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

It's fairly common that we have both interaction effects and nonlinear effects from log transformation in the same model. The interpretation of these effects is done by plotting as well and you need to take that into account when you construct the plot.

So let's first review the log transformation. The idea of log transformation is that we take a log of either our dependent variable or any of the independent variables and that changes the interpretation of that variable to relative units. For example, if we're saying that the prestige depends on a log of income then the interpretation of beta2 here would be how much prestige will increase if income increases 1% relative to the current level.

So we're talking about relative effects of income changes to prestige. We can also do it the other way around, so we have a log of income as the dependent variable we have prestige as the dependent variable. Then the interpretation would be how much income increases relative to the current level when prestige increases by one point.

So log transformation makes a lot of sense for certain kind of variables for example income. The raises that you get are usually in relative terms and if you think what's the utility of each additional euro it diminishes as your salary goes up so you need more raises. If you have a thousand euros per month then adding a 1000 more is a huge effect. If you have a 5,000 euros per month salary then increasing that by a thousand euros, it's a lot, but it's not as huge difference as for somebody who makes just a thousand euros per month.

So relative effects are done with log transformation. So how do you combine these with interaction effects? This is the model estimated with Stata. So we have prestige and women we have education, prestige and percentages women, we have income and log of income as the dependent variables. So we know this far that interpreting this model requires that you plot.

So you calculate what is the fitted value for prestige for income as a function of prestige holding education at the mean and comparing different levels of percentage woman. So we could calculate the marginal prediction of percentage woman is 0 50 and 100, holding education at the mean and varying the prestige.

The log transformation here complicates things a bit but not by much. So instead of calculating the predictions directly we calculate predictions using the exact same procedure and then we just take exponential of those predictions. So instead of predicting lines we predict a line and then we take an exponential of that line.

It looks like that. This is from Stata again, margins plot command and we have linear effects here and we have curvilinear effects here. So these are relative effects. We have the effect of increasing prestige on income for male-dominated professions and women-dominated the professions and here we have the same effects with lines. As you can see the interpretations are quite different. So here women get no income at all as a function of prestige or no increase at all.

Here they get a relative increase but the absolute increase is less than for men dominated professions. How do we know which one of these lines set of three lines fits the data best? We can do that by simply adding observations to this plot. So we can have plots like that and each circle here is one profession.

So we have prestige for that profession and we have income for that profession.The size of the circle presents the number of women. The smallest circles are no women in that profession, the largest circles are all women in that profession and here we can see that this set of lines explains the data a lot better because here, for example, there are no observations here. So we are extrapolating here so it doesn't really fit and these are way too up for this line and particularly if you look at the confidence intervals or prediction intervals.

Then we have here, we can see the prediction intervals here are large which means that some of the observations can be up here and also we have no observations here. So one way of ruling out outlier as an explanation for lines or assessing which set of lines explains the data better is to just plot the data and the lines in the same plot and that allows you to compare.