Interpretation Of Regression Coefficients Research Articles

An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations

The Weibull distribution is a versatile probability distribution widely applied in modeling the failure times of objects or systems. Its behavior is shaped by two essential parameters: the shape parameter and the scale parameter. By manipulating these parameters, the Weibull distribution adeptly captures diverse failure patterns observed in real-world scenarios. This flexibility and broad applicability make it an indispensable tool in reliability analysis and survival modeling. This manuscript explores five parameterizations of the Weibull distribution, each based on different moments, like mean, quantile, and mode. It meticulously characterizes each parameterization, introducing a novel one based on the model’s mode, along with its hazard and survival functions, shedding light on their unique properties. Additionally, it delves into the interpretation of regression coefficients when incorporating regression structures into these parameterizations. It is analytically established that all five parameterizations define the same log-likelihood function, underlining their equivalence. Through Monte Carlo simulation studies, the performances of these parameterizations are evaluated in terms of parameter estimations and residuals. The models are further applied to real-world data, illustrating their effectiveness in analyzing material fatigue life and survival data. In summary, this manuscript provides a comprehensive exploration of the Weibull distribution and its various parameterizations. It offers valuable insights into their applications and implications in modeling failure times, with potential contributions to diverse fields requiring reliability and survival analysis.

The regression trap: why regression analyses are not suitable for selecting determinants to target in behavior change interventions

ABSTRACT Objective Regression analyses are commonly used for selecting determinants to target in behavior change interventions, but the aim of this article is to explain why regression analyses are not suitable for this purpose (i.e. the regression trap). Methods This aim is achieved by providing (1) a theoretical rationale based on overlap among determinants; (2) a mathematical rationale based on the formulas that are used to calculate regression coefficients; and (3) examples based on real-world data. Results First, the meaning of regression coefficients is commonly explained as expressing the association between a determinant and a target behavior ‘holding all other predictors constant.’ We explain that this often boils down to ‘neglecting a part of the psyche.’ Second, we demonstrate that the interpretation of regression coefficients is distorted by correlations between determinants. Third, the examples provided demonstrate the impact this has in practice. This results in interventions targeting determinants that are less relevant and, thereby, have less impact on behavior change. Conclusion There are theoretical, mathematical, and practical reasons why regression analyses, and by extension multivariate analyses relying on correlations, are not suitable to select determinants to target in behavior change interventions. Instead, intervention developers should consider univariate distributions and bivariate association estimates simultaneously and there are freely accessible tools available to do so.

Multilevel modeling in single-case studies with count and proportion data: A demonstration and evaluation.

The outcomes in single-case experimental designs (SCEDs) are often counts or proportions. In our study, we provided a colloquial illustration for a new class of generalized linear mixed models (GLMMs) to fit count and proportion data from SCEDs. We also addressed important aspects in the GLMM framework including overdispersion, estimation methods, statistical inferences, model selection methods by detecting overdispersion, and interpretations of regression coefficients. We then demonstrated the GLMMs with two empirical examples with count and proportion outcomes in SCEDs. In addition, we conducted simulation studies to examine the performance of GLMMs in terms of biases and coverage rates for the immediate treatment effect and treatment effect on the trend. We also examined the empirical Type I error rates of statistical tests. Finally, we provided recommendations about how to make sound statistical decisions to use GLMMs based on the findings from simulation studies. Our hope is that this article will provide SCED researchers with the basic information necessary to conduct appropriate statistical analysis of count and proportion data in their own research and outline the future agenda for methodologists to explore the full potential of GLMMs to analyze or meta-analyze SCED data. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Plot3logit: Ternary Plots for Interpreting Trinomial Regression Models

This paper presents the R package plot3logit which enables the covariate effects of trinomial regression models to be represented graphically by means of a ternary plot. The aim of the plot is helping the interpretation of regression coefficients in terms of the effects that a change in values of regressors has on the probability distribution of the dependent variable. Such changes may involve either a single regressor, or a group of them (composite changes), and the package permits both cases to be handled in a user-friendly way. Moreover, plot3logit can compute and draw confidence regions of the effects of covariate changes and enables multiple changes and profiles to be represented and compared jointly. Upstream and downstream compatibility makes the package able to work with other R packages or applications other than R.

That’s classified! Inventing a new patent taxonomy

AbstractInnovation researchers currently make use of various patent classification schemas, which are hard to replicate. Using machine learning techniques, we construct a transparent, replicable and adaptable patent taxonomy, and a new automated methodology for classifying patents. We contrast our new schema with existing ones using a long-run historical patent dataset. We find quantitative analyses of patent characteristics are sensitive to the choice of classification; our interpretation of regression coefficients is schema dependent. We suggest much of the innovation literature should be carefully interpreted in light of our findings.

FTIR assessment of compositional changes in lignocellulosic wastes during cultivation of Cyclocybe cylindracea mushrooms and use of chemometric models to predict production performance

Solid-state fermentation of various lignocellulosic residues through the use of mushroom fungi leads to the production of edible/medicinal biomass. However, several aspects of this process remain obscure, while the potential exploitation of a wide range of agro-industrial wastes as mushroom substrates is of questionable usefulness. This study attempts to develop a fast and inexpensive method based on Fourier transform infrared (FTIR) spectroscopy for (a) determining compositional changes in lignocellulosic wastes used for the cultivation of Cyclocybe cylindracea mushrooms, (b) identifying compounds in substrates which are related to enhanced productivity and (c) developing a chemometric model through the use of multivariate analysis to predict biological efficiency in mushroom production process. High prediction scores were obtained (R2CAL: 0.95, R2CV: 0.70, RMSECV: 24%), while interpretation of regression coefficients was congruent to results of principal component analysis demonstrating a positive correlation of C. cylindracea biological efficiency values to the initial materials content in lignocellulosic compounds, and a negative correlation to their protein and phenolics content. The combined use of FTIR data and chemometrics provides valuable information on structural modifications of major substrate components during C. cylindracea growth and fructification, and could be exploited for evaluating substrates’ suitability prior to their use for mushroom cultivation.

That's Classified! Inventing a New Patent Taxonomy

We investigate how patent classification influences the interpretation of patent statistics. Innovation researchers currently make use of various patent classification schemas, which are hard to replicate. Using machine learning techniques, we construct a transparent, replicable and adaptable patent taxonomy, and a new automated methodology for classifying patents. We then contrast our new schema with existing ones using a long-run historical patent dataset. We find quantitative analysis of patent characteristics are sensitive to the choice of classification; our interpretation of regression coefficients is schema-dependent. We suggest much of the innovation literature should be carefully interpreted in light of our findings.

Effect Modification and Interaction Terms: It Takes Two to Tango

In this Investigators' Corner I look more deeply into the previously discussed phenomenon of effect modification. I revisit an explanation and examples of the phenomenon and then examine how to account for it statistically. Specifically, I show, in detail, how to write a regression equation that includes interaction terms that account for the effect modification. Finally, I look at interpretation of regression coefficients both with and without the presence of effect modification, and the associated interaction terms.

Research involving limited dependent variables: issues in the literature and recommendations for improvement

Despite previous recommendations for improvement, a literature review reveals a minority of recent papers in management journals provide correct interpretations of regression coefficients for analyses of limited dependent variables. Furthermore, the use of marginal effects to interpret relationships has resulted in confusing and inaccurate conclusions. This paper recommends simpler and more informative alternatives to the calculation and reporting of marginal effects. In particular, two key recommendations involve choosing and explicitly stating a suitable measurement scale for dependent variables and explicitly stating whether relationships with independent variables are multiplicative or additive effects. These recommendations for reporting hypotheses, analysis and interpretations will not only improve the precision of future research but also provide superior interpretations of past literature. Significantly, this paper shows how standard regression coefficients can be used to interpret relationships between variables for any values of all variables. Other approaches such as the recommended inclusion of marginal effects and plots requires fixing other variables to specific values (such as their mean value) and so are of less value to readers.

Representation of exposures in regression analysis and interpretation of regression coefficients: basic concepts and pitfalls

Regression models are being used to quantify the effect of an exposure on an outcome, while adjusting for potential confounders. While the type of regression model to be used is determined by the nature of the outcome variable, e.g. linear regression has to be applied for continuous outcome variables, all regression models can handle any kind of exposure variables. However, some fundamentals of representation of the exposure in a regression model and also some potential pitfalls have to be kept in mind in order to obtain meaningful interpretation of results. The objective of this educational paper was to illustrate these fundamentals and pitfalls, using various multiple regression models applied to data from a hypothetical cohort of 3000 patients with chronic kidney disease. In particular, we illustrate how to represent different types of exposure variables (binary, categorical with two or more categories and continuous), and how to interpret the regression coefficients in linear, logistic and Cox models. We also discuss the linearity assumption in these models, and show how wrongly assuming linearity may produce biased results and how flexible modelling using spline functions may provide better estimates.

