Understanding complex relationships: Asking for the when and why
What you’ll learn
We are living in an increasingly complex and multifaceted world. In order to successfully navigate modern business challenges, we therefore need to understand and adequately capture complex relationships. This chapter discusses complex relationships where one thing influences the relationship between two other things (moderation) or where the relationship between two things is explained by another thing (mediation). We will illustrate with practical examples why managers and experts profit from knowledge of moderation and mediation analysis.
Data conversation
It was Kenny’s and James’ first opportunity to prove their communication expertise to their colleagues. Ten months ago, Kenny and James joined the non-profit organisation as trainees and ever since they had been passionate about their job. It didn’t take long until the Head of Communication entrusted them with the task to develop a communication concept for the website of the organisation. To do so, Kenny and James conducted an extensive literature research which led them to the conclusion that visual images play a crucial role in online communication. Previous studies consistently indicated that emotional images (i.e., images showing either happy or sad people) attract users’ attention and increase their engagement with the web content.
‘But does it matter whether we use happy or sad images?’, Kenny asked his colleague. The question turned into a lengthy discussion that resulted in no clear answer. However, both were curious to find out whether happy or sad images were more effective in retaining users on the website. They therefore set up two versions of the website, one with only happy images and the other with only sad images. They then invited 80 people to visit the website and randomly assigned them to either the happy image version or the sad image version. All participants had to fill out a questionnaire. The subsequent analysis suggested that the type of imagery did not have a substantial impact on the length of the website visit (Figure 4.1). ‘The findings indicate that we can use both happy and sad images on our website’, Kenny figured. But by chance, he remembered that the questionnaire also captured whether the participants were members of the non-profit organisation or not. Did their membership status possibly influence whether they were more responsive to sad or happy images?
An interesting pattern emerged from Kenny’s and James’ experiment. The findings suggested that people’s membership status (member vs non-member) influenced whether they were more responsive to sad or happy images. The data showed that members spent significantly more time on the website when they were assigned to the website version with only sad images. In contrast, non-members remained significantly longer on the website when they were directed to the version with only happy images (Figure 4.6). ‘This is interesting and might have practical implications’, James said. The results suggest that we might want to use sad images for subpages that primarily address our members and happy images for subpages that are directed at a general public.
Although they had to validate the findings with more data, taking into account the potential influence of membership status – a moderating variable – helped Kenny and James to develop a more sophisticated communication approach. At the end of the day, both felt tired but satisfied. They were looking forward to presenting their communication strategy to the Head of Communication the next day.
Often it is the simple things in life that make us happy: going for a walk, enjoying a colourful sunset or spending time with our loved ones. But as pleasant as these simple things can be, the world is often quite complicated. A lot of things are contingent upon each other. For instance, the extent to which we become more generous with age may depend on our socioeconomic status. And sometimes the effect of one thing is explained by another thing: we may feel happier, the better feedback we get. This may be due to the fact that good feedback increases our self-esteem, which then translates into more happiness.
Fortunately, statistics allow us to examine these kinds of complex relationships (Field, 2018). Whenever we want to find out to what extent one thing (e.g., socioeconomic status) influences the relationship between two other things (e.g., age and generosity), we use moderation analysis. Whenever we aim to understand to what extent the relationship between two things (e.g., feedback and happiness) is explained by another thing (e.g., self-esteem), we use mediation analysis. Put in more analytical terms, moderation and mediation both enable us to better understand the relationship between a predictor and an outcome by testing how a third variable fits into this relationship (Figure 4.2).
When we conduct a moderation analysis (Figure 4.3), we are interested in something that is called an interaction effect (Field, 2018). We want to find out to what extent a variable affects the strength or direction of the relationship between a predictor and an outcome variable (Fritz and Arthur, 2017). Engaging with moderator variables is crucial for several reasons. First, it allows for a more nuanced understanding of the world. Moderation analysis is a data-based way to say ‘it depends’. Second, some effects are ‘hidden’ in that they only show when you consider the influence of a moderator variable. Without taking into account the moderator variable, you may wrongfully conclude that there is no effect. This can have a detrimental impact on the quality and robustness of your decisions.
