TU-L0022 - Statistical Research Methods D, Lecture, 25.10.2022-29.3.2023

Interaction models allow us to study the effect of a third variable on the strength of the relationship between two other variables or alternatively nonlinear effects of one variable on another, where the effect first goes up and then goes down, or vice versa. So why are these kinds of models interesting? This is the normal way of drawing a moderation model. And we have the M here, which influences the strength of the relationship between X and Y. And this kind of model allows us to answer the question under which conditions the effect of X and Y works, and under which conditions it does not.

So let's say X is the amount of weights that you lift through the week, how many times a week you go to the gym, and Y is your weight gain, so how much muscle mass you gain. That relationship could be moderated by the amount of food that you eat. If you eat a lot while at the same time going to the gym, then you will gain muscle mass. If you go to a gym a lot and you don't eat much, then there is no muscle gain. So you need both, and the effect of one variable depends on the presence of another variable, on the third variable. So these models allow us to study the effects of context and the effects of two variables influencing the dependent variable together.

Moderation models come in two typical variants that are both presented in the Deephouse's paper and in the Hekman's paper. Let's start with the Hekman paper. So the Hekman paper has a pretty traditional moderation hypothesis. They're saying that the relationship between customer performance, sorry, the provider performance, and customer satisfaction depends on the provider's race. So for example minorities are rewarded less for their good performance, than whites, in this particular scenario. So that's the traditional case of moderation effect, you have a third variable, called the moderator, which influences the relationship between these two variables.

Then we also have this another type of interaction effect, called a U-shaped effect.  And Deephouse says that there's a curvilinear concave down relationship, which basically means that the effect of strategic deviation on ROA is first positive, but once you get too deviant, then it starts to go down, so it's negative. So it's positive first then it turns negative, so it looks like a U, that is drawn upside down. Why this is an interaction effect is because the effect of strategic deviation on ROA depends on itself, so that initially it's positive, but when strategic deviation value increases, then this relationship turns negative. So that's a way of making U-shaped effects using interactions.

A typical way of drawing these models is to draw boxes and then you have an arrow, which presents a causal relationship, or a regression relationship, and then you have these arrows from the third box, that go to the middle of this arrow. And this particular paper studies the effect of service provider performance on rating, and then there is a customer gender/racial bias, that acts as a moderator for this relationship. So the strength of these relationships depends on the customers' possible bias against the service provider.

How we estimate these kinds of models can be understood by writing the model in this kind of form. So we're saying here that the effect of X has some base value beta1, and then it depends also on the value of M, so it's beta1 + beta2m. If beta2 is a large number, then it means that the M has a strong moderating effect, if it's a value that is close to zero, then there is no moderation effect. We can't estimate that kind of model directly in the regression analysis, but if you write it differently then we can. So we can rewrite it without the parentheses, and it becomes beta0 + beta1x + beta2mx + beta3m. So the idea of, how we estimate these kinds of moderation models is that, we multiply the moderator and the interesting variable together, and then we add all two variables and their product as independent variables to the regression analysis. Here this equation shows that the effect of X on Y is no longer constant. So it's not a constant effect like we had in a regression analysis because it depends on the value of M.

And to understand, how we interpret these effects that are not constant but depend on another variable, we need to introduce the concept of marginal effect. So the marginal effect is the idea that the effect of one variable on another depends on other variables, and it's constant at a certain point but it can vary between points. So let's take a look at regression analysis example. So normal regression analysis gives you a line, and the marginal effect is, how much Y changes, when X changes a little. So it's a derivative or a tangent for this line. And because this is a line, the derivative or the direction of the line is always constant. And the marginal effect is a line. So marginal effect is, how much Y changes when X changes a little, or a very small amount at a particular point. When we have nonlinear effects, for example a log-transformed dependent variable, then the marginal effect is no longer constant. We can see here that the direction of the line is different, it goes up but less strongly as it goes here. So the regression line here, if we draw it here, then it's much steeper than here. So the marginal effect for nonlinear effects depends on, which part of the curve we are looking at. And typically when we do interactions, we are interested in interpreting the marginal effects.