Wolfgang Wiedermann

Heteroscedasticity as a Basis of Direction Dependence in Reversible Linear Regression Models

ABSTRACT Heteroscedasticity is a well-known issue in linear regression modeling. When heteroscedasticity is observed, researchers are advised to remedy possible model misspecification of the explanatory part of the model (e.g., considering alternative functional forms and/or omitted variables). The present contribution discusses another source of heteroscedasticity in observational data: Directional model misspecifications in the case of nonnormal variables. Directional misspecification refers to situations where alternative models are equally likely to explain the data-generating process (e.g., x → y versus y → x). It is shown that the homoscedasticity assumption is likely to be violated in models that erroneously treat true nonnormal predictors as response variables. Recently, Direction Dependence Analysis (DDA) has been proposed as a framework to empirically evaluate the direction of effects in linear models. The present study links the phenomenon of heteroscedasticity with DDA and describes visual diagnostics and nine homoscedasticity tests that can be used to make decisions concerning the direction of effects in linear models. Results of a Monte Carlo simulation that demonstrate the adequacy of the approach are presented. An empirical example is provided, and applicability of the methodology in cases of violated assumptions is discussed.

Asymmetric properties of the Pearson correlation coefficient: Correlation as the negative association between linear regression residuals

ABSTRACT An interpretation of the Pearson correlation coefficient as the negative association between linear regression residuals is used to develop asymmetric formulas, which allow researchers to decide upon directional dependence. Model selection based on residuals extends direction dependence methodology (originally proposed for non normal variables) to normally distributed predictors. Simulation results on the robustness of the methods and an empirical example are presented. We discuss potential advantages of a change in perspective in which non normality is not treated as a source of bias, but as a valuable characteristic of variables, which can be used to gain further insights into bi- and multivariate relations.

Direction of dependence in measurement error models

Methods to determine the direction of a regression line, that is, to determine the direction of dependence in reversible linear regression models (e.g., x→y vs. y→x), have experienced rapid development within the last decade. However, previous research largely rested on the assumption that the true predictor is measured without measurement error. The present paper extends the direction dependence principle to measurement error models. First, we discuss asymmetric representations of the reliability coefficient in terms of higher moments of variables and the attenuation of skewness and excess kurtosis due to measurement error. Second, we identify conditions where direction dependence decisions are biased due to measurement error and suggest method of moments (MOM) estimation as a remedy. Third, we address data situations in which the true outcome exhibits both regression and measurement error, and propose a sensitivity analysis approach to determining the robustness of direction dependence decisions against unreliably measured outcomes. Monte Carlo simulations were performed to assess the performance of MOM-based direction dependence measures and their robustness to violated measurement error assumptions (i.e., non-independence and non-normality). An empirical example from subjective well-being research is presented. The plausibility of model assumptions and links to modern causal inference methods for observational data are discussed.

Decisions Concerning the Direction of Effects in Linear Regression Models Using Fourth Central Moments

Direction dependence analysis is attracting growing attention in the social sciences for its potential to help decide concerning the direction of effects of linear regression models. Direction dependence analysis assumes that observed data deviate from normality. Various tests have been proposed that can be applied when observed variables are skewed. However, these tests cannot be used when data are nonnormal and symmetric. The present chapter discusses direction dependence approaches for symmetric nonnormal data based on the fourth central moment. A new direction dependence approach based on regression residuals obtained from competing linear regression models is proposed. Three significance tests are described which can be used to test hypotheses compatible with direction dependence when data are nonnormal and symmetric. Results of a Monte Carlo simulation are reported which suggest that the significance tests perform well under various data scenarios. An empirical example from research on intimate partner violence is given to illustrate the application of the direction dependence tests.

Beautiful small: Misleading large randomized controlled trials? The example of colloids for volume resuscitation

In anesthesia and intensive care, treatment benefits that were claimed on the basis of small or modest-sized trials have repeatedly failed to be confirmed in large randomized controlled trials. A well-designed small trial in a homogeneous patient population with high event rates could yield conclusive results; however, patient populations in anesthesia and intensive care are typically heterogeneous because of comorbidities. The size of the anticipated effects of therapeutic interventions is generally low in relation to relevant endpoints. For regulatory purposes, trials are required to demonstrate efficacy in clinically important endpoints, and therefore must be large because clinically important study endpoints such as death, sepsis, or pneumonia are dichotomous and infrequently occur. The rarer endpoint events occur in the study population; that is, the lower the signal-to-noise ratio, the larger the trials must be to prevent random events from being overemphasized. In addition to trial design, sample size determination on the basis of event rates, clinically meaningful risk ratio reductions and actual patient numbers studied are among the most important characteristics when interpreting study results. Trial size is a critical determinant of generalizability of study results to larger or general patient populations. Typical characteristics of small single-center studies responsible for their known fragility include low variability of outcome measures for surrogate parameters and selective publication and reporting. For anesthesiology and intensive care medicine, findings in volume resuscitation research on intravenous infusion of colloids exemplify this, since both the safety of albumin infusion and the adverse effects of the artificial colloid hydroxyethyl starch have been confirmed only in large-sized trials.