Let’s illustrate this with an example (Figure 4.4). Suppose that we collect data to better understand the relationship between people’s involvement in humanitarian issues (such as fighting famine in Africa) and the amount of donations that they make. We analyse this data and state that there is a positive relationship between involvement and the amount of donation: the more people are involved in humanitarian issues (say have friends or family members in an affected region), the more they donate. Moreover, the relationship does not seem to be that strong (the line is rather flat).
Assume that we had also collected data on people’s gender and that we can therefore classify them as female or male. Doing the same analysis but with gender as a (categorical) moderator variable, we would see that the relationship between involvement and amount of donation varies depending on whether people identify as male or female (Figure 4.5). There is no relationship between involvement and the amount of donation for men (because the line is almost completely flat in our fictitious example), whereas for women, there is a strong positive relationship. As women’s involvement increases, so does the amount of donations that they make (as the dotted line is quite steep).
This is an example for a moderator variable that influences the strength of the relationship between a predictor and an outcome. In the dialogue box (below) you see what it means if a moderator variable changes the direction of the relationship between a predictor and an outcome.
The examples provided illustrate different ways of how a third variable can influence the relationship between two other variables. In the donation example, it has been shown that gender (moderator) influences the strength of the relationship between involvement and donation, such that there is an effect for women but not for men. In the website example, however, membership status (moderator) has flipped the entire interpretation, such that the positive effect of sad images for members is completely reversed for non-members.
But how does moderation analysis fit into the general linear model? As we have seen previously, we can add predictors to the linear model. To test for moderation, we add both predictors and then add a third term that includes the interaction of the two predictors. The interaction can be calculated by multiplying the two predictors. The model then reads as follows:
Equation: Moderation
where y is the outcome value, a denotes the intercept, b1 is the slope of the first predictor, b2 is the slope of the second predictor, x1i is the ith value of the first predictor variable, x2i is the ith value of the second predictor variable and εi is the error.
In contrast, mediation occurs when the relationship between a predictor and outcome can be explained by a third variable – a mediator (Field, 2018). A mediator provides an explanation for why the relationship between a predictor and an outcome exists. Mediation analyses basically enable you to say ‘here’s why’. Such analyses are important because they help to uncover the processes and mechanisms by which a predictor influences an outcome (Agler and De Boeck, 2017; Rucker et al., 2011). This way, mediation analyses allow you to understand the behaviours and attitudes of your employees, customers or political allies on a more profound level. The better you understand your stakeholders, the better you can adjust to their needs and the more successful you are in gaining their support.
The following example should clarify the concept of mediation. Imagine that your HR team had discovered that the number of holidays taken by your staff was positively related to their job performance. The more holidays your employees have, the better their job performance (Figure 4.7). You may, of course, wonder why this positive relationship exists. The number of holidays presumably has a positive impact on job performance because employees perceive their work–life-balance to be very good. Figure 4.8 shows the respective mediation model. The model suggests that the relationship between holidays and job performance operates through an increase in the perceived work–life balance.
But how can we determine if a variable explains – that is, mediates – the relationship between two other variables? According to Baron and Kenny (1986), mediation can be identified based on a three-step approach. First, the starting point consists of testing for a significant relationship between the predictor and the outcome variable (e.g., number of holidays and job performance). Second, there must be a significant relationship between the predictor and the mediator variable. That means that the number of holidays must significantly influence people’s perceived work–life balance. And third, there must be a significant relationship between the mediator and the outcome variable (i.e., perceived work–life balance has to significantly predict job performance).
We speak of full mediation (or, complete mediation) when the predictor no longer shows a significant relationship with the outcome variable after the mediator variable has been entered into the analysis. In our example, this would imply that perceived work–life balance fully explains the effect of the number of holidays on job performance.
