Regression without regrets -initial data analysis is a prerequisite for multivariable regression.

Statistical regression models are used for predicting outcomes based on the values of some predictor variables or for describing the association of an outcome with predictors. With a data set at hand, a regression model can be easily fit with standard software packages. This bears the risk that data analysts may rush to perform sophisticated analyses without sufficient knowledge of basic properties, associations in and errors of their data, leading to wrong interpretation and presentation of the modeling results that lacks clarity. Ignorance about special features of the data such as redundancies or particular distributions may even invalidate the chosen analysis strategy. Initial data analysis (IDA) is prerequisite to regression analyses as it provides knowledge about the data needed to confirm the appropriateness of or to refine a chosen model building strategy, to interpret the modeling results correctly, and to guide the presentation of modeling results. In order to facilitate reproducibility, IDA needs to be preplanned, an IDA plan should be included in the general statistical analysis plan of a research project, and results should be well documented. Biased statistical inference of the final regression model can be minimized if IDA abstains from evaluating associations of outcome and predictors, a key principle of IDA. We give advice on which aspects to consider in an IDA plan for data screening in the context of regression modeling to supplement the statistical analysis plan. We illustrate this IDA plan for data screening in an example of a typical diagnostic modeling project and give recommendations for data visualizations.

The Impact of Functional Form Complexity on Model Overfitting for Nonlinear Mixed-Effects Models.

Nonlinear mixed-effects models (NLMEMs) allow researchers to model curvilinear patterns of growth, but there is ambiguity as to what functional form the data follow. Often, researchers fit multiple nonlinear functions to data and use model selection criteria to decide which functional form fits the data "best." Frequently used model selection criteria only account for the number of parameters in a model but overlook the complexity of intrinsically nonlinear functional forms. This can lead to overfitting and hinder the generalizability and reproducibility of results. The primary goal of this study was to evaluate the performance of eight model selection criteria via a Monte Carlo simulation study and assess under what conditions these criteria are sensitive to model overfitting as it relates to functional form complexity. Results highlighted criteria with the potential to capture overfitting for intrinsically nonlinear functional forms for NLMEMs. Information criteria and the stochastic information complexity criterion recovered the true model more often than the average or conditional concordance correlation. Results also suggest that the amount of residual variance and sample size have an impact on model selection for NLMEMs. Implications for future research and recommendations for application are also provided.

Reproductive biology of the bullseye puffer Sphoeroides annulatus: Gonadosomatic index and its suitability for estimating length at maturity.

This study aimed to describe updated information on the basic reproductive biology of Sphoeroides annulatus females and potential use of the gonadosomatic index I G as input data to assess length at 50% maturity ( L 50 ) and identify the breeding season. From April 2014 to December 2015, 767 females were analysed along the north-west coast of the Mexican Pacific Ocean. Both immature and mature females were represented in the total length ( L T ), which ranged from 81 to 440 mm, and the smallest mature female was 92 mm (new record for the species). Gonadal maturity phases were assigned through histological analysis. A I G threshold value I cut was established using the lower whisker value for maturity phase spawning capable (2.06%). The monthly variation of I G supported the annual spawning seasonality, and the presence of reproductively active mature females I G ≥ I cut coincided with the observed periods with histological data and the peaks for both seasons 2014 and 2015. Also the presence of reproductively inactive I G < I cut mature females confirmed the nonreproductive season observed from January to April 2015. Estimates of L 50 were based on the cumulative proportion of reproductively active mature females: both datasets showed a 'bumpy trajectory'. Considering the functional form of data and models, a second objective was to verify if each dataset presented a simple or double sigmoid trajectory and select the most suitable model through the Akaike information criterion. For both datasets, the best fitted model was the double sigmoidal M DS , while the single sigmoidal M SS had low performance and did not have statistical support. The M DS provided a L 50 value ca. 254 mm L T for both data sets. A trade-off between the model's correct fit and a reliable L 50 value was found, but this fact is not always considered when calculating the maturity, resulting in biased parameters. Correct data fitting may seem trivial, but the consequences for the conservation of immature individuals could be dramatic, affecting the population recruitment. In conclusion, the histologically calibrated I cut was suitable for classifying females of S. annulatus into reproductively inactive or active types, with an agreement percentage of 99%. The I cut  = 2.06% can be useful for subsequent studies for this species.

