My client would like to know which, of many different kinds of product dimensions, are most important for perceived quality. Some of the dimensions are as follows: price, material, country of origin, whether or not your colleagues/friends have the product, and so on. Most of these dimensions include many levels (price, for example, can be from $1 - $1000; material can include steel, plastic, and so on; country of origin includes 10 possible countries).
We want to be able to answer general questions like: which of these factors are most important in predicting customer appraisals of quality? My client has created a survey in which a random sample of, say, three of these variables are shown to a customer (e.g. a 100 dollar, plastic, widget from China; a 500 dollar metal drum from Cambodia, etc.), and the customer rates how high they perceive the quality to be.
Coming from an econometric background, it's not clear how to answer this kind of question using my normal tools. There seems to be an enormous number of combinations of variables, and the potential for countless interaction effects seems overwhelming to interpret.
I've come across literature on conjoint analysis and taguchi methods, which seem relevant, but the articles I've found seem to broadly describe design principles, rather than the mechanics of the analytical strategy.
Random forests with variable importance seems promising, but it's not clear how to recover regression-esque effect sizes from forests, nor is it clear how to get a sense of which interactions are most relevant.
Perhaps some kind of Lasso regression? Would I fully specify all the interactions, and run a Lasso procedure? I'm worried it may select non-sensical interactions.
Apologies if the question is poorly specified. I'd like to be able to say, "If your product is coming from China, characteristics X, Y, Z are most important. If you are selling a plastic tool, characteristics A, B, C, are most important."