Modelling explanatory variables which depend on each other

Question

I'm trying to estimate the value of an apartment, by doing a regression through similar apartments. The regression model looks now like this lmRob(price ~ ., data = data), but this is clearly wrong.

My problem is modelling / defining explanatory variables which depend on each other. For example elevator and floor number depend on each other. If the apartment is at ground floor, the existence of a elevator is irrelevant. But if the floor number is higher, the importance of the existence of the elevator is also higher. How could I model this in my regression model?

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[UPDATE]

I also don't want a model that requires an apartment which is located at ground floor and an elevator exists to be equally valuable as a similar apartment also located at ground floor but without elevator. Instead, what I want is to have a variable which substitutes the two variables floor number and elevator and in which the importance of an elevator depending on the floor number is shown.

Answer 1

Some commenters wanted to close this because they thought it is a duplicate of What is the effect of having correlated predictors in a multiple regression model? but that is not entirely correct. Even if there is little correlation (or colinearity) in your data, there is the problem that whether the building has an elevator or not does not seem to be relevant for the buyer who is not going to use the elevator, because he is living on the ground floor!

There are other problems: it does seem strange to use floor number as variable in a linear regression, for why should value depend linearly on floor number? If floor 0, no need to use stairs, if floor 1, you need stairs but still burglars can reach the windows, if higher not much difference, but maybe from the highest floors there is better sun and sight? So no reason to expect a linear effect. You try try a quadratic, or even use a spline ...

Or you could compare with results from tree modelling ... but for now, lets see some options for a linear model. Some commenters argue for an interaction between floor and elevator, we write that floor*elevator (which includes separate terms for each variable). Then you can see if the effect of floor is different when there is an elevator or not. Or you could use a quadratic model for floor, then you need to cross each term with elevator, so both the linear and quadratic part in floor can have effects differing if elevator or not. Or you could build some new discrete variable with categories depending on both variables floor and elevator. Such a variable could be informed from results of a tree modelling exercise. If you really think elevator is completely irrelevant if on the ground floor, have a look at How do you deal with "nested" variables in a regression model?

Answer 2

You can create another variable:

new_elevator_var = num_of_elevators * floor_number  

This will reflect the presence and importance of elevator(s). It will be 0 even if there are many elevators but floor number is 0 and will increase as the number of floors increase.

Answer 3

^{This answer purely shows how to add a linear term. It is also interesting to model this different. For example an isotonically increasing function of floor number, a spline, or a decision tree model}

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Asside from defining variables manually, you can do this by introducing an interaction effect in the model, and drop the main effect.

This will cause the influence/effect of elevator to be zero when the the floor level is zero as well.

The example below demonstrates this by showing a linear model with an included interaction where the main effect is left out.

set.seed(1)
n = 50
elevator = sample(0:1, n, replace = T)
floor = sample(0:10, n, replace = T)
z = rnorm(n)

mod = lm(z ~ 1 + floor + elevator:floor)
plot(floor, predict(mod), col = 1+elevator, 
     main = "lm(z ~ 1 + floor + elevator:floor)", cex.main = 1)
legend(0,0.6,c("without elevator", "with elevator"), col = c(1,2), cex = 0.7, pch = 1)

mod = lm(z ~ 1 + elevator + floor + elevator:floor)
plot(floor, predict(mod), col = 1+elevator, 
     main = "lm(z ~ 1 + elevator + floor + elevator:floor)", cex.main = 1)
legend(0,0.6,c("without elevator", "with elevator"), col = c(1,2), cex = 0.7, pch = 1)