Statistical significance of rank regression

Rank regression, which is quite simple to use some form of monotonic relationship between X and Y. Since the rank regression is a nonparametric approach so there are essentially no confidence interval, hypothesis tests, prediction intervals, and interpretation of regression coefficients. In this article, we proposed a bootstrap statistical significance measure of the rank regression by formulating a bootstrap interval for the rank regression parameters. If the rank regression parameters from the original data are not within the bootstrap interval, the rank regression parameters are considered significance. Numerical examples show that the merit of using this proposed bootstrap interval.

Statistical Methods to Study Timing of Vulnerability with Sparsely Sampled Data on Environmental Toxicants

Statistical Methods to Study Timing of Vulnerability with Sparsely Sampled Data on Environmental Toxicants

Generalised Concept of Molar Absorption Coefficient for Non-Uniform Water Absorbance Patterns

Aquaphotomics elucidates the use of the water–light interaction at all frequencies of the electromagnetic spectrum as a molecular mirror. The aquaphotome is the entire complement of water molecules' absorbance bands in aqueous and biological systems, i.e. a comprehensive spectral assignment of all existing water absorbance bands. development of the aquaphotome means constructing a data base of characteristic water absorbance bands and patterns in order to identify respective molecules surrounded by water. as it requires a huge amount of work and contributions from various scientific fields, development of new experimental designs and methods of spectral data analysis, the aquaphotomics column has started inviting contributions from scientists active in various fields. our first guest is Dr Ikehata from the National food research Institute in Japan. for a generalised analysis of absorption spectra of non-ideal mixtures, such as aqueous systems, he and his colleagues have recently proposed the use of partial molar absorption (PMA) and excess partial molar absorption (EPMA) with no wavelength limit. Calculating the generalised absorptivity is going to assist water band assignment and the interpretation of regression coefficients and loading plots in each study.— Roumiana Tsenkova

Scaling regression inputs by dividing by two standard deviations

Interpretation of regression coefficients is sensitive to the scale of the inputs. One method often used to place input variables on a common scale is to divide each numeric variable by its standard deviation. Here we propose dividing each numeric variable by two times its standard deviation, so that the generic comparison is with inputs equal to the mean +/-1 standard deviation. The resulting coefficients are then directly comparable for untransformed binary predictors. We have implemented the procedure as a function in R. We illustrate the method with two simple analyses that are typical of applied modeling: a linear regression of data from the National Election Study and a multilevel logistic regression of data on the prevalence of rodents in New York City apartments. We recommend our rescaling as a default option--an improvement upon the usual approach of including variables in whatever way they are coded in the data file--so that the magnitudes of coefficients can be directly compared as a matter of routine statistical practice.

Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors)

Hierarchical or ‘‘multilevel’’ regression models typically parameterize the mean response conditional on unobserved latent variables or ‘‘random’’ effects and then make simple assumptions regarding their distribution. The interpretation of a regression parameter in such a model is the change in possibly transformed mean response per unit change in a particular predictor having controlled for all conditioning variables including the random effects. An often overlooked limitation of the conditional formulation for nonlinear models is that the interpretation of regression coefficients and their estimates can be highly sensitive to difficult-to-verify assumptions about the distribution of random effects, particularly the dependence of the latent variable distribution on covariates. In this article, we present an alternative parameterization for the multilevel model in which the marginal mean, rather than the conditional mean given random effects, is regressed on covariates. The impact of random effects model violations on the marginal and more traditional conditional parameters is compared through calculation of asymptotic relative biases. A simple two-level example from a study of teratogenicity is presented where the binomial overdispersion depends on the binary treatment assignment and greatly influences likelihood-based estimates of the treatment effect in the conditional model. A second example considers a three-level structure where attitudes toward abortion over time are correlated with person and district level covariates. We observe that regression parameters in conditionally specified models are more sensitive to random effects assumptions than their counterparts in the marginal formulation.

A note on the robust interpretation of regression coefficients

It is known that ratios of regression coefficients have an interpretation more stable under model perturbation than regression coefficients themselves. This is explored in more detail for a special case of linear and log linear regression.