A note on fourth moment-based direction dependence measures when regression errors are non normal

ABSTRACT Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression errors are at least mesokurtic. Direction dependence measures based on the co-kurtosis of variables are proposed that relax this assumption. Simulations suggest that co-kurtosis-based measures perform equally well as existing kurtosis-based methods when distributional assumptions of the latter are fulfilled. However, kurtosis-based approaches are sensitive to platy- or leptokurtic errors, while co-kurtosis-based measures protect Type I error and power rates. Data requirements necessary for causal inference and recommendations for selecting proper direction dependence measures are discussed.

Direction dependence analysis: A framework to test the direction of effects in linear models with an implementation in SPSS

In nonexperimental data, at least three possible explanations exist for the association of two variables x and y: (1) x is the cause of y, (2) y is the cause of x, or (3) an unmeasured confounder is present. Statistical tests that identify which of the three explanatory models fits best would be a useful adjunct to the use of theory alone. The present article introduces one such statistical method, direction dependence analysis (DDA), which assesses the relative plausibility of the three explanatory models on the basis of higher-moment information about the variables (i.e., skewness and kurtosis). DDA involves the evaluation of three properties of the data: (1) the observed distributions of the variables, (2) the residual distributions of the competing models, and (3) the independence properties of the predictors and residuals of the competing models. When the observed variables are nonnormally distributed, we show that DDA components can be used to uniquely identify each explanatory model. Statistical inference methods for model selection are presented, and macros to implement DDA in SPSS are provided. An empirical example is given to illustrate the approach. Conceptual and empirical considerations are discussed for best-practice applications in psychological data, and sample size recommendations based on previous simulation studies are provided.

Testing Event-Based Forms of Causality

Three fundamental types of causal relations are those of necessity, sufficiency, and necessity and sufficiency. These types are defined in contexts of categorical variables or events. Using statement calculus or Boolean algebra, one can determine which patterns of events are in support of a particular form of causal relation. In this article, we approach the analysis of these forms of causality taking the perspective of the analyst of empirical data. It is proposed using Configural Frequency Analysis (CFA) to test hypotheses about type of causal relation. Models are proposed for two-variable and multi-variable cases. Two CFA approaches are proposed. In the first, individual patterns (configurations) are examined under the question whether they are in support of a particular type of causal relation. In the second, patterns that are in support are compared with corresponding patterns that are not in support. In an empirical example, hypotheses are tested on the prediction of sustainability of change in dietary fat intake habits.

Fellow Scholars: Let’s Liberate Ourselves from Scientific Machinery

In science, a well-oiled machinery exists that can be used to obtain research funding, to create information for data analysis, and to publish results. Innovation does take place. However, there exist basic assumptions, beliefs, and convictions that are neither questioned nor reconsidered, but can confine innovation within the frame provided. In this article, two examples are given to encourage researchers to reconsider these quasi-factual assumptions. The first example is that of person-oriented research. Results of person-oriented research allow researchers to validly describe both individuals and aggregates, and, therefore, devise intervention plans that are more likely to succeed. The second example is that of causality. New statistical methods to evaluate causal hypotheses allow one to make decisions about causal processes that, before the development of these methods, were impossible to justify statistically. Both examples show that reconsidering commonly held beliefs can result in most useful and applicable results.

Item Response Models for Dependent Data: Quasi-exact Tests for the Investigation of Some Preconditions for Measuring Change

The Rasch model has several advantages for the psychometric investigation of item quality (e.g., specific objectivity). One approach to testing model fit uses quasi-exact tests which are well suited to test the validity of the Rasch model when sample sizes are rather small. Application of these tests is not restricted to Rasch modeling. In this chapter, we show that these tests can be used to test preconditions for measuring change such as measurement invariance, unidimensionality, and local independence across time points. For example, if items are unidimensional across time points (i.e., all items measure the same latent construct across time) and groups (e.g., control and training groups), it follows that there are no significant interindividual differences within groups and over time. All individuals in a group change in the same direction. On the other hand, significant results across time but not within groups suggest group differences in change, such as training effects. In this chapter, we first give an introduction to quasi-exact tests. Then, we demonstrate the applicability of three test statistics for the investigation of preconditions for measuring change using empirical power analysis and an empirical example concerning spatial ability.