Is there a dose-response relationship? Investigating the functional form between COVID-19 incidence rates and life satisfaction in a multilevel framework.

While there is plenty of research linking the effects of the global COVID-19 pandemic to a drastic reduction of life satisfaction in the population, there is little information on the functional form of this relationship. Until now, one could suspect that this association is linear and a higher number of COVID-19 infections in a region leads to a continuous decline of satisfaction. However, there are reasons to assume that this interrelation is indeed more complex and deserves further attention. To resolve this question, high-quality panel data of the first wave of COVID-19 from Germany are analysed in a fixed-effect multilevel framework. With information from more than 6,000 respondents (after imputation) nested in 339 federal districts, we estimate linear models with higher-order terms up to the fifth degree of median COVID-19 incidence rates and random intercepts for districts to describe the functional form. The results indicate that even regions with very low incidences are affected and a linear decline of satisfaction is only apparent for rather low incidence levels, quickly reaching a plateau, which is then quite constant, even for higher incidence levels. These findings indicate that at least in rich and industrialized countries like Germany, assuming a strictly linear relation between incidences and change of satisfaction is not appropriate.

Robust estimation in the nested case-control design under a misspecified covariate functional form.

The Cox proportional hazards model is typically used to analyze time-to-event data. If the event of interest is rare and covariates are difficult or expensive to collect, the nested case-control (NCC) design provides consistent estimates at reduced costs with minimal impact on precision if the model is specified correctly. If our scientific goal is to conduct inference regarding an association of interest, it is essential that we specify the model a priori to avoid multiple testing bias. We cannot, however, be certain that all assumptions will be satisfied so it is important to consider robustness of the NCC design under model misspecification. In this manuscript, we show that in finite sample settings where the functional form of a covariate of interest is misspecified, the estimates resulting from the partial likelihood estimator under the NCC design depend on the number of controls sampled at each event time. To account for this dependency, we propose an estimator that recovers the results obtained using using the full cohort, where full covariate information is available for all study participants. We present the utility of our estimator using simulation studies and show the theoretical properties. We end by applying our estimator to motivating data from the Alzheimer's Disease Neuroimaging Initiative.

A review of spline function procedures in R.

BACKGROUND: With progress on both the theoretical and the computational fronts the use of spline modelling has become an established tool in statistical regression analysis. An important issue in spline modelling is the availability of user friendly, well documented software packages. Following the idea of the STRengthening Analytical Thinking for Observational Studies initiative to provide users with guidance documents on the application of statistical methods in observational research, the aim of this article is to provide an overview of the most widely used spline-based techniques and their implementation in R. METHODS: In this work, we focus on the R Language for Statistical Computing which has become a hugely popular statistics software. We identified a set of packages that include functions for spline modelling within a regression framework. Using simulated and real data we provide an introduction to spline modelling and an overview of the most popular spline functions. RESULTS: We present a series of simple scenarios of univariate data, where different basis functions are used to identify the correct functional form of an independent variable. Even in simple data, using routines from different packages would lead to different results. CONCLUSIONS: This work illustrate challenges that an analyst faces when working with data. Most differences can be attributed to the choice of hyper-parameters rather than the basis used. In fact an experienced user will know how to obtain a reasonable outcome, regardless of the type of spline used. However, many analysts do not have sufficient knowledge to use these powerful tools adequately and will need more guidance.

Religiosity and Premarital Sexual Behaviors among Adolescents: An Analysis of Functional Form.

