# Interpreting multiple linear regression models with layered effects

For voter research, I want to understand how people form a certain opinion on a hot topic, measured on a five step opinion scale Y. Prior theory suggests 10 independent variables, (refering to them as x) might explain Y. Number of observations is 250.

My goal is to understand which of these 10 independent variables have the greatest effect on Y.

The 10 independent variables are highly correlated with each other but not higher than r=0.7, mostly around r=0.4. In detail, correlating each x with Y and summing up the correlation coefficients brings a sum of 5. Putting all x into one model yields r=0.8 but makes me uncomfortable, because that will have real world effects divided between the x (right?), obscuring my real world understanding of Y. And because of prior theoretic research I think the independent variables are not just correlated by chance. I think they are layered effects. Like x1 and x2 directly influence Y, but x1 itself is influenced by x4 and x7. That model would make me much more comfortable, because it adds a lot more understanding to how Y is formed.

Questions: How do I find out this layered effects structure? My idea: For Layer 1, could I do a lasso with the 10 variables and increase lambda until only a few variables remain that make sense logically and by the prior research? Now assuming x1, x3 and x4 were selected for layer 1. Could I now run a lasso with x1 as the dependent variable and the other 7 sorted out variables as the independent variables to find out layer 2 and so on? Can this approach detect real world layered effects or is this just an arbitrary way to slice the model, guided by "expert knowledge"?

• Maybe you can also give me some keywords to search for this topic, because I am not sure which is the right terminology to use. Dec 25, 2022 at 12:52
• I don’t think lasso will give you the causes of outcomes. Having said that, a three way interaction may pick this up. Dec 25, 2022 at 19:47
• ok three way anova? Or do I have to search for causal interference? Like .. what is the right terminology for this layered effects structure? Is it path analysis? people.exeter.ac.uk/SEGLea/multvar2/pathanal.html Dec 26, 2022 at 9:55
• No, an interaction between three variables in regression/lasso. Dec 26, 2022 at 13:39
• The other approach would be to estimate a system of equations. Dec 26, 2022 at 15:05