Estimation Of Interaction Effects Research Articles

Best Practices for Estimating, Interpreting, and Presenting Nonlinear Interaction Effects

Many effects of interest to sociologists are nonlinear. Additionally, many effects of interest are interaction effects—that is, the effect of one independent variable is contingent on the level of another independent variable. The proper way to estimate, interpret, and present these two types of effects individually are well known. However, many analyses that combine these two—that is, tests of interaction when the effects of interest are nonlinear—are not properly interpreted or tested. The consequences of approaching nonlinear interaction effects the way one would approach a linear interaction effect are severe and can often result in incorrect conclusions. I cover both nonlinear effects in the context of linear regression, and—most thoroughly—nonlinear effects in models for categorical outcomes (focusing on binary logit/probit). My goal in this article is to synthesize an evolving methodological literature and to provide straightforward advice and techniques to estimate,interpret, and present nonlinear interaction effects.

E-tailer website attributes and trust: understanding the role of online reviews

PurposeThe purpose of this paper is to report the results of an experimental study designed to better understand the role of online product reviews, both valence and volume, in the formation of initial online trust during a consumer’s exploratory stage with an e-tailer. This is done within the context of simultaneously varying e-tailer reputation and product type.Design/methodology/approachParticipants take part in a conjoint task that involves viewing fictitious web pages and indicating their level of trust in using the site to purchase the product displayed. The web pages are developed by manipulating four attributes (e-tailer reputation, product type, summary product star rating and number of online reviews) according to a full factorial design. Conjoint models are estimated to determine the relative influence of each attribute on trust perceptions, the significance of selected two-way interactions among the four attributes, and potential moderating effects of varied prior online experiences, including previous usage frequency of online reviews.FindingsResults reveal that e-tailer reputation has the greatest impact on initial trust perceptions, followed by the summary review star rating of the product. Significant two-way interactions show that a large number of reviews enhance the effect of a positive summary review on trust while shopping for high priced experience products diminishes the positive influence of e-tailer reputation. Prior online experiences moderate the relationship between these website attributes and perceived trust by interacting with the two strongest trust cues in the model. The effects of these attributes on trust perceptions are less for those with higher levels of prior online experiences.Originality/valueThe study uses conjoint analysis, which requires participants to implicitly “tradeoff” among website attributes in making overall judgments about e-tailer trustworthiness. Consequently, the relative influence of online reviews (both valence and volume) on initial trust perceptions is derived empirically in a realistic setting that involves online shopping contexts with different risk (by varying product type). Moreover, the authors are able to estimate interaction effects. A significant interaction between summary product star rating and number of reviews implies that online review volume may be more important to perceived e-tailer trustworthiness than earlier studies suggest.

Pharmacokinetic interaction between bedaquiline and clofazimine in patients with drug-resistant tuberculosis.

Bedaquiline (BDQ) and clofazimine (CFZ) are both recommended for treating drug-resistant tuberculosis (DR-TB). As CFZ is an inhibitor of the cytochrome P450 isoenzyme 3A4 (CYP3A4) in vitro, and BDQ a substrate of CYP3A4, there is a potential for pharmacokinetic (PK) drug-drug interaction that may result in increased BDQ exposure when co-administered with CFZ, which could increase the toxicity of BDQ. We assessed the effect of co-administered CFZ on BDQ bioavailability, or on clearance of BDQ and its N-monodesmethyl metabolite (M2), in patients with DR-TB using a population PK model developed from data of patients with DR-TB. This was a secondary analysis of a study designed to explore drug-drug interactions between BDQ and antiretrovirals. Of 46 participants, 30 were on concomitant CFZ when intensive PK sampling of BDQ was done. CFZ did not have a statistically significant effect on BDQ bioavailability (-9.1%, 90%CI -22.8 to +7.1; P = 0.19) or on BDQ and M2 clearance (+12.2%, 90%CI -13.7 to +38; P = 0.32). We did not find a statistically significant PK drug-drug interaction between BDQ and CFZ, but cannot exclude a potentially clinically relevant interaction due to the wide confidence intervals of the estimated interaction effects.