Partial mediation occurs when the predictor’s effect on the outcome is reduced but is still significant after the mediator variable has been added. The occurrence of partial mediation can be viewed as an indication that there might be further mediator variables. In our example, partial mediation would imply that perceived work–life balance is not the only mechanism explaining the relationship between the number of holidays and job performance and that we might want to include additional mediator variables in our analysis (e.g., we might have reason to assume that the relationship between number of holidays and job performance is also explained by happiness such that the more holidays people have, the happier they are and, as a result, the better they work).
Keep it short and simple – but not at the expense of clarity
We often find ourselves in situations where we have to simplify and leave out details. This may be because we are given time constraints for our presentations or space constraints for written reports. Such constraints can be helpful as they force you to focus on the things that really matter. Nonetheless, they may also seduce you to remove visualisations that are vital to a thorough understanding of your data analysis or your findings. You may think that certain charts take too much time to be explained or too much space in your report and that it is better to drop them. But be careful about removing visualisations – especially when presenting complex relationships. Moderation and mediation are abstract terms and without proper explanation your audience might easily get lost. If this happens, you will have a hard time maintaining their attention, let alone getting across your message. To avoid confusion and disinterest, it is often very helpful to present the conceptual model underlying your moderation and mediation analysis. Conceptual models offer an at-a-glance summary of the variables that you included in your analysis and visualise how they relate to each other. That means, conceptual models give your audience an immediate overview of what is the predictor, the moderator/mediator and the outcome. You can find examples of conceptual models in Figure 4.3 (Conceptual moderation model) and Figure 4.8 (Conceptual mediation model).
Moreover, if you intend to remove a visualisation, ask yourself: What is the value added of this visualisation? To what extent does this visualisation help my audience understand the relationship that I have investigated? Or to what extent does it enhance people’s understanding of my findings? What would happen if I omitted this visualisation? Could I have the same effect by just using words?
If you see that words cannot substitute the benefit of a visualisation, then keep it. You may also want to ask two of your colleagues (with similar levels of data literacy) for feedback: one of them receives the version with the diagram and the other receives the version with plain text. Ask them about their understanding of your analysis and check whether the inclusion of the visualisation led to greater comprehension and memorability of your findings.
Key take-aways
Successfully navigating the complex world we live in requires that we understand under which conditions certain effects occur (moderation) and what the key mechanisms driving these effects (mediation) are. Concepts of moderation and mediation are thus a ‘must know’ for anyone involved with data analysis. You may find the following questions helpful when dealing with moderation or mediation analysis.
- 1. Do you have reason to assume that the strength or direction of a relationship between a predictor and an outcome is influenced by a third variable (i.e., a moderator variable)? ➤ moderation analysis.
- 2.If you find no relationship between a predictor and an outcome variable: is it plausible to assume that there is a ‘hidden effect’ – that means, an effect which is contingent upon a third variable (i.e., a moderator variable)? ➤ moderation analysis.
- 3.Do you have reason to expect that the relationship between a predictor and an outcome is carried through another variable (i.e., a mediator variable)? ➤ mediation analysis.
Traps
Analytics traps
- ●Overlooking a ‘hidden effect’: wrongfully concluding that there is no effect when in fact a moderator variable influences the relationship between two things.
- ●Using the same strategy or approach for your target audience without realising that some audience members are more responsive to another strategy or approach (i.e., failure to consider moderator variables).
- ●Stating a relationship between two things without understanding the underlying explanatory mechanisms and processes (i.e., failure to capture and engage with mediator variables).
Communication traps
- ●Speaking of moderation or mediation without explaining what these concepts mean and what a moderator or mediator ‘does’.
- ●Removing visualisations that would substantially help your audience understand your data analysis and your findings.
Further resources
Find a helpful video on the difference between moderators and mediators at:
https://www.youtube.com/watch?v=WZr1jlKi_s0
To visualise moderation effects, go to this website and download the (free) Stats Tools Package and click on the ‘2 way interactions’ tab: