管理学领域的曲线效应及统计检验方法

Curvilinear Effect and Statistical Test Method in the Management Research

一般管理学研究在探讨变量间的关系时,通常假定该关系为线性的,即正相关或负相关的关系。然而越来越多实证研究及对客观世界的观察结果表明,变量之间的关系很可能不是简单线性的,而是曲线的。当一个积极的因素投入过度将不能带来额外的收益,甚至可能会带来负面的影响,这种现象称为“过犹不及”效应。相反,如果出现某种因素投入在高水平或低水平时,收益都要比将该因素投入在中等水平时要好,这种现象则称为“居中不良”效应。“过犹不及”和“居中不良”都是典型的曲线效应,这些曲线效应广泛存在于管理学领域及其他社会科学领域,如心理学、社会学、经济学等。本文探究了国内外对于曲线效应现象的论述,并以二次函数曲线关系为例,介绍非线性关系的几种模型(简单非线性关系、非线性调节效应、非线性中介效应)及其统计检验方法,以期为研究者对非线性关系的研究提供参考。

Previous research mainly presumed a linear relationship between two constructs(i.e.,X is positively or negatively related to Y).However,the increasing number of mixed findings and the complexity of management phenomena suggest that such a linear presumption may not always be true.The curvilinear paradigm can be traced to both Eastern and Western cultures.For example,Confucius and Aristotle both advocated the Doctrine of the Mean,which means that a medium level of investment results in the best.Although scholars have begun to notice the curvilinear effect in management field,the empirical studies based on curvilinear presumptions are still few,and people’s understanding of the curvilinear effect is far from adequate.This paper aims to help scholars expand the understanding of curvilinear effect in the management field and provide support to statistical tests for the curvilinear effect model.The curvilinear effect manifests in different forms in the management field,and the most common kinds are the too-much-of-a-goodthing(TMGT)effect and the too-confusing-to-be-a-good-thing(TCGT)effect.Specifically,the TMGT effect refers to the phenomenon that a beneficial antecedent may result in detrimental outcomes when taken too far.On the contrary,the TCGT effect refers to the phenomenon that the medium level of an antecedent may result in the worst outcomes compared with its higher or lower level.Those curvilinear effects are ubiquitous in the management field and other fields of social science(e.g.,psychology,sociology,and economics).In order to provide a direct and intuitive understanding of curvilinear effect in the management field,this paper includes examples of empirical research on curvilinear effect both in China and abroad in four subfields of management(i.e.,organizational behavior,human resource management,entrepreneurship,and strategic management).In contrast to the linear relationship model,there are several mathematical models that can be used to conduct statistical test for the curvilinear effect.The quadratic function model is the simplest and most widely used mathematical model in the management field.This paper,taking the quadratic function model as an example,introduces several commonly used statistical models of curvilinear effect(i.e.,the simple curvilinear effect model,the curvilinear moderation model,and the curvilinear mediation model),and provides guidance on statistical tests for these models.First,we use the quadratic function to establish a mathematical model for the curvilinear relationship in the management research,and this relationship is in a U shape or inverted-U shape in the figure,depending on the coefficientβ2.Ifβ2 is significantly positive,the relationship between X and Y is in a U shape(i.e.,a TCGT effect);on the contrary,ifβ2 is significantly negative,the relationship between X and Y is in an inverted-U shape(i.e.,a TMGT effect).Second,we add a moderating factor Z on the basis of the quadratic function model to test the moderation effect of the curvilinear relationship,and the moderation effect of the moderator Z on the curvilinear relationship is reflected in that the moderator can change the position of the critical point and the shape of the(inverted)U shaped curve.Third,we apply the quadratic function model to the mediation model and introduce two widely used curvilinear mediation effect model.The first model is the first-stage curvilinear mediation model(i.e.,the relationship between X and M is curvilinear and the relationship between M and Y is linear).The second model is the second-stage curvilinear mediation model(i.e.,the relationship between X and M is linear and the relationship between M and Y is curvilinear).In the discussion section,we discuss the theoretical contribution of the curvilinear effect research.Firstly,the curvilinear effect paradigm can be utilized to reconcile mixed findings in previous research.Secondly,the curvilinear effect paradigm can help refine theories in the management field.Thirdly,the curvilinear effect paradigm has better explanatory power than the linear effect paradigm.Furthermore,this paper points out the practice implications of the curvilinear effect paradigm.First,the TMGT effect reveals that the increasing investment of some beneficial elements does not always result in desirable outcomes.Therefore,we suggest organizations to maintain the input of these beneficial elements within an appropriate range.Second,the curvilinear effect paradigm provides more details to guide management practices.In contrast to the linear effect research,the curvilinear effect research not only reveals the direction of the relationship between two constructs but also figures out the critical point,which provides guidance to reach the optimal value or optimal interval of input in management practices.Meanwhile,we discuss the contribution of this paper itself.First,we systematically expound the curvilinear effect in order to help readers better understand the connotation and nature of the curvilinear effect and inspire their research interest in the curvilinear effect.Second,we review empirical research on curvilinear effect both in China and abroad in four subfields of management to show that curvilinear effect is ubiquitous in the management field.However,the vast majority of management research still adopts the linear effect paradigm,and the curvilinear effect studies are few and far between.This paper aims to help scholars to understand the curvilinear effect and encourage them to adopt the curvilinear model in the management research and other fields of social science.Third,we explore the issue of how to conduct statistical tests on the curvilinear effect model.In this paper,we introduce in detail how to statistically test the simple curvilinear model,the curvilinear moderation model,and the curvilinear mediation model.We also discuss the difference and relationship between the curvilinear effect model and the linear effect model,in order to provide a mathematical foundation and statistical support to management scholars.At the end,we discuss the potential limitations and future direction of this paper.