Is standardizing in multiple linear regression necessary if underlying features already possess the same units?

Question

I want to preface this post by saying I have limited statistical experience. To explain my issue, I will propose a hypothetical example. Suppose my boss wants the company to increase its profits and asks me to choose between two different strategies: increase advertising or increase corporate training. So, I collected all of the company data I could find on other companies. Conveniently, I can find information on their yearly profits (in dollars), advertising spending (in dollars), and corporate training spending (in dollars). In my eyes, all I would need to do is run a multiple linear regression, where yearly profits are my dependent variable and my two feature variables are advertising spending and corporate training spending. In this case, since each feature variable is measured in dollars, we can interpret the feature coefficients as the change in profits expected as a result of a one-dollar increase in advertising or corporate training spending, respectively. So, based on this logic, I feel as if it were to be reasonable to directly compare the coefficient values to determine which has a stronger effect on profits.

While this logic seems sound to me, I notice common objections online to this strategy. Most sources indicate that you cannot simply compare the coefficients because there potentially exists a unit difference that could be manipulated; the solution is to convert the feature variables into unitless values using standardization and then compare the coefficients of the standardized regression. However, in my case, aren't the units of each coefficient the same? Therefore, should we not be able to compare directly? If not, why?

Answer 1

You are correct about the unit issue. Since everything is in dollars, there isn't a unit problem. Standardizing will simply change the units from dollars to standard deviations. If the levels of spending on advertising and training are very different, then the coefficients may change a lot. Here, it seems to me that leaving things as dollars makes more sense. I'm no expert on business, but I think they make decisions in dollars, not sd.

However, your idea of a multiple regression with only two independent variables is problematic. First, you will want some covariates (type of industry, size of company, and so on). Taking logs of all the variables may be helpful, especially if the companies are of very different sizes. Also, you may want to look at splines of the independent variables, as the relation to profit may be nonlinear (my intuition is that this is likely; advertising, at some point, may saturate the market).

Second, and more important, you have repeated measures and you will, therefore, you will violate the assumption of independent errors. There are ways to deal with this including multilevel models and generalized estimating equations.