Linear Statistical Models

Preface. 1 Linear Algebra, Projections. 1.1 Introduction. 1.2 Vectors, Inner Products, Lengths. 1.3 Subspaces, Projections. 1.4 Examples. 1.5 Some History. 1.6 Projection Operators. 1.7 Eigenvalues and Eigenvectors. 2 Random Vectors. 2.1 Covariance Matrices. 2.2 Expected Values of Quadratic Forms. 2.3 Projections of Random Variables. 2.4 The Multivariate Normal Distribution. 2.5 The chi2, F, and t Distributions. 3 The Linear Model. 3.1 The Linear Hypothesis. 3.2 Confidence Intervals and Tests on eta = c 1 ss 1 + ... + c k ss k . 3.3 The Gauss-Markov Theorem. 3.4 The Gauss-Markov Theorem For The General Case. 3.5 Interpretation of Regression Coefficients. 3.6 The Multiple Correlation Coefficient. 3.7 The Partial Correlation Coefficient. 3.8 Testing H 0 : theta epsilon V 0 V. 3.9 Further Decomposition of Subspaces. 3.10 Power of the F-Test. 3.11 Confidence and Prediction Intervals. 3.12 An Example from SAS. 3.13 Another Example: Salary Data. 4 Fitting of Regression Models. 4.1 Linearizing Transformations. 4.2 Specification Error. 4.3 Generalized Least Squares. 4.4 Effects of Additional or Fewer Observations. 4.5 Finding the Best Set of Regressors. 4.6 Examination of Residuals. 4.7 Collinearity. 4.8 Asymptotic Normality. 4.9 Spline Functions. 4.10 Nonlinear Least Squares. 4.11 Robust Regression. 4.12 Bootstrapping in Regression. 4.13 Quantile Regression. 5 Simultaneous Confidence Intervals. 5.1 Bonferroni Confidence Intervals. 5.2 Scheffe Simultaneous Confidence Intervals. 5.3 Tukey Simultaneous Confidence Intervals. 5.4 Comparison of Lengths. 5.5 Bechhofer's Method. 6 Two-and Three-Way Analyses of Variance. 6.1 Two-Way Analysis of Variance. 6.2 Unequal Numbers of Observations Per Cell. 6.3 Two-Way Analysis of Variance, One Observation Per Cell. 6.4 Design of Experiments. 6.5 Three-Way Analysis of Variance. 6.6 The Analysis of Covariance. 7 Miscellaneous Other Models. 7.1 The Random Effects Model. 7.2 Nesting. 7.3 Split Plot Designs. 7.4 Mixed Models. 7.5 Balanced Incomplete Block Designs. 8 Analysis of Frequency Data. 8.1 Examples. 8.2 Distribution Theory. 8.3 Conf. Ints. on Poisson and Binomial Parameters. 8.4 Log-Linear Models. 8.5 Estimation for the Log-Linear Model. 8.6 Chi-Square Goodness-of-Fit Statistics. 8.7 Limiting Distributions of the Estimators. 8.8 Logistic Regression. The Statistical Language R. Answers. Index.

Estimating effects of probands' characteristics on familial risk: I. Adjustment for censoring and correlated ages at onset

Family studies with age at onset of the disease as the endpoint face two important problems: censoring and correlation of age at onset among relatives. We present a multivariate survival model for ages at onset of relatives which incorporates the problems cited above. The interpretations of regression coefficients and association parameter in the context of family studies are emphasized. The present paper describes a statistical method for estimating these parameters. In a companion paper [Pulver and Liang, Genet Epidemiol 8:339-350, 1991] this model is applied to a genetic epidemiologic study of schizophrenia.

A Note on the Interpretation of Regression Coefficients within a Class of Truncated Distributions

DESPITE THE INTENSE ACTIVITY in the area of estimation of limited dependent variable (LDV) models (see, for example, the Fall 1976 issue of the Annals of Economic and Social Measurement), the interpretation of regression coefficients in truncated regression models has been largely ignored. This issue should not be taken lightly since the obvious interpretation which equates regression coefficients to partial derivatives of the conditional mean of the dependent variable is unfortunately incorrect. One specific implication of the results presented here is that whenever the dependent variable y in the usual classical linear normal regression model is truncated above and/or below, then the effect of the fth regressor on the conditional mean of y is proportional to, but not equal to, the fth regression coefficient. However, more generally this note presents a simple expression for the effect of the fth regressor on any conditional moment of y in terms of a generalized LDV model introduced by Poirier [3]. Poirier introduced a general LDV model which permitted skewness in a pre-truncated variable by transforming it within the class of transformations suggested by Box and Cox

Coding for Correlation and Regression

Data coding for analyzing the relationships between variables can often be a critical step relative to tests of significance or interpretation of regression coefficients. Resultant correlations can sometimes range from "trivial" to "large" depending on the data and the methods of coding. This paper focuses on the procedures available for substituting a special predictive coding method for some of the more complex general regression procedures.