By Richard Haans

Work in strategic management often investigates curvilinear relationships. In such relationships, some variable first has a positive effect, which then turns into a negative effect (or vice versa) on an outcome variable. However, theorizing and testing for such relationships is often substantially more complex than for linear relationships—especially when adding moderation to the mix. Motivated by a sense that these complexities were under-appreciated, Zi-Lin He, Constant Pieters, and I wrote our 2016 article reviewing all 110 articles investigating curvilinear relationships published in the Strategic Management Journal to evaluate the state of past work and to offer recommendations for future work. In this blog post, I offer some reflections on where, whether, and how the article has been used since its publication.

In absolute terms, our article has seen considerable use: as of August 2020, it has 491 citations on Google Scholar and 294 on Web of Science.[i] This level of attention is something that continues to amaze me. The project emerged from a Research Methods course taught by Zi-Lin that Constant and I took, and at the time of publication we were unsure whether the paper would attract an audience. As shown in Figure 1, the article has largely been cited in the fields of Business and Management, with 158 and 205 of the citing articles in the Web of Science belonging to a journal in those categories, respectively. Journals in thirty-two other Web of Science categories have cited the article at least once. In total, our citations in the Web of Science come from 127 unique journals. In all, these numbers suggest that our article has seen some use outside of our core fields but that the bulk of our impact has remained within the fields of Business and Management.

Figure 1: Citations from different Web of Science categories.

To then assess whether or not our work has actually had a more substantive impact on the field, I continued by replicating our original keyword query in the titles, abstracts, and author keywords on the Web of Science for articles published between 2017 and 2019 in the fields of Business and Management. This query (admittedly crude and prone to potential over- and under-counting) yielded 1,740 hits—representing a sample of articles investigating curvilinear effects during this period; 116 of these articles cite our paper. Though the total number of articles published per year has remained more or less constant (584 in 2017; 568 in 2018; 588 in 2019), I do observe our article increasingly gets cited (25, 34, and 57 times, respectively).

Some interesting patterns emerge from these data. First, I find that articles that cite our paper obtain significantly higher impact. Indeed, articles that cite (and, presumably, make use of) our paper obtain an average of 6.49 citations, versus 4.74 citations for those that do not (p = 0.017). This effect is especially pronounced for those articles published in 2017, where those that cite us obtain 12.56 citations on average versus 6.89 for those that do not. Figure 2 illustrates these effects (p-values obtained from simple t­­-tests). Of course, these patterns are presented with an array of caveats (the list of omitted variables is obviously huge) and are not intended to offer any sort of causal (or even serious correlational) evidence. Nevertheless, the patterns do suggest that those articles that (even potentially superficially) follow the recommendations set out in our paper have obtained higher impact—I hope that this indicates that our work has led to higher quality research.

Figure 2: Citations to articles investigating curvilinear effects.

Finally, to assess how the original paper has been used I downloaded all articles published in one of the Strategic Management Society journals which also cite our article (10 in SMJ, 1 in SEJ, and 2 in GSJ) and coded how they cited us. Noting that this check was informal in nature and limited only to the direct reference to our work (such that I may have missed other engagement with the article), I find that all except one of the 13 articles refers to our paper in their empirical sections. Most commonly, this is for the three-step procedure that we describe for testing for an (inverted) U-shaped effect and which was originally developed by Jo Thori Lind and Halvor Mehlum in 2010 in the Oxford Bulletin of Economics and Statistics. I also observe several papers that follow our recommendations for testing of moderation of curvilinear relationships, with most focusing on flattening or steepening of their curves. Although the contribution of our paper therefore mainly seems to be of an empirical nature, I was also happy to see many references to the recommendations that we set out for theorizing curvilinear relationships: six of the 13 articles refer to these recommendations—most commonly building on our framework for utilizing different combinations of latent mechanisms to build up the observed curvilinear effect. Personally, I find these theoretical recommendations the most interesting aspect of our paper (and likely the most future-proof, given that methodological approaches tend to get outdated, fast), such that I hope that these recommendations get picked up more over time.

One final note is that publication year is strongly, negatively, correlated to the number of references made in these articles to our work (the correlation equals -0.444). I see two potential explanations for this pattern: one the one hand, it may be that with the increasing popularity of our article also came an increase in the number of papers that more ceremoniously cite it once or twice for a single check. On the other hand, it may also be that the various recommendations that were set out became more institutionalized, reducing the need to explicitly cite our paper. My hunch is that the latter is more applicable, as I also observe various other recommendations—such as graphical representation of the observed effects—being present in these articles and in the wider literature investigating curvilinear effects. Nevertheless, there are many recommendations that we made both for the theorization and the testing of these complex relationships that I rarely see implemented, such as the thorough investigation of potential alternative functional forms to ensure an (inverted) U-shape is indeed the best fit, making use of graphs to make explicit the theorized latent mechanisms that drive the effects, and making sure that the full curve is theorized (rather than just one half). As such, while I am obviously thrilled about the positive impact that our article has achieved since its publication, I also continue to see tremendous opportunities for deeper engagement with the full range of recommendations that we set out. Going by the presented numbers, this might be a win-win for all involved.

[i] Editor’s note: The paper by Haans, Pieters, and He is the most cited article in Strategic Management Journal since 2016 on Web of Science.