While numerous studies show a persistent inverse association between religion variables and adolescents' sexual behaviors, the nature of this relationship is not well understood. Specifically, many previous studies presuppose that the associations between adolescent religiosity and sexual behaviors are linear. However, a number of studies have also identified important nonlinearities of religious influence during adolescence, with highly religious individuals being distinct from their peers. Incorporating this knowledge into a theoretically-motivated modeling approach, this article provides a comparative analysis of functional forms describing the relationships between religiosity and adolescent sexual behaviors. Using data from two waves of the National Study of Youth and Religion, a linear functional form is compared with nonlinear alternatives that link multiple religion measures to outcomes of sexual activity. Results show that the majority of these relationships are best defined by nonlinear functional forms, suggesting that the influence of religiosity increases as individuals become more religious.

Performance of two formal tests based on martingales residuals to check the proportional hazard assumption and the functional form of the prognostic factors in flexible parametric excess hazard models.

Net survival, the one that would be observed if the disease under study was the only cause of death, is an important, useful, and increasingly used indicator in public health, especially in population-based studies. Estimates of net survival and effects of prognostic factor can be obtained by excess hazard regression modeling. Whereas various diagnostic tools were developed for overall survival analysis, few methods are available to check the assumptions of excess hazard models. We propose here two formal tests to check the proportional hazard assumption and the validity of the functional form of the covariate effects in the context of flexible parametric excess hazard modeling. These tests were adapted from martingale residual-based tests for parametric modeling of overall survival to allow adding to the model a necessary element for net survival analysis: the population mortality hazard. We studied the size and the power of these tests through an extensive simulation study based on complex but realistic data. The new tests showed sizes close to the nominal values and satisfactory powers. The power of the proportionality test was similar or greater than that of other tests already available in the field of net survival. We illustrate the use of these tests with real data from French cancer registries.

On modeling.

Mapping tissue microstructure with MRI holds great promise as a noninvasive window into tissue organization at the cellular level. Having originated within the realm of diffusion NMR in the late 1970s, this field is experiencing an exponential growth in the number of publications. At the same time, model-based approaches are also increasingly incorporated into advanced MRI acquisition and reconstruction techniques. However, after about two decades of intellectual and financial investment, microstructural mapping has yet to find a single commonly accepted clinical application. Here, we suggest that slow progress in clinical translation may signify unresolved fundamental problems. We outline such problems and related practical pitfalls, as well as review strategies for developing and validating tissue microstructure models, to provoke a discussion on how to bridge the gap between our scientific aspirations and the clinical reality. We argue for recalibrating the efforts of our community toward a more systematic focus on fundamental research aimed at identifying relevant degrees of freedom affecting the measured MR signal. Such a focus is essential for realizing the truly revolutionary potential of noninvasive three-dimensional in vivo microstructural mapping.

Very Long-Term Complete Remission Can Be Achieved in Men With High-Risk Localized Prostate Cancer and a Very High PSA Value: An Analysis of the GETUG 12 Phase 3 Trial.

INTRODUCTION: Serum prostate specific antigen (PSA) is a well-known prognostic parameter in men with prostate cancer. The treatment of men with very high PSA values and apparently no detectable metastases is not fully established. PATIENTS AND METHODS: Ancillary analysis from the GETUG 12 phase 3 trial. Patients with non-metastatic high-risk prostate cancer by bone and computerized tomography (CT) scan were randomly assigned to receive androgen deprivation therapy (ADT) and docetaxel plus estramustine or ADT alone. Relapse-free survival (RFS), clinical RFS, metastases-free survival (MFS), overall survival (OS), and prostate cancer-specific survival (PCSS) were estimated using the Kaplan-Meier method for different levels of PSA (50 ng/mL, 75 ng/mL, and 100 ng/mL). The relationship between PSA and outcomes was studied using residual-based approaches and spline functions. RESULTS: The median follow-up was 12 years (range: 0-15.3). Baseline PSA (<50 ng/mL, n = 328; ≥50ng/mL, n = 85) was associated with improved RFS (P = .0005), cRFS (P = .0024), and MFS (P = .0068). The 12-year RFS rate was 46.33% (CI 40.59-51.86), 33.59% (CI 22.55-44.97), and 11.76% (1.96-31.20) in men with PSA values <50 ng/mL (n = 328), 50-100 ng/mL (n = 68), and ≥100 ng/mL (n = 17), respectively. Exploratory analyses revealed no deviation from the linear relationship assumption between PSA and the log hazard of events. CONCLUSIONS: Men with apparently localized prostate cancer and a high baseline PSA value have a reasonable chance of being long-term disease-free when treated with curative intent combining systemic and local therapy.

