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.     

Fibromyalgia impact and depressive symptoms: Can perceiving a silver lining make a difference?

Individuals with fibromyalgia are at greater risk for depressive symptoms than the general population, and this may be partially attributable to physical symptoms that impair day-to-day functioning. However, individual-level protective characteristics may buffer risk for psychopathology. For instance, the ability to perceive a “silver lining” in one’s illness may be related to better mental and physical health. We examined perceived silver lining as a potential moderator of the relation between fibromyalgia impact and depressive symptoms. Our sample of persons with fibromyalgia (N = 401) completed self-report measures including the Fibromyalgia Impact Questionnaire-Revised, Depression Anxiety Stress Scales, and the Silver Lining Questionnaire. Moderation analyses covaried age, sex, and ethnicity. Supporting hypotheses, increasing impact of disease was related to greater depressive symptoms, and perceptions of a silver lining attenuated that association. Despite the linkage between impairment and depressive symptoms, identifying positive aspects or outcomes of illness may reduce risk for psychopathology. Therapeutically promoting perception of a silver lining, perhaps via signature strengths exercises or a blessings journal, and encouraging cognitive reframing of the illness experience, perhaps via Motivational Interviewing or Cognitive Behavioral Therapy, may reduce depressive symptoms in persons with fibromyalgia.     

How prevalent is contextual information in research on school bullying?

Empirical articles on school bullying need to give contextual details of the study, including on participants (number, age, gender), the country in which data was gathered, and the year in which it was gathered. We argue that these are important aspects of information, and that country and year of data collection cannot be inferred unambiguously unless they are explicitly stated. We report an analysis of contextual information on a sample of 201 articles, from 1976 to 2015, on school bullying. The great majority of studies gave information on number and age of participants, and most on gender balance. Most also gave explicit information on the country in which data was gathered. However only about one quarter of articles gave information on the date (year) in which data was gathered. For those that did, the average gap from data gathering to publication was 4 years, with a range of 1 to 11 years. We argue that the date of data collection is an important historical aspect, as many societal changes, even over a period of a few years, can impact on prevalence and nature of bullying. We recommend that besides participant and country information, year of data collection is routinely given in empirical articles on school bullying.     

The more you have,the more you want? Higher social class predicts a greater desire for wealth and status

Traditional theories have focused on the intentions of lower-class individuals to climb on the social ladder, yet they have paid relatively little attention to the motivations of upper-class individuals to ascend even higher. Addressing this issue, Studies 1 and 2 provided cross-national evidence that higher social class is associated with a greater desire for wealth and status. Moreover, by manipulating perceived social class, Studies 3 and 5 experimentally confirmed that compared to people in the lower-class group, those in the upper-class group express a stronger desire for wealth and status. Furthermore, in line with self-categorization theory predictions, Studies 3–5 showed that upper-class individuals tend to see and use wealth and status as important attributes in defining and categorizing self, and this tendency explains the effect of social class on desire for wealth and status. Together, our findings demonstrate a “having more—wanting more” relationship, and its consequences are further discussed.     

Zero-sum beliefs shape advantaged allies’ support for collective action

Three studies (N₁ = 1,019; N₂ = 312; N₃ = 494) tested whether seeing intergroup relations as inherently antagonistic shaped advantaged social groups’ allyship intentions. More specifically, we tested whether endorsing zero-sum beliefs related to their willingness to support system-challenging and system-supporting collective action. Zero-sum beliefs were negatively correlated with system-challenging and positively correlated with system-supporting collective action intentions. Zero-sum beliefs were more common among advantaged than disadvantaged groups and translated into lower allyship intentions. Advantaged group members with higher levels of zero-sum beliefs were also more likely to experience anger and fear when considering the demographic racial shift in the United States. Increased fear was associated with greater support for system-supporting and lower support for system-challenging collective action. We find consistent evidence that advantaged group members see intergroup relations as a zero-sum game and that these beliefs are negatively related to their intentions to become allies.     

Standing up for whom? Targets’ different goals in the confrontation of discrimination

We challenge the common interpretation of targets’ immediate confrontation in reaction to discrimination as self-serving behavior and propose different underlying motivations for this phenomenon. In five online scenario studies (N_overall = 1,447), we demonstrate across different samples and contexts that targets indicate a distinct pursuit of the following self-reported confrontation goals: individual-benefit (e.g., perpetrator apologizes); group-benefit (e.g., prejudice reduction); and distancing (e.g., demonstrating that one is different from typical group members). Furthermore, meaningful associations of the pursuits of individual-benefitting goals and group-benefitting goals with group identification, disidentification, and further collective action intentions indicate that they represent different confrontation motivations: Individual-benefitting confrontation serves to cope with the individual mistreatment of discrimination, whereas group-benefitting confrontation represents a form of collective action. Distancing goals were associated with disidentification and—unexpectedly—group identification. Our results show that the phenomenon of confrontation in reaction to discrimination can be the result of different underlying psychological processes.