 Interpretation of interaction effects, or what does the interaction coefficient actually mean, is it large or not, or requires some special consideration. Let's first take a look at why that's the case, and then how we actually interpret this kind of models. In HECMAS paper, they have lots of interactions here, and let's start with the interpretation of this first interaction coefficient, which is productivity and tenure. So we multiply productivity and tenure together, and we get the regression coefficient. We cannot interpret that set there is paribus, because interpreting set there is paribus, keeping all other factors constant, means that we would increase this interaction term while keeping the first order terms that form the interaction constant. So we can't say that our productivity and quality, their interaction results in certain increase of customer satisfaction holding productivity and tenure constant. So you can't do that, because you can't change this interaction, and at the same time hold on the tenure and productivity constant. So this varies only when tenure or productivity are both vary. So we can't interpret that directly. So how do we do that? And to do that we need to consider the definition of an interaction model. So what does that actually coefficient mean? And we can write an interaction model such as that one in a slightly different way. So we can write it in that way, and here we can see that the regression coefficient of X is actually not stable. It's not constant anymore. Instead it's beta 1 plus beta 2 times M. So the effect of X on Y depends on M. For example, the effects of productivity on customer satisfaction depends on tenure. We can't say that holding tenure constant, which it's some constant effect, because the effect varies with tenure. So we address this problem by calculating marginal effects or marginal predictions or simple slopes. So the idea of a simple slope is that you calculate the regression slope on different levels of the moderation variable. In here they had a gender and race of the physician as moderators. And they were interested in the quality attributes and their effects on satisfaction. So we calculate one regression line where we hold the gender variable at zero. And then we calculate one other regression line where we hold the gender variable at one, for example. And that allows us to plot the interactions. So we can see here that we have males versus females. So this is our physician quality for males leads to a strong increase in satisfaction for females. The simple slope, slope holding females constant, holding females at zero or one depending on how this variable is coded, results in no increase in customer satisfaction. There is also a bit weird result here. So we have this result that for non-wide physicians, physician productivity actually decreases satisfaction. My guess for this result is that this is a result of an outlier. They only had 13 non-wide physicians in the sample, so that's very little. So it could be that there is one observation that tilts this regression line down. We don't know because we don't have the data. Let's see how Heckman actually explained this effect. So they are first saying that this is a crossover effect. The crossover effect simply means that regression lines are going to different directions. So there is the line for whites goes up and the line for non-whites goes down. So that's a crossover. And then they know that there's an effect and they don't really explain it. So in an ideal case when you have that kind of a funny result, then you really have to go and check your data if there's something wrong with the data. And then you have to say that we check that it is not a data problem, rule out alternative explanations such as an outlier. And then you can say that we don't know why that happens. They just say that they don't really know why it happens without explaining how they try to do diagnostics. To understand what these plots quantify, let's calculate one plot ourselves. So let's do the plot for physician quality, the first of the four, and patient satisfaction. The relationship between customer physician quality and patient satisfaction depends on the physician gender. So physician gender here is the moderator, this is the independent variable, physician quality and satisfaction is the dependent variable. So we do the calculation. You can actually do this from published results. And if you have a paper that doesn't include this kind of interaction plots, and then interpretation of the results would be pretty difficult. So if you want to interpret the results, you can do this yourself in a few minutes when you know how to do it. So let's calculate based on model two. We have satisfaction is the intercept, a gender effect plus quality effect plus gender times quality. Normally we hold all other independent variables at their means, and in this case it means that everything else is zero, because they are standard as coefficients, the data are standardized, everything in standardization has mean of zero. So it's pretty simple to calculate the effects of satisfaction. So that's the equation for satisfaction, for fitted values of satisfaction, and how do we come, go from this equation to this plot? Well, we first have to define how we plot it. So what are the categories or values of the independent variable and the moderated variable that we are using? So we have to define some values for quality and gender. And let's start with gender. So we have gender coded zero and one, let's say one is female, zero is male, and we want to know compared to those two categories. The problem is that these coefficients are standardized. So it's not actually the zero and one coded variable that enters into the