Investigating the Performance of Alternate Regression Weights by Studying All Possible Criteria in Regression Models with a Fixed Set of Predictors

We describe methods for assessing all possible criteria (i.e., dependent variables) and subsets of criteria for regression models with a fixed set of predictors, x (where x is an n×1 vector of independent variables). Our methods build upon the geometry of regression coefficients (hereafter called regression weights) in n-dimensional space. For a full-rank predictor correlation matrix, R _xx, of order n, and for regression models with constant R ² (coefficient of determination), the OLS weight vectors for all possible criteria terminate on the surface of an n-dimensional ellipsoid. The population performance of alternate regression weights—such as equal weights, correlation weights, or rounded weights—can be modeled as a function of the Cartesian coordinates of the ellipsoid. These geometrical notions can be easily extended to assess the sampling performance of alternate regression weights in models with either fixed or random predictors and for models with any value of R ². To illustrate these ideas, we describe algorithms and R (R Development Core Team, 2009) code for: (1) generating points that are uniformly distributed on the surface of an n-dimensional ellipsoid, (2) populating the set of regression (weight) vectors that define an elliptical arc in ℝ ⁿ , and (3) populating the set of regression vectors that have constant cosine with a target vector in ℝ ⁿ . Each algorithm is illustrated with real data. The examples demonstrate the usefulness of studying all possible criteria when evaluating alternate regression weights in regression models with a fixed set of predictors.     

Locating the Extrema of Fungible Regression Weights

In a multiple regression analysis with three or more predictors, every set of alternate weights belongs to an infinite class of “fungible weights” (Waller, Psychometrica, in press) that yields identical SSE (sum of squared errors) and R ² values. When the R ² using the alternate weights is a fixed value, fungible weights (a _i ) that yield the maximum or minimum cosine with an OLS weight vector (b) are called “fungible extrema.” We describe two methods for locating fungible extrema and we report R code (R Development Core Team, 2007) for one of the methods. We then describe a new approach for populating a class of fungible weights that is derived from the geometry of alternate regression weights. Finally, we illustrate how fungible weights can be profitably used to gauge parameter sensitivity in linear models by locating the fungible extrema of a regression model of executive compensation (Horton & Guerard, Commun. Stat. Simul. Comput. 14:441–448, 1985).     

The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

Ezekiel’s classic estimator of the population squared multiple correlation coefficient: Monte Carlo-based extensions and refinements

Abstract

Ezekiel’s adjusted R² is widely used in linear regression analysis. The present study examined the statistical properties of Ezekiel’s measure through a series of Monte Carlo simulations. Specifically, we examined the bias and root mean squared error (RMSE) of Ezekiel’s adjusted R² relative to (a) the sample R² statistic, and (b) the sample R² minus the expected value of R². Simulation design factors consisted of sample sizes (N?=?50, 100, 200, 400), number of predictors (2, 3, 4, 5, 6), and population squared multiple correlations (ρ² = 0, .10, .25, .40, .60). Factorially crossing these design factors resulted in 100 simulation conditions. All populations were normal/Gaussian, and for each condition, we drew 10,000 Monte Carlo samples. Regarding systematic variation (bias), results indicated that with few exceptions, Ezekiel’s adjusted R² demonstrated the lowest bias. Regarding unsystematic variation (RMSE), the performance of Ezekiel’s measure was comparable to the other statistics, suggesting that the bias-variance tradeoff is minimal for Ezekiel’s adjusted R². Additional findings indicated that sample size-to-predictor ratios of 66.67 and greater were associated with low bias and that ratios of this magnitude were accompanied by large sample sizes (N?=?200 and 400), thus suggesting that researchers using Ezekiel’s adjusted R² should aim for sample sizes of 200 or greater in order to minimize bias when estimating the population squared multiple correlation coefficient. Overall, these findings indicate that Ezekiel’s adjusted R² has desirable properties and, in addition, these findings bring needed clarity to the statistical literature on Ezekiel’s classic estimator.     