Biocrusts mediate the plant community composition of dryland restoration ecosystems.

Degradation and expansion are current threats associated with drylands. During natural or artificial restoration, dryland ecosystems tend to contain a unique community, namely, biocrusts. Studies have shown that biocrusts serve multiple functions and have the potential to accelerate or inhibit degradation. However, the mechanisms by which biocrusts mediate dryland plant community structure and composition remain largely unknown. In this study, we performed a series of pot experiments under greenhouse conditions at three disturbance levels (strong, moderate and no disturbance) and at three rainfall gradients (- 50 %, normal, + 50 %) for different biocrust types. Our objective was to examine how biocrusts influence the introduction and establishment of different functional forms of plant species during the different succession stages of dryland recovery. Our results showed that biocrusts have significantly different effects on the seed germination and survival of the tested plant species. Although strong disturbances to the biocrusts and increasing rainfall alleviated the inhibitory effects of biocrusts on the perennial herb (S. glareosa) and subshrub (C. latens) species, these factors could only promote the temporary survival of these two plants in the revegetated area. These results indicate that biocrusts in revegetated systems play important sieving functions during plant species establishment. Additionally, the survival of S. viridis, a plant species with small seeds adapted to the current habitat, was promoted in the community, but two plant species that are currently unsuitable, S. glareosa and C. latens, were filtered out by the biocrusts. Our conclusions suggest that biocrusts play an ecological filtering role in plant species introduction and establishment in revegetated ecosystems and demonstrate the scientific significance of biocrusts in maintaining the health and stability of dryland ecosystems.

Assessment of the Functional Form of the Relationship between Balance Control and Physical Activity Regarding Demographic, Anthropometrical, and Eye Impairment Explanatory Covariates in 9- to 11-Year-Old Children: Results of Polynomial and Cluster Analyses.

Explaining the causal and functional relationship between balance control and physical activity (PA) when comparing demographic, anthropometric, and eyesight impairment is uncharted. This study aimed to assess the shape of the relationships between balance control and PA and to verify the usefulness of explanatory variables (sex, chronological age, biological age, myopia, and BMI) in explaining the formation of functional forms between both abovementioned variables. The current contribution evaluated data from 9-11-year-old children (47 boys, 58 girls) and attempts to explain the shape of the relationship between the overall stability index and PA synthetic index, sorting children into clusters depending on their balance features and PA and comparing the separated groups in terms of explanatory variables. The analysis demonstrates four larger clusters that displayed distinct functional relationships. Only BMI and myopia turned out to be useful in explaining cluster memberships. Children in clusters with a linear-constant decline functional form were mostly non-myopic girls, thin, and less mature. Meanwhile, it becomes clear that children with an L-shape were myopic, early-maturated boys with a BMI in the middle of the range for normal weight. The pattern of an inverted U-shaped functional form was distinctive for myopes with rather high BMIs compared to their peers but normally matured.

Functional redundancy and stability in a subtidal macroalgal community in the Southwestern Atlantic coast.

Functional redundancy can stabilize ecological functions as asynchronous fluctuations among functionally similar species may buffer environmental changes. We investigated the temporal dynamics of a subtidal macroalgal community in the warm temperate Southwestern Atlantic coast (SWA) to evaluate whether functional redundancy stabilize ecosystems functions through compensatory dynamics under realistic environmental scenarios. Despite temporal variations in the community structure occurred, a high stability in macroalgal coverage was found at the community-level driven by taxa asynchronous fluctuations. No relationship between functional redundancy and stability occurred, suggesting that functional compensation cannot surpass the influence of environmental fluctuations on the performance of ecological functions. Declines in Sargassum species abundance, along with its low functional redundancy, indicate that this canopy-forming algae must be prioritized in conservation efforts in the SWA. Our study adds to the comprehension and generalization of biodiversity-stability findings in natural systems across distinct geographical areas, also contributing to their operationalization in marine ecosystems.