analysis, instead it is the standardized version. So we have to calculate our standardized version of the gender variable. So we call it Z-gender. Z or Z-score is typically used to refer to standardized variable. So we are standardized by eliminating the mean, subtracting the mean from gender, and dividing by standard deviation. We get these values from the correlations and descriptive statistics table, and we get the Z-scores for gender. Then we need to define what values we have for the quality variable. These are high and low conditions, are typically plus and minus ones standard deviation from the mean. So we set physical quality to either minus one or plus one. So we have here minus one for a man. So that's a bad man. This is a good male physician. This is a bad female physician, and this is a good female physician. Then we simply plug these values into that equation here, and that produces us the fitted values for satisfaction. So now we have these satisfaction values. Which go to Y. We have these X values. These are the quality values here, and we can do the plotting. So we plot four points. So this is the Y coordinate, and this is the X coordinate. We plot two points, and then we connect the points with lines. So we connect these two points for male with a line, and these two points for female with a line, and that gives us a plot like that. So we have here the points connected by a line, and if we put this plot right over the original plot, we can see that it's the same. So we can calculate the interaction plot ourselves, and this line here is the regression line for females, and this is the regression line for males, and we can see that it's quite substantially different. These significant stars are tests for simple slope. So they test whether this regression line is flat, whether the slope is zero. And we reject that hypothesis here. We can reject it for the female physician's line because it's so close to being just flat horizontal line. Let's take a look at deep houses paper that has a different kind of interaction model. So deep houses paper doesn't have a moderation model. Instead they have this curved effect model. So they have this u-shape model. And again, interpreting these are strategic deviation squared effect, and strategic deviation is nearly impossible without plotting how the line goes. So we need to take a look at the model 3 here, and we plug in the regression estimates to this model 3. So we have the model 3 equation here. Then we have the numbers here, the years that's intercept. Then we have regression coefficients here. And then we have means of these different variables. So mean of market series are 0.008, and total expense ratio is 0.085. So we hold everything else at their means. And then we have a strategic deviation squared. Then we plot calculate different values for relative ROA going from 0, which is the absolute minimum of strategic deviation all the way to about 30, which are inferred to be the range of strategic deviation based on their results. They don't report it, but this kind of range would make sense. And then we can see that the effect is u-shape goes up and then down. And if we just fit a linear model, the model 2, then it shows that all the observations are here. So we have some observations that are very low ROA, very little deviation. We have some observations that have very high deviation, very low ROA. So these observations here and here bend very much in line to be a curve. Most of the observations are probably somewhere around here, but again without having access to the raw data, we can't really say. So this is what they report in the paper. They say that they did an analysis of the estimated functional relationship and they found this inverted u-shape. So it's u upside down. And then they say that it is for the full range of dvs from 0 to 30. So the range here is important, because if we only look at the range from 0 to for example 10, then we would conclude that it's pretty linear, just slightly decreasing, but always increasing effect. Slightly, the increasing trend just slightly, it's slightly less here, but it's still increasing. So that's how we do dvs shapes. Summary of moderation, these are pretty simple to estimate and plot. So you first estimate the main effects model. It's not absolutely necessary, but it's a convention. So you just have the linear effect and no products whatsoever. Then we estimate model 2, which includes the interaction term and both lower level terms that you use to form the interaction. It is very important to have both x and m, if you have the product of x and m, because otherwise the product of x and m will be confounded with the omitted variable m, if you don't leave that into the model. Some people recommend that you center your variables before doing the interaction term. That's a bad idea. Don't center. I'll explain why in another video. Then you compare model 1 and model 2 using the F test. This is actually not necessary, because the significance test for the T test for the interaction tells you the exact same thing, but it's a very common practice to do so and it doesn't take much space to have the delta R squared, the R squared change and the F statistic in the Richardson table after the R squared values. Then you do an interaction plot and you calculate the marginal prediction. So how does the Richardson line go for one group? How does it go for other group? How does it go as a U-shape? How does it vary as a function of, for example, the amount of women in a profession? Then you interpret what do the results mean. Doing these plots is very useful for any kind of nonlinear models and even if you have a linear model, if you don't understand whether it's a strong effect or not, then plotting can help you.