Fungible Weights in Multiple Regression

Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungiblebecause they yield identical SSE (sum of squared errors) and R ² values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression. The author wishes to thank Drs. Robyn Dawes, William Grove, Markus Keel, Leslie Yonce, Joe Rausch, the editor, and three anonymous reviewers for helpful comments on earlier versions of this article.     

Illness representations predict adherence in adolescents and young adults with type 1 diabetes

Objective: Most adolescents and young adults (AYAs) with type 1 diabetes struggle with diabetes self-management and exhibit suboptimal glycemic control. This study examined two models of association between illness representations, a modifiable predictor of suboptimal outcomes, and adherence and glycemic control in AYAs with type 1 diabetes.

Design and main outcome measures: Ninety-nine AYAs (ages 15–20?years) completed measures of illness representations and adherence at two visits. Blood glucose monitoring frequency and haemoglobin A1c were obtained via chart review. Relationships were examined using structural equation modelling.

Results: Illness representations accounted for a significant proportion of the variance in blood glucose monitoring frequency (ΔR^2?=?.23, p?<?.01) and adherence to emergency precautions at Time 1 (ΔR^2?=?.07, p?=?.03). Illness representations also accounted for significant variance in blood glucose monitoring frequency (ΔR^2?=?.08, p?=?.01), adherence to recommendations for insulin and food (ΔR²?=?.08, p?=?.02) and exercise (ΔR^2?=?.10, p?<?.01), and adherence to emergency precautions (ΔR^2?=?.16, p?<?.01) at Time 2.

Conclusion: Illness representations are salient predictors of adherence in this population. Interventions targeting adherence promotion and glycemic control in AYAs with type 1 diabetes may be enhanced by efforts to modify illness representations.     

Insomnia symptoms are associated with elevated C-reactive protein in young adults

Abstract

Objective: Insomnia is associated with elevated inflammation; however, studies have not investigated if this relationship is confounded with depression and neuroticism, which are associated with insomnia and inflammation. The current study examined the association of insomnia symptoms with C-reactive protein (CRP) and with interleukin-6 (IL-6), independently and after controlling for depressive symptoms and neuroticism. Design: Fifty-two young adults (mean age?=?25.2?±?3.9 years, 52% female) completed a baseline survey to assess psychological characteristics, followed by a plasma blood draw. Main outcome measures: Plasma CRP and IL-6. Results: When examined alone, insomnia symptoms were significantly associated with elevated CRP (β?=?0.52; R²?=?0.27), as was neuroticism (β?=?0.41, R²?=?0.17), but not depressive symptoms (β?=?0.21, R²?=?0.05). The association between insomnia symptoms and CRP remained significant when depressive symptoms and neuroticism were entered into the model simultaneously; this model did not explain more variance than the model with insomnia symptoms alone. No variables were associated with IL-6. Conclusions: Results suggest that insomnia symptoms are independently associated with elevated CRP in young adults, even after controlling for presumed overlapping psychological constructs. Findings highlight the potential importance of treating insomnia to reduce systemic inflammation.     

Fractal growth of graphite nodules in iron

It has been found from a large number of statistical tests that graphite nodules in malleable iron grow with time according to a power law. The growth of the nodules and their fractal dimension have been investigated experimentally as a function of annealing time. Based on the assumption of carbon-diffusion-controlled growth in the initial stage, the growth equation is R _G = K ₁ t ^1/(D-1), where R _G is the radius of the graphite nodules, t the time, D the fractal dimension and K ₁ a constant. Assuming cementite-dissolution-controlled growth in the later stages of growth, the relation is R _G = K ₃ t ^3/D , where K ₃ is a constant. The fractal dimension, or the aggregate state of the graphite nodules, strongly influences the growth process.     

A simplification of a completeness proof of Guaspari and Solovay

The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R ^– are improved and the relationship between R and R ^– is clarified.     