Impact of globalization on the environment in major CO2-emitting countries: Evidence using bootstrap ARDL with a Fourier function.

Alongside sustainable development as a major global aim, the contribution made by globalization to environmental issues has become crucial in recent decades. Prior studies have focused on how trade in globalization influences the environment. However, multiple economic, social, and political factors are also important, the integration of which needs to be considered in sustainable development. Sharp and smooth breaks in time series models are the consequence of real-world structures. Using the bootstrap autoregressive-distributed lag test with a Fourier function, the present study reexamined the nexus between globalization and the environment in China, the United States, and India. The empirical results indicate that in the United States, the nexus between globalization and the environment is cointegrated in the long-term. In the short term, globalization is improving the environment in the United States and India. However, in China, globalization is resulting in environmental degradation. This research will assist policymakers in developing comprehensive strategies for sustainable development.

Improving functional form in cross-sectional regression studies to capture the non-linear safety effects of roadway attributes-Freeway median width case study.

Crash modification factors (CMFs) for several roadway attributes are based on cross-sectional regression models, in the main because of the lack of data for the preferred observational before-after study. In developing these models, little attention has been paid to those functional forms that reflect the reality that CMFs should not be single-valued, as most available ones are, but should vary with application circumstance. Using a full Bayesian Markov Chain Monte Carlo (MCMC) approach, this study aimed to improve the functional forms used to derive CMFs in cross-sectional regression models, with a focus on capturing the variability inherent in crash modification functions (CMFunctions). The estimated CMFunction for target crashes for freeway median width, used for a case study, indicates that the approach is capable of developing a function that can capture the logical reality that the CMF for a given change in a feature's value depends not only on the amount of the change but also on the original value. The results highlight the importance of using the functional forms that can capture non-linear effects of road attributes for CMF estimation in cross-sectional models. The case study provides credible CMFs for assessing the safety implications of decisions on freeway median width that could be used in improving current design practice.

Privileged (Default) Causal Cognition: A Mathematical Analysis.

Causal cognition is a key part of human learning, reasoning, and decision-making. In particular, people are capable of learning causal relations from data, and then reasoning and planning using those cognitive representations. While there has been significant normative work on the causal structures that ought to be learned from evidence, there has been relatively little on the functional forms that should (normatively) be used or learned for those qualitative causal relations. Moreover, empirical research on causal inference-learning causal relations from observations and interventions-has found support for multiple different functional forms for causal connections. This paper argues that a combination of conceptual and mathematical constraints leads to a privileged (default) functional form for causal relations. This privileged function is shown to provide a theoretical unification of the widely-used noisy-OR/AND models and linear models, thereby showing how they are complementary rather than competing. This unification thus helps to explain the diverse empirical results, as these different functional forms are "merely" special cases of the more general, more privileged function.

Examining the influence of link function misspecification in conventional regression models for developing crash modification factors.

This study further examined the use of regression models for developing crash modification factors (CMFs), specifically focusing on the misspecification in the link function. The primary objectives were to validate the accuracy of CMFs derived from the commonly used regression models (i.e., generalized linear models or GLMs with additive linear link functions) when some of the variables have nonlinear relationships and quantify the amount of bias as a function of the nonlinearity. Using the concept of artificial realistic data, various linear and nonlinear crash modification functions (CM-Functions) were assumed for three variables. Crash counts were randomly generated based on these CM-Functions. CMFs were then derived from regression models for three different scenarios. The results were compared with the assumed true values. The main findings are summarized as follows: (1) when some variables have nonlinear relationships with crash risk, the CMFs for these variables derived from the commonly used GLMs are all biased, especially around areas away from the baseline conditions (e.g., boundary areas); (2) with the increase in nonlinearity (i.e., nonlinear relationship becomes stronger), the bias becomes more significant; (3) the quality of CMFs for other variables having linear relationships can be influenced when mixed with those having nonlinear relationships, but the accuracy may still be acceptable; and (4) the misuse of the link function for one or more variables can also lead to biased estimates for other parameters. This study raised the importance of the link function when using regression models for developing CMFs.

