrix. Correct specification of the underlying structure is a potentially beneficial goal, in terms of improving efficiency and enhancing scientific understanding. We consider two sets of criteria that have previously been suggested, respectively, for selecting an appropriate working correlation structure, and for ruling out a particular structure(s), in the GEE analysis of longitudinal studies with binary outcomes. The first selection criterion chooses the structure for which the model-based and the sandwich-based estimator of the covariance matrix of the regression parameter estimator are closest, while the second selection criterion chooses the structure that minimizes the weighted error sum of squares. The rule out criterion deselects structures for which the estimated correlation parameter violates standard constraints for binary data that depend on the marginal means. In addition, we remove structures from consideration if their estimated parameter values yield an estimated correlation structure that is not positive definite. We investigate the performance of the two sets of criteria using both simulated and real data, in the context of a longitudinal trial that compares two treatments for major depressive episode. Practical recommendations are also given on using these criteria to aid in the efficient selection of a working correlation structure in GEE analysis of longitudinal binary data.

Relative influence of Gleason score and pretreatment PSA in predicting survival following brachytherapy for prostate cancer

PURPOSE To evaluate 10-year survival rates after prostate brachytherapy and to assess the relative importance of pretreatment prostate-specific antigen (PSA) and Gleason score in predicting cancer death. MATERIALS AND METHODS A retrospective review was performed on all patients treated with permanent brachytherapy for stage T1 or T2 primary prostate cancer at a single institution from December 1988 through June 30, 1998. The study cohort consisted of 1266 patients with a median follow-up of 4.1 years and a maximum of 12.6 years. Actuarial survival and cause-specific survival rates were calculated as the primary endpoints, and compared at 5 and 10 years. Groups studied consist of PSA<or=4, PSA 4.1-10, and PSA>or=10 as well as Gleason 2-4, 5-6, and 7-10. Multivariate and univariate analysis were performed looking at stage, grade, PSA, and risk group as variables. RESULTS The median age at the time of treatment was 73 years and at the time of analysis 603 patients were known to be alive. Overall survival is 38% at 10 years, however most deaths were unrelated to prostate cancer. Cause specific survival at 5 and 10 years is 98% and 87%. Both grade (>or=Gleason 7) and PSA (>or=10 ng/ml) predict adversely for cancer death within 10 years. Patients with low grade or PSA at presentation reveal prostate cancer-specific survival of 91% and 98%, respectively. By contrast, men with high grade or high PSA presentation have survival of 66% and 69% at 10 years. In multivariate analysis, the presence of one of these adverse features carries a hazard ratio of cancer death of 4.7 and 6.4, while the presence of multiple risk factors places patients in an unfavorable risk group with a hazard ratio of 27. CONCLUSIONS Biochemical disease-free survival is a useful tool to assess prostate cancer treatments and is predicted based on established pretreatment risk groups. Long-term cancer-specific survival is ultimately a more important endpoint. Brachytherapy is reported here to be an excellent therapeutic alternative for selected early stage patients with prostate cancer. This is based on 10-year cause specific survival, which may also be predicted by stage, grade, PSA, and risk group. Of these, the risk group remains the most powerful parameter to identify those patients at highest risk of biochemical failure and death from prostate cancer.

Female hurricanes are deadlier than male hurricanes

Significance Meteorologists and geoscientists have called for greater consideration of social science factors that predict responses to natural hazards. We answer this call by highlighting the influence of an unexplored social factor, gender-based expectations, on the human toll of hurricanes that are assigned gendered names. Feminine-named hurricanes (vs. masculine-named hurricanes) cause significantly more deaths, apparently because they lead to lower perceived risk and consequently less preparedness. Using names such as Eloise or Charlie for referencing hurricanes has been thought by meteorologists to enhance the clarity and recall of storm information. We show that this practice also taps into well-developed and widely held gender stereotypes, with potentially deadly consequences. Implications are discussed for understanding and shaping human responses to natural hazard warnings. Do people judge hurricane risks in the context of gender-based expectations? We use more than six decades of death rates from US hurricanes to show that feminine-named hurricanes cause significantly more deaths than do masculine-named hurricanes. Laboratory experiments indicate that this is because hurricane names lead to gender-based expectations about severity and this, in turn, guides respondents’ preparedness to take protective action. This finding indicates an unfortunate and unintended consequence of the gendered naming of hurricanes, with important implications for policymakers, media practitioners, and the general public concerning hurricane communication and preparedness.

Generalized Linear Models

In this chapter we shall discuss a class of statistical models that generalize the well-understood normal linear model. A normal or Gaussian model assumes that the response Y is equal to the sum of a linear combination X T β of the d-dimensional predictor X and a Gaussian distributed error term. It is well known that the least-squares estimator \(\hat \beta \) of β performs well under these assumptions. Moreover, extensive diagnostic tools have been developed for models of this type.

The overlooked potential of Generalized Linear Models in astronomy - I: Binomial regression

Revealing hidden patterns in astronomical data is often the path to fundamental scientific breakthroughs; meanwhile the complexity of scientific inquiry increases as more subtle relationships are sought. Contemporary data analysis problems often elude the capabilities of classical statistical techniques, suggesting the use of cutting edge statistical methods. In this light, astronomers have overlooked a whole family of statistical techniques for exploratory data analysis and robust regression, the so-called Generalized Linear Models (GLMs). In this paper ‐ the first in a series aimed at illustrating the power of these methods in astronomical applications ‐ we elucidate the potential of a particular class of GLMs for handling binary/binomial data, the so-called logit and probit regression techniques, from both a maximum likelihood and a Bayesian perspective. As a case in point, we present the use of these GLMs to explore the conditions of star formation activity and metal enrichment in primordial minihaloes from cosmological hydro-simulations including detailed chemistry, gas physics, and stellar feedback. Finally, we highlight the use of receiver operating characteristic curves as a diagnostic for binary classifiers, and ultimately we use these to demonstrate the competitive predictive performance of GLMs against the popular technique of artificial neural networks.

Mathematica 5.2: A Review

Mathematica is characterized by Wolfram Research, manufacturers of the software, as a “technical computing system.” Most of those who use it, however, likely think of it as a comprehensive numeric and symbolic computational package with extensive associated graphical capabilities and as a programming language with an interactive document or notebook interface. Regardless of how the package is characterized, Mathematica is perhaps the most well-known and well-used technical computing package on the market. Current estimates show that Mathematica has a user base exceeding that of its two closest competitors—Maple and MathCAD. The purpose of this review is to provide the reader with an overview of Mathematica 5.2’s scope and capabilities, its limitations and any points needing development, and a sense of how the Mathematica program actually works. To achieve the latter end, I shall provide several worked out examples demonstrating the package at work. Given the fact that the Notebook interface can be saved in LaTEX format, replicating actual example input and output should clearly represent how the screen display appears. I shall first provide details of license costs, together with a listing of what comes with the package. Following this I’ll provide a brief history of Wolfram Research and of the evolution of the software. Next I’ll list some of the most important enhancements that appear in version 5.2, which was released in July 2005. Thereafter I’ll discuss the general capabilities of the package, followed by various examples. Finally, I’ll mention any shortcomings or features of the package that I think need further development.