Home- and community-based growth monitoring to reduce early life growth faltering: an open-label, cluster-randomized controlled trial

Background: Despite the continued high prevalence of faltering growth, height monitoring remains limited in many low- and middle-income countries.Objective: The objective of this study was to test whether providing parents with information on their child’s height can improve children’s height and developmental outcomes.Design: Villages in Chipata District, Zambia (n = 127), were randomly assigned with equal probability to 1 of 3 groups: home-based growth monitoring (HBGM), community-based growth monitoring including nutritional supplementation for children with stunted growth (CBGM+NS), and control. Primary study outcomes were individual height-for-age z score (HAZ) and overall child development assessed with the International Fetal and Newborn Growth Consortium for the 21st Century Neurodevelopment Assessment tool. Secondary outcomes were weight-for-age z score (WAZ), protein consumption, breastfeeding, and general dietary diversity.Results: We enrolled a total of 547 children with a median age of 13 mo at baseline. Estimated mean difference (β) in HAZ was 0.127 (95% CI: −0.107, 0.361) for HBGM and −0.152 (95% CI: −0.341, 0.036) for CBGM+NS. HBGM had no impact on child development [β: −0.017 (95% CI: −0.133, 0.098)]; CBGM+NS reduced overall child development scores by −0.118 SD (95% CI: −0.230, −0.006 SD). Both interventions had larger positive effects among children with stunted growth at baseline, with estimated interaction effects of 0.503 (95% CI: 0.160, 0.846) and 0.582 (95% CI: 0.134, 1.030) for CBGM+NS and HBGM, respectively. HBGM increased mean WAZ [β = 0.183 (95% CI: 0.037, 0.328)]. Both interventions improved parental reports of children’s protein intake.Conclusions: The results from this trial suggest that growth monitoring has a limited effect on children’s height and development, despite improvements in self-reported feeding practices. HBGM had modest positive effects on children with stunted growth. Given its relatively low cost, this intervention may be a cost-effective tool for increasing parental efforts toward reducing children’s physical growth deficits. This trial was registered at clinicaltrials.gov as NCT02242539.

Blood mercury concentrations are associated with decline in liver function in an elderly population: a panel study

BackgroundMercury is a toxic heavy metal and is known to affect many diseases. However, few studies have examined the effects of mercury exposure on liver function in the general population. We examined the association between blood mercury concentrations and liver enzyme levels in the elderly.MethodsWe included 560 elderly participants (60 years or older) who were recruited from 2008 to 2010 and followed up to 2014. Subjects visited a community welfare center and underwent a medical examination and measurement of mercury levels up to five times. Analyses using generalized estimating equations model were performed after adjusting for age, sex, education, overweight, alcohol consumption, smoking, regular exercise, high-density lipoproteins cholesterol, and total calorie intake. Additionally, we estimated interaction effects of alcohol consumption with mercury and mediation effect of oxidative stress in the relationship between mercury levels and liver function.ResultsThe geometric mean (95% confidence interval (CI)) of blood mercury concentrations was 2.81 μg/L (2.73, 2.89). Significant relationships were observed between blood mercury concentrations and the level of liver enzymes, including aspartate aminotransferase (AST), alanine aminotransferase (ALT), and gamma glutamyl transferase (GGT), after adjusting for potential confounders (P < 0.05). The odds ratios of having abnormal ALT levels were statistically significant in the highest mercury quartile compared to those with the lowest quartile. Particularly, regular alcohol drinkers showed greater effect estimates of mercury on the liver function than non-drinkers groups. There was no mediation effect of oxidative stress in the relationship between blood mercury concentrations and liver function.ConclusionsOur results suggest that blood mercury levels are associated with elevated liver enzymes and interact with alcohol consumption for the association in the elderly.

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

ABSTRACTResearchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.

Comparison of 2 models for gene-environment interactions: an example of simulated gene-medication interactions on systolic blood pressure in family-based data.