Modeling the effects of AADT on predicting multiple-vehicle crashes at urban and suburban signalized intersections.

Annual Average Daily Traffic (AADT) is often considered as a main covariate for predicting crash frequencies at urban and suburban intersections. A linear functional form is typically assumed for the Safety Performance Function (SPF) to describe the relationship between the natural logarithm of expected crash frequency and covariates derived from AADTs. Such a linearity assumption has been questioned by many researchers. This study applies Generalized Additive Models (GAMs) and Piecewise Linear Negative Binomial (PLNB) regression models to fit intersection crash data. Various covariates derived from minor-and major-approach AADTs are considered. Three different dependent variables are modeled, which are total multiple-vehicle crashes, rear-end crashes, and angle crashes. The modeling results suggest that a nonlinear functional form may be more appropriate. Also, the results show that it is important to take into consideration the joint safety effects of multiple covariates. Additionally, it is found that the ratio of minor to major-approach AADT has a varying impact on intersection safety and deserves further investigations.

Trace element seasonality in marine macroalgae of different functional-form groups.

Novel information on the seasonality of element accumulation in seaweeds is provided. Seasonal patterns of As, Ba, Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb, Se, Sr, U, V and Zn concentrations in macroalgae belonging to different functional-form groups (Ulva intestinalis, Ulva rigida, Codium fragile, Gracilaria gracilis) from the Thessaloniki Gulf, Aegean Sea were determined and compared. Uni- and multivariate data analyses were applied. Element concentrations generally decreased during spring and/or summer, probably due to the growth effect, but a reverse trend, particularly in Ulva species, was also observed. Most elements (Cd, Co, Cr, Cu, Mo, Ni, Pb, Sr) in Ulva species displayed a comparatively low monthly variability, indicating that the extent of seasonal variation is closely related to thallus morphology and growth strategy. In particular, these data suggest that Cd, Co, Cr, Cu, Mo, Ni, Pb and Sr contents in fast-growing, sheet-like macroalgae are less influenced by the season, compared to their contents in coarsely-branched and thick-leathery macroalgae; therefore, sheet-like macroalgae may be more appropriate to be used in biomonitoring of coastal waters. The data presented could be utilized in the development of biomonitoring programmes for the protection of coastal environments.

Trends and group differences in the association between educational attainment and U.S. adult mortality: implications for understanding education's causal influence.

Has the shape of the association between educational attainment and U.S. adult mortality changed in recent decades? If so, is it changing consistently across demographic groups? What can changes in the shape of the association tell us about the possible mechanisms in play for improving health and lowering mortality risk over the adult life course? This paper develops the argument that societal technological change may have had profound effects on the importance of educational attainment - particularly advanced education - in the U.S. adult population for garnering health advantages and that these changes should be reflected in changes in the functional form of the association between educational attainment and mortality. We review the historical evidence on the changing functional form of the association, drawing on studies based in the United States, to assess whether these changes are consistent with our argument about the role of technological change. We also provide an updated analysis of these functional form patterns and trends, contrasting data from the early 21st Century with data from the late 20th Century. This updated evidence suggests that the shape of the association between educational attainment and U.S. adult mortality appears to be reflecting lower and lower adult mortality for very highly educated Americans compared to their low-educated counterparts in the 21st Century. We draw on this review and updated evidence to reflect on the question whether education's association with adult mortality has become increasingly causal in recent decades, why, and the potential research, policy, and global implications of these changes.