Errors-in-variables models

Abstract: The successful application of a conceptual rainfall-runoff (CRR) model depends on how well it is calibrated. Despite the popularity of CRR models, reports in the literature indicate that it is typically difficult, if not impossible, to obtain unique optimal values for their parameters using automatic calibration methods. Unless the best set of parameters associated with a given calibration data set can be found, it is difficult to determine how sensitive the parameter estimates (and hence the model forecasts) are to factors such as input and output data error, model error, quantity and quality of data, objective function used, and so on. Results are presented that establish clearly the nature of the multiple optima problem for the research CRR model SIXPAR. These results suggest that the CRR model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets. Next, the performance of three existing global search procedures are evaluated on the model SIXPAR. Finally, a powerful new global optimization procedure is presented, entitled the shuffled complex evolution (SCE-UA) method, which was able to consistently locate the global optimum of the SIXPAR model, and appears to be capable of efficiently and effectively solving the CRR model optimization problem.

Abstract: Many economic researchers have attempted to measure the effect of aggregate market or public policy variables on micro units by merging aggregate data with micro observations by industry, occupation, or geographical location, then using multiple regression or similar statistical models to measure the effect of the aggregate variable on the micro units. The methods are usually based upon the assumption of independent disturbances, which is typically not appropriate for data from populations with grouped structure. Incorrectly using ordinary least squares can lead to standard errors that are seriously biased downward. This note illustrates the danger of spurious regression from this kind of misspecification, using as an example a wage regression estimated on data for individual workers that includes in the specification aggregate regressors for characteristics of geographical states. Copyright 1990 by MIT Press.

Abstract: Summary 1. Linear regression models are an important statistical tool in evolutionary and ecological studies. Unfortunately, these models often yield some uninterpretable estimates and hypothesis tests, especially when models contain interactions or polynomial terms. Furthermore, the standard errors for treatment groups, although often of interest for including in a publication, are not directly available in a standard linear model. 2. Centring and standardization of input variables are simple means to improve the interpretability of regression coefficients. Further, refitting the model with a slightly modified model structure allows extracting the appropriate standard errors for treatment groups directly from the model. 3. Centring will make main effects biologically interpretable even when involved in interactions and thus avoids the potential misinterpretation of main effects. This also applies to the estimation of linear effects in the presence of polynomials. Categorical input variables can also be centred and this sometimes assists interpretation. 4. Standardization (z-transformation) of input variables results in the estimation of standardized slopes or standardized partial regression coefficients. Standardized slopes are comparable in magnitude within models as well as between studies. They have some advantages over partial correlation coefficients and are often the more interesting standardized effect size. 5. The thoughtful removal of intercepts or main effects allows extracting treatment means or treatment slopes and their appropriate standard errors directly from a linear model. This provides a simple alternative to the more complicated calculation of standard errors from contrasts and main effects. 6. The simple methods presented here put the focus on parameter estimation (point estimates as well as confidence intervals) rather than on significance thresholds. They allow fitting complex, but meaningful models that can be concisely presented and interpreted. The presented methods can also be applied to generalised linear models (GLM) and linear mixed models.

Abstract: Preface. Acknowledgments. Acronyms. 1. Introduction. 1.1 Advantages of Longitudinal Studies. 1.2 Challenges of Longitudinal Data Analysis. 1.3 Some General Notation. 1.4 Data Layout. 1.5 Analysis Considerations. 1.6 General Approaches. 1.7 The Simplest Longitudinal Analysis. 1.8 Summary. 2. ANOVA Approaches to Longitudinal Data. 2.1Single-Sample Repeated Measures ANOVA. 2.2 Multiple-Sample Repeated Measures ANOVA. 2.3 Illustration. 2.4 Summary. 3. MANOVA Approaches to Longitudinal Data. 3.1 Data Layout for ANOVA versus MANOVA. 3.2 MANOVA for Repeated Measurements. 3.3 MANOVA of Repeated Measures-s Sample Case. 3.4 Illustration. 3.5 Summary. 4. Mixed-Effects Regression Models for Continuous Outcomes. 4.1 Introduction. 4.2 A Simple Linear Regression Model. 4.3 Random Intercept MRM. 4.4 Random Intercept and Trend MRM. 4.5 Matrix Formulation. 4.6 Estimation . 4.7 Summary. 5. Mixed-Effects Polynomial Regression Models. 5.1 Introduction. 5.2 Curvilinear Trend Model. 5.3 Orthogonal Polynomials. 5.4 Summary. 6. Covariance Pattern Models. 6.1 Introduction. 6.2 Covariance Pattern Models. 6.3 Model Selection. 6.4 Example. 6.5 Summary. 7. Mixed Regression Models with Autocorrelated Errors. 7.1 Introduction. 7.2 MRMs with AC Errors. 7.3 Model Selection. 7.4 Example. 7.5 Summary. 8. Generalized Estimating Equations (GEE) Models. 8.1 Introduction. 8.2 Generalized Linear Models (GLMs). 8.3 Generalized Estimating Equations (GEE) Models. 8.4 GEE Estimation. 8.5 Example. 8.6 Summary. 9. Mixed-Effects Regression Models for Binary Outcomes. 9.1 Introduction. 9.2 Logistic Regression Model. 9.3 Probit Regression Models. 9.4 Threshold Concept. 9.5 Mixed-Effects Logistic Regression Model. 9.6 Estimation. 9.7 Illustration. 9.8 Summary. 10. Mixed-Effects Regression Models for Ordinal Outcomes. 10.1 Introduction. 10.2 Mixed-Effects Proportional Odds Model. 10.3 Psychiatric Example. 10.4 Health Services Research Example. 10.5 Summary. 11. Mixed-Effects Regression Models for Nominal Data. 11.1 Mixed-Effects Multinomial Regression Model. 11.2 Health Services Research Example. 1 1.3 Competing Risk Survival Models. 11.4 Summary. 12. Mixed-effects Regression Models for Counts. 12.1 Poisson Regression Model. 12.2 Modified Poisson Models. 12.3 The ZIP Model. 12.4 Mixed-Effects Models for Counts. 12.5 Illustration. 12.6 Summary. 13. Mixed-Effects Regression Models for Three-Level Data. 13.1 Three-Level Mixed-Effects Linear Regression Model. 13.1.1 Illustration. 13.2 Three-Level Mixed-Effects Nonlinear Regression Models. 13.3 Summary. 14. Missing Data in Longitudinal Studies. 14.1 Introduction. 14.2 Missing Data Mechanisms. 14.3 Models and Missing Data Mechanisms. 14.4 Testing MCAR. 14.5 Models for Nonignorable Missingness. 14.6 Summary. Bibliography. Topic Index.