BackgroundNearly half of adults in the United States who are diagnosed with hypertension use blood-pressure-lowering medications. Yet there is a large interindividual variability in the response to these medications. Two complementary gene–environment interaction methods have been published and incorporated into publicly available software packages to examine interaction effects, including whether genetic variants modify the association between medication use and blood pressure. The first approach uses a gene–environment interaction term to measure the change in outcome when both the genetic marker and medication are present (the “interaction model”). The second approach tests for effect-size differences between strata of an environmental exposure (the “med-diff” approach). However, no studies have quantitatively compared how these methods perform with respect to 1 or 2 degree of freedom (DF) tests or in family-based data sets. We evaluated these 2 approaches using simulated genotype–medication response interactions at 3 single nucleotide polymorphisms (SNPs) across a range of minor allele frequencies (MAFs 0.1–5.4 %) using the Genetic Analysis Workshop 19 family sample.ResultsThe estimated interaction effect sizes were on average larger in the interaction model approach compared to the med-diff approach. The true positive proportion was higher for the med-diff approach for SNPs less than 1 % MAF, but higher for the interaction model when common variants were evaluated (MAF >5 %). The interaction model produced lower false-positive proportions than expected (5 %) across a range of MAFs for both the 1DF and 2DF tests. In contrast, the med-diff approach produced higher but stable false-positive proportions around 5 % across MAFs for both tests.ConclusionsAlthough the 1DF tests both performed similarly for common variants, the interaction model estimated true interaction effects with less bias and higher true positive proportions than the med-diff approach. However, if rare variation (MAF <5 %) is of interest, our findings suggest that when convergence is achieved, the med-diff approach may estimate true interaction effects more conservatively and with less variability.

Estimation of interaction effects using pooled biospecimens in a case-control study.

Pooling, or physically mixing biospecimens, prior to evaluating biomarkers dramatically reduces biomarker evaluation cost, reduces the quantity of biospecimens required of each individual, and may reduce the percentage of laboratory measurements below the lower limit of detection. Motivated by a case-control study on miscarriage (binary outcome) and cytokines (continuous exposures), we are interested in estimating parameters in a logistic regression, where individuals with the same disease status (with or without a miscarriage) are paired and their pooled cytokine concentrations are assessed. Previous research has proposed a set-based logistic model to evaluate the relationship between a disease and pooled exposures. While the set-based logistic model is very useful for estimating main effects, it cannot estimate interactions of continuous exposures when both are measured in pools. Therefore, we propose using the expectation maximization (EM) algorithm to obtain estimators of all parameters in logistic regression model, including interactions effects. Using a simulation study, we present comparisons of efficiency under different scenarios where exposures have been measured in pools and individually. Our simulations show that randomly sampling half of the available biospecimens has less efficiency than pooling pairs of biospecimens stratified by disease status. The EM algorithm provides a method for estimating interaction effects when biospecimens have already been pooled for other reasons such as the gain in efficiency for estimating main effects demonstrated by previous research. This manuscript demonstrates that the EM algorithm offers a promising approach to estimate interaction effects of pooled biospecimens.

The choice of product indicators in latent variable interaction models: post hoc analyses.

The unconstrained product indicator (PI) approach is a simple and popular approach for modeling nonlinear effects among latent variables. This approach leaves the practitioner to choose the PIs to be included in the model, introducing arbitrariness into the modeling. In contrast to previous Monte Carlo studies, we evaluated the PI approach by 3 post hoc analyses applied to a real-world case adopted from a research effort in social psychology. The measurement design applied 3 and 4 indicators for the 2 latent 1st-order variables, leaving the researcher with a choice among more than 4,000 possible PI configurations. Sixty so-called matched-pair configurations that have been recommended in previous literature are of special interest. In the 1st post hoc analysis we estimated the interaction effect for all PI configurations, keeping the real-world sample fixed. The estimated interaction effect was substantially affected by the choice of PIs, also across matched-pair configurations. Subsequently, a post hoc Monte Carlo study was conducted, with varying sample sizes and data distributions. Convergence, bias, Type I error and power of the interaction test were investigated for each matched-pair configuration and the all-pairs configuration. Variation in estimates across matched-pair configurations for a typical sample was substantial. The choice of specific configuration significantly affected convergence and the interaction test's outcome. The all-pairs configuration performed overall better than the matched-pair configurations. A further advantage of the all-pairs over the matched-pairs approach is its unambiguity. The final study evaluates the all-pairs configuration for small sample sizes and compares it to the non-PI approach of latent moderated structural equations.