Religious Orientation and Ethnic Identity as Predictors of Religious Coping Among Bereaved Individuals

Religious orientation and ethnic identity inform the religious coping process, but research on this topic is scarce. The authors collected data on these constructs from a sample (N = 319) of bereaved adults. A canonical correlation analysis showed that individuals who engage in traditional spiritual practices and strive to achieve ordinary and transcendental spiritual goals are more likely to engage in positive religious coping (Wilks's Λ = .36, R_c² = .62, p < .001). Also, a multiple regression analysis revealed that individuals with higher levels of ethnic identity development are more likely to engage in positive religious coping (β = .12, t < .05). Finally, a discriminant analysis indicated that ethnic identity and a conservative religious orientation discriminated between Whites and ethnic minority individuals, Wilks's Λ = .71, χ²(4, N = 204) = 70.10, p < .001, R_c² = .26. The authors encourage counselors to strengthen their multicultural and spiritual competencies to provide effective services to a culturally and religiously diverse clientele.     

Right and Wrong Reasons in Folk‐Psychological Explanation

Abstract

Davidson argued that the fact we can have a reason for acting, and yet not be the reason why we act, requires explanation of action in terms of the agent's reasons to be causal. The present paper agrees with Dickenson (Pacific Philosophical Quarterly, 2007) in taking this argument to be an inference to the best explanation. However, its target phenomenon is the very existence of a case in which an agent has more than one reason, but acts exclusively becaue of one reason. Folk psychology appears to allow for this phenomenon. However, appreciation of ‘rationalization’ as a form of contrastive explanation reveals the existence of the Davidsonian possibility to the problematic. Claims that ‘I did it because of R₁, not because of R₂’ are entertained in folk psychology, and may be sincere or insincere. But as reports of conscious practical reasoning, even when sincere, they are not authoritative about the mechanism of motivation.     

When Home Is Where the Harm Is: Family Betrayal and Posttraumatic Outcomes in Young Adulthood

Research on institutional betrayal has found that institutional wrongdoing that fails to prevent or respond supportively to victims of abuse adds to the burden of trauma. In this two-study investigation with young adult university students, we demonstrated parallels between institutional betrayal and ways that families can fail to prevent or respond supportively to child abuse perpetrated by a trusted other, a phenomenon we call family betrayal (FB). In Study 1, psychometric analysis of a new FB questionnaire provided evidence of its internal consistency, unidimensionality, and convergent and discriminant validity. The majority (approximately 72%) of young adults abused in childhood reported a history of FB, with an average of 4.26 FB events (SD = 4.45, range 0–14). Consistent with betrayal trauma theory, Study 2 revealed that FB was 4× more likely to occur in relation to childhood abuse by someone very close to the victim (vs. non-interpersonal victimization), with a particularly strong effect for female participants. FB history predicted significant delay to disclosure of a self-identified worst traumatic event (η_p² = .017) and significant increases in dissociation (?R² = .05) and posttraumatic stress (?R² = .07) symptoms in young adulthood. Moreover, with FB in the regression models, only FB—not child abuse nor recent interpersonal victimization—predicted dissociation and clinically significant elevations in posttraumatic stress. Findings suggest that FB is a prevalent phenomenon among young adults abused as children and that it explains unique, clinically significant variance in posttraumatic distress, warranting increased attention from trauma researchers and clinicians.     

Hybridization-induced wide-band bipolaron model for high-Tc superconductors

Abstract

A microscopic theory based on the idea of hybridization-induced wide-band bipolarons is proposed to explain the occurrence of high-temperature superconductors such as La_2?x-Ba_xCuO₄, and La_2?xSr_xCuO₄. In our view, these are such that bipolarons are formed whose singlet ground state possesses a large binding energy (Δ) with bandwidth W (>Δ) as a result of two distinct electron-phonon mechanisms and two types of hybridization between Cu²⁺ d electrons and other electrons. This wide-band bipolaron gas has features similar to Cooper pairs and leads to a superconducting state with high T_c in which all occupied electron states participate. The large electron-phonon coupling and a large W both cause the transition temperature T_c, to rise spectacularly.     