Interaction Effects in Latent Growth Models: Evaluation of Alternative Estimation Approaches

The purpose of this investigation is to compare and evaluate 2 approaches for estimating interaction effects in latent growth models (LGMs): the unconstrained approach and the latent moderated structural equations (LMS) approach. To reduce the complexity of modeling interactions in LGMs, we created difference-product indicators to replace the traditional product indicators used in the unconstrained approach because these differences in original indicators represent changes over time. Our focus was to verify the performance of this simplified interaction model of LGMs with difference-product indicators by using the unconstrained approach and comparing it with the LMS approach. Our simulation study showed that the LMS approach generally resulted in smaller biases in the estimated parameters and was consistently more precise than the unconstrained approach under normal conditions, particularly when the sample size was small. When normality assumptions were violated, however, the unconstrained approach was shown to be more robust than the LMS approach in terms of the bias of estimation. In summary, we generally recommend both the unconstrained approach and the LMS approach if the indicators are normally distributed and when the sample size is large enough (e.g., not less than 500). Under normal conditions, the LMS approach is preferred. If normality assumptions are violated, however, the unconstrained approach is recommended. Under the most stringent conditions when normality assumptions are severely violated and the sample size is small, the results from both the unconstrained approach and the LMS approach should be treated with caution, and alternative procedures might be considered.

Causal inference of interaction effects with inverse propensity weighting, G‐computation and tree‐based standardization

AbstractGiven the recent interest of subgroup‐level studies and personalized medicine, health research with causal inference has been developed for interaction effects of measured confounders. In estimating interaction effects, the inverse of the propensity weighting (IPW) method has been widely advocated despite the immediate availability of other competing methods such as G‐computation estimates. This paper compares the advocated IPW method, the G‐computation method, and our new Tree‐based standardization method, which we call the Interaction effect Tree (IT). The IT procedure uses a likelihood‐based decision rule to divide the subgroups into homogeneous groups where the G‐computation can be applied. Our simulation studies indicate that the IT‐based method along with the G‐computation works robustly while the advocated IPW method needs some caution in its weighting. We applied the IT‐based method to assess the effect of being overweight or obese on coronary artery calcification (CAC) in the Chicago Healthy Aging Study cohort.

Predictors of Successful Weight Loss Maintenance: A Qualitative Comparative Analysis

Individuals successful at weight loss maintenance (WTLM) utilize similar behaviors but in varying amounts and combinations. Research identifying characteristics which predict WTLM success could be used to develop effective WTLM treatment programs. The purpose of this retrospective analysis was to determine behavioral (e.g., self monitoring of dietary intake) and biological factors (e.g., sex) which may influence WTLM success. Methods: In a previously conducted 12-month WTLM study, weight-reduced middle-aged and older men and women (n = 39) were assigned behavioral goals and asked to daily self-monitor body weight, fruit and vegetable intake, water consumption, and physical activity. This investigation utilized a crisp set qualitative comparative analysis (QCA) to characterize individuals’ weight changes and behaviors. Sex difference in clinically significant weight loss (WL) (≥5% WL) at 12 months was investigated, and growth curve models estimated interaction effects between sex and WTLM predictors. Results: QCA findings suggest patterns of weight change during the initial three months of WTLM treatment may dictate 12-month weight outcomes. No sex difference was found in clinically significant WL or in the interaction of sex and behaviors on weight change. Conclusions: WTLM treatments should provide more intensive support during transition from WL to WTLM, particularly during the first three months.

Estimating interaction effects with incomplete predictor variables.

The existing missing data literature does not provide a clear prescription for estimating interaction effects with missing data, particularly when the interaction involves a pair of continuous variables. In this article, we describe maximum likelihood and multiple imputation procedures for this common analysis problem. We outline 3 latent variable model specifications for interaction analyses with missing data. These models apply procedures from the latent variable interaction literature to analyses with a single indicator per construct (e.g., a regression analysis with scale scores). We also discuss multiple imputation for interaction effects, emphasizing an approach that applies standard imputation procedures to the product of 2 raw score predictors. We thoroughly describe the process of probing interaction effects with maximum likelihood and multiple imputation. For both missing data handling techniques, we outline centering and transformation strategies that researchers can implement in popular software packages, and we use a series of real data analyses to illustrate these methods. Finally, we use computer simulations to evaluate the performance of the proposed techniques.

Genetic variation in multiple biologic pathways, flavonoid intake, and breast cancer

We previously reported an inverse association between flavonoid intake and breast cancer incidence, which has been confirmed by others, but no studies have considered simultaneously potential interactions of flavonoids with multiple genetic polymorphisms involved in biologically relevant pathways (oxidative stress, carcinogen metabolism, DNA repair, and one-carbon metabolism). To estimate interaction effects between flavonoids and 13 polymorphisms in these four pathways on breast cancer risk, we used population-based data (n = 875 cases and 903 controls) and several statistical approaches, including conventional logistic regression and semi-Bayesian hierarchical modeling (incorporating prior information on the possible biologic functions of genes), which also provides biologic pathway-specific effect estimates. Compared to the standard multivariate model, the results from the hierarchical model indicate that gene-by-flavonoid interaction estimates are attenuated, but more precise. In the hierarchical model, the average effect of the deleterious versus beneficial gene, controlling for average flavonoid intake in the DNA repair pathway, and adjusted for the three other biologically relevant pathways (oxidative stress, carcinogen metabolism, and one-carbon metabolism), resulted in a 27 % increase risk for breast cancer [odds ratio = 1.27; 95 % confidence interval (CI) = 0.70, 2.29]. However, the CI was wide. Based on results from the semi-Bayesian model, breast cancer risk may be influenced jointly by flavonoid intake and genes involved in DNA repair, but our findings require confirmation.

Model-Based Estimates of the Effects of Efavirenz on Bedaquiline Pharmacokinetics and Suggested Dose Adjustments for Patients Coinfected with HIV and Tuberculosis

Safe, effective concomitant treatment regimens for tuberculosis (TB) and HIV infection are urgently needed. Bedaquiline (BDQ) is a promising new anti-TB drug, and efavirenz (EFV) is a commonly used antiretroviral. Due to EFV's induction of cytochrome P450 3A4, the metabolic enzyme responsible for BDQ biotransformation, the drugs are expected to interact. Based on data from a phase I, single-dose pharmacokinetic study, a nonlinear mixed-effects model characterizing BDQ pharmacokinetics and interaction with multiple-dose EFV was developed. BDQ pharmacokinetics were best described by a 3-compartment disposition model with absorption through a dynamic transit compartment model. Metabolites M2 and M3 were described by 2-compartment models with clearance of BDQ and M2, respectively, as input. Impact of induction was described as an instantaneous change in clearance 1 week after initialization of EFV treatment and estimated for all compounds. The model predicts average steady-state concentrations of BDQ and M2 to be reduced by 52% (relative standard error [RSE], 3.7%) with chronic coadministration. A range of models with alternative structural assumptions regarding onset of induction effect and fraction metabolized resulted in similar estimates of the typical reduction and did not offer a markedly better fit to data. Simulations to investigate alternative regimens mitigating the estimated interaction effect were performed. The results suggest that simple adjustments of the standard regimen during EFV coadministration can prevent reduced exposure to BDQ without increasing exposures to M2. However, exposure to M3 would increase. Evaluation in clinical trials of adjusted regimens is necessary to ensure appropriate dosing for HIV-infected TB patients on an EFV-based regimen.

Latent interaction effects in the theory of planned behaviour applied to quitting smoking

This study applies three latent interaction models in the theory of planned behaviour (TPB; Ajzen, 1988, Attitudes, personality, and behavior. Homewood, IL: Dorsey Press; Ajzen, 1991, Organ. Behav. Hum. Decis. Process., 50, 179) to quitting smoking: (1) attitude × perceived behavioural control on intention; (2) subjective norms (SN) × attitude on intention; and (3) perceived behavioural control × intention on quitting behaviour. The data derive from a longitudinal Internet survey of 939 smokers aged 15-74 over a period of 4 months. Latent interaction effects were estimated using the double-mean-centred unconstrained approach (Lin et al., 2010, Struct. Equ. Modeling, 17, 374) in LISREL. Attitude × SN and attitude × perceived behavioural control both showed a significant interaction effect on intention. No significant interaction effect was found for perceived behavioural control × intention on quitting. The latent interaction approach is a useful method for investigating specific conditions between TPB components in the context of quitting behaviour. Theoretical and practical implications of the results are discussed.