Approximant phases,phason planes and domain-boundary networks in Al−Ni−Rh alloys

Structurally complicated ξ′- and ξ-phases have been found, for the first time, in as-cast Al₇₃Ni₅Rh₂₂ and Al₇₅Ni₁₅Rh₁₀ alloys. The lattice parameters of these two phases were determined by means of electron diffraction and high-resolution transmission electron microscopy (HREM). These two phases have similar orthorhombic structures but with different lattice parameters of a?=?23.2?Å, b?=?16.4?Å, c?=?12.0?Å for the ξ′-phase and a?=?20.3?Å, b?=?16.4?Å, c?=?14.8?Å for the ξ-phase. A new two-dimensional domain-boundary network has also been observed in these two phases. Domain boundaries with normals closely parallel to the [001] direction are actually phason planes represented by a translation vector of r?=(1/2)a?+(1/2τ)c in the ξ′-phase and r=±(1/2τ²) a?+(τ/2)c in the ξ-phase, whereas the newly-found wide and zigzag boundaries perpendicular to the above set were attributed from the step-like boundary structures of domains related by a translation vector of r?=(1/τ)((1/2)a?+(1/2τ)c). The structural difference between the two types of planar faults is discussed.     

Internal Cue Theory: Calibration and Resolution of Confidence in General Knowledge

A theory is proposed in which beliefs in the form of internal cue validities mediate the processing of ecological cue validities in the assessment of confidence. The conditions necessary for perfect calibration are specified: (a) correspondence between ecological and internal validity, (b) perfect translation of internal validity into a confidence assessment, and (c) consistent utilization of cues. Process errors are then added to these conditions to investigate how calibration is affected by error variance of confidence assessments. To accomplish this, the calibration score (C) is decomposed into three additive parts: D² = bias, i.e., the squared difference between mean confidence and proportion correct; R² = resolution, i.e., the squared difference between the standard deviations of confidence and proportion correct; L = linearity, i.e., how closely the calibration curve follows a linear function. In the equation C = D² + R² + L, R² (resolution) reflects the subject′s ability to discriminate cue validities. Selection of items is a critical factor in studies of confidence. Informal selection with a tendency to avoid easy items results in overconfidence. Internal cue theory predicts both that overconfidence should disappear (in accordance with previous research) and that resolution should improve when item selection is made representative of the natural environment. Both predictions are confirmed by data from published studies on confidence in general knowledge. It is noteworthy that resolution is still poor and accounts for the major portion of miscalibration under representative item selection.     

A Logic for Multiple-source Approximation Systems with Distributed Knowledge Base

The theory of rough sets starts with the notion of an approximation space, which is a pair (U,R), U being the domain of discourse, and R an equivalence relation on U. R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U. The focus then is on approximations of concepts on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base, which is a tuple (U,R_P)_{Pßf N}(U,R_P)_{P\ss_f N}, where N is a set of sources and P ranges over all finite subsets of N. Each R _P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS ^D is proposed. In LMSAS ^D , quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS ^D are discussed and it is shown that the modal system KTB is embedded in LMSAS ^D . The epistemic logic S5^D_nS5^D_n is also embedded in LMSAS ^D , and cannot replace the latter to serve our purpose. The relationship of LMSAS ^D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed.     

Psychological need satisfaction and thwarting: A test of Basic Psychological Needs Theory in physical activity contexts

ObjectivesTo test Basic Psychological Needs Theory (BPNT; Deci & Ryan, 2002) to determine if psychological need thwarting experienced when physically active contributes to the understanding of well-being and ill-being.Design/methodParticipants (N = 155, 67.70% female, M_age = 37.46 years; SD_age = 19.89 years) completed assessments of psychological need satisfaction and thwarting, subjective vitality and positive/negative affect during separate testing sessions separated by 6 months.ResultsScores from the modified version of the Psychological Need Thwarting Scale (PNTS-PA; Bartholomew, Ntoumanis, Ryan, & Thøgersen-Ntoumani, 2011) demonstrated discriminant evidence of validity, evidence of internal structure and minimal error variance. Changes in psychological need satisfaction positively predicted positive affect (R² = .16, p < .05), subjective vitality (R² = .13, p < .05) and negatively predicted negative affect (R² = .12, p < .05). Additional regression analyses revealed that changes in psychological need thwarting predicted negative affect (ΔR² = .11, p < .05), but not positive affect (ΔR² = .01, p > .05) or subjective vitality (ΔR² = .04, p > .05) beyond contributions made by psychological need satisfaction.ConclusionsOverall, these results extend the potential utility of the PNTS-PA as an instrument for use with BPNT beyond sport and support Deci and Ryan's (2002) contentions regarding the critical role of psychological need thwarting.     