Testing for Two-Way Interactions in the Multigroup Common Factor Model

In this article, a 2-way multigroup common factor model (MG-CFM) is presented. The MG-CFM can be used to estimate interaction effects between 2 grouping variables on 1 or more hypothesized latent variables. For testing the significance of such interactions, a likelihood ratio test is presented. In a simulation study, the robustness of the likelihood ratio test under different sample size conditions is studied and the likelihood ratio test is compared to the Wilks's lambda F test in multivariate analysis of variance (MANOVA) with respect to the approximated power to detect a 2-way interaction effect on a single latent variable. The manipulated factors are the number of indicators, the values of factor loadings, the sample size, and the interaction effect size. The results of the simulation study show that the Type I error rate of the likelihood ratio test is satisfactory and that under all conditions, the approximated power of the likelihood ratio test is considerably higher than that of the Wilks's lambda F test in MANOVA.

The Pitfalls Of Multiple Dummy Variables In A Regression

This study reports long-been-neglected pitfalls when using multiple dummy variables in a regression model. We conduct simulation analysis to explain the mathematical meaning of a product term of two dummy variables, and find that the interaction dummy term indicates only the extra contribution, not the additive contribution of two dummies. The results suggest that testing and estimating interaction effects of dummy variables are not meaningful, and that dummy variable techniques should be handled carefully when introducing two or more dummies in a multiple regression model.

Which factors drive product sales in OTC markets?

PurposeThe purpose of this paper is to determine sales drivers for different OTC product categories.Design/methodology/approachThe study uses data from both consumer and retail panels, which are gathered for various product categories. These long‐term data are analyzed per product category with two specific regression models, mainly time‐series analysis with VAR models and Shapley value regression.FindingsIt is found that purchase intention drives sales a lot in general. Besides, it is very important to distinguish seasonal vs non‐seasonal markets. The trend coefficient, which implies the stage of maturity of the market, indicates more or less saturated markets for the examples. The proposed models can be easily applied to different OTC categories without a lot of customization.Research limitations/implicationsThe study does not take into account different outlets (e.g. online, supermarkets) and does not estimate interaction effects between the single drivers.Practical implicationsThe paper provides the market researcher with a guideline on how to proceed to model OTC product categories, e.g. which data are to be used, which models are to be estimated, which conclusions can be drawn.Originality/valueThe study develops an analysis approach which is readily applicable to different OTC product categories, which exhibit very distinct market characteristics. The advantage of this approach is that it applies a standardized tool kit of methods to analyze highly varying markets.

Fractions of Rechtschaffner matrices as supersaturated designs in screening experiments aimed at evaluating main and two-factor interaction effects

Optimal fractions of resolution V design matrices proposed by Rechtschaffner in 1967 are developed and applied as supersaturated designs in screening experiments. Rechtschaffner matrices allow evaluation of all main factors and two-factor interactions, which in many real-world studies are of practical significance. However, the number of experimental runs increases rapidly with the number of factors in the matrices, which are therefore impractical for more than 5–6 factors. On the contrary, saturated fractions based on Hadamard matrices, which are commonly applied in screening studies, cannot evaluate the interaction effects. Here, a procedure for selecting the optimum fractions of Rechtschaffner matrices is presented and provides supersaturated matrices that are well adapted to a variety of problems, thus allowing the development of screening studies with a relatively small number of experiments. The procedures developed to derive the size-reduced matrices and to evaluate the active factors are discussed and compared in terms of efficiency and reliability, by means of simulation studies and application to a real problem. These fractions are the first supersaturated design matrices capable of estimating interaction effects. Additionally, one important advantage of these supersaturated matrices is that they enable development of follow-up procedures in cases of inconclusive results, by enlarging the matrix and eventually resolving the full Rechtschaffner matrix of departure when it is necessary to evaluate the active factors and their interactions.