Use of correlational data to examine the effects of risk perceptions on precautionary behavior

Abstract

The idea that perceptions of high personal risk lead people to adopt precautionary behavior (the “motivational hypothesis”) is mainly tested with correlational data. A review of studies from selective journals reveals a high proportion with methodological and conceptual problems that make them invalid as tests of this hypthesis. Three problems arc emphasized: (1) the misinterpretation of correlations from cross-sectional studies as testing the motivational hypothesis when they actually indicate the accuracy of risk perceptions; (2) the failure to control for prior behavior in prospective studies; and (3) the we of prospective studies in situations of little behavior change. Path models are used to help explain these problems. Recommendations for selecting research designs and for calculating the least problematic correlations are given, along with warnings about the many assumptions needed to interpret even these correlations.

Summary

Overall, 27 of the 61 cross-sectional analyses listed in Tables 2 and 3 were conducted using clearly inappropriate variables. The appropriateness of the remaining correlations rest on one or two rather questionable assumptions: (1) that people do not distort their risk perceptions to justify their behavior or intentions and (2) that negative screening results a problem like breast cancer do not affect perceptions of future risk. The correlations preferred for testing the motivational hypothesis are listed in Table 4.

Summary

When the amount of precautionary behavior in a population has become relatively stable and behavior at time t+1 is well-predicted from behavior at time t, no independent variable other than B_t will have much predictive value in a prospective design. To avoid this problem, research should be conducted at a time when a substantial change in behavior is occurring, such as soon after the risk is recognized. If this is not possible, interventions that lead people to change their behavior (e.g., by raising risk perceptions or by lowering barriers to action) are required. In effect, they remove the system from equilibrium and allow one to watch what happens as people seek a new equilibrium. The least satisfactory choice is to use a cross-sectional correlation, such as R _H?B, that represents a summation over previous changes in behavior. The assumptions required to interpret such a correlation have been discussed earlier.     

Effect of spin-orbit interaction on the metal-insulator transition in doped silicon

Abstract

We show that in disordered metallic systems, spin-orbit interactions lead to the existence of two metallic phases separated in temperature. The low-temperature metallic phase arises from spin-orbit interactions. The crossover temperature is proportional to (Z-Z′)² (Z is the atomic number of the donor and Z′ that of the matrix) and is expected to be observable for donors with large atomic numbers. The presence of two metallic phases implies two mobility edges, existing at different temperatures, and affecting the conductivity when the Fermi energy E _F lies below the mobility edge E _c We thus predict anomalies in the conductivity in this range of concentration, consistent with the measurements of Long and Pepper on Si:Sb.     

Forecasting the influence of customer-related micro-events on employees’ emotional,attitudinal and physiological responses

The affective events theory proposes that daily events elicit affective reactions on workers that, over time, influence affective and judgement-driven behaviours. It also suggests that this relation is moderated by dispositions and appraisals. On the other hand, the social interaction model argues that the impact of emotions is moderated by how individuals regulate them. This study aimed to: (1) investigate what customer-related events elicit affect; (2) test the moderating role of workers’ susceptibility for emotional contagion on the relation events-affect; and (3) explore whether affective states influence cardiovascular efficiency and turnover intentions. We conducted a longitudinal study in an inbound call centre by following 48 workers during 10 working days, gathering 267 events and 1,232 affective reactions. We combined diaries, questionnaires and physiological data. Data was analysed qualitatively and quantitatively. We extracted 13 event categories and, using artificial neural networks (ANN), found support for the moderating role of emotional contagion. At daily level, fear was the stronger predictor of cardiovascular efficiency, whereas anger was the stronger predictor of turnover. ANN models showed satisfactory predictive values (R²_Turnover = .51, p < .01; R²_{Cardiovascular efficiency} = .32, p < .01). The importance of results for theory and practice is discussed.