When is least squares better than reduced major axis? Consider two linear regression methods:

*

*least squares regression (LSR)

*reduced major axis (RMA)

I know the definitions of both regression methods but I would like to know when is the LSR better than RMA. Does it depend only on properties  of the dataset (normality, variability, correlation) or there is something else?
 A: To start with, the  major theoretical advantage of RMA estimation is that it is symmetric with respect to the inputs --- i.e., if you exchange the response variable in the model with an explanatory variable, the estimated coefficients from RMA will remain consistent (in the sense that the predicted regression equations from the two resulting models are just rearrangements of one another).  This is a nice desideratum for the estimation method, and it is a useful theoretical advantage in some cases.  However, this advantage comes with a cost: it involves looking at the problem from a perspective that is neutral with respect to the status of the variables (as response/explanatory variables), which undercuts the nature of regression analysis as an analysis that only cares about the conditional distribution of the response.  It also comes with the corollary cost that the behaviour of the estimator is now more complicated, and more sensitive to variations in the explanatory variables.
The performance of both estimators will of course depend on the properties of the underlying data and will be affected by all aspects of the joint distribution of the data.  It is possible to conduct a comparative examination of the properties of the two estimators by deriving standard properties like their expectation, variance, etc., under a stipulated true joint distribution.  Depending on your assumptions, sometimes OLS will perform better and sometimes RMA will perform better.  While the results are heavily dependent on the assumed true model for the data, as a general rule, OLS estimation is generally more accurate if the true model is "close to" the assumed regression form.  OLS estimation is designed to minimised squared-error-loss, with errors measured wholely with respect to the response variable; it is generally quite a good estimator for the conditional expectation of the response variable.  RMA estimation is designed to minimised squared-error-loss, but with "errors" measured with respect to deviations of all variables from the multidimensional regression line; it is generally more sensitive to the explanatory variables and is often an inferior estimator of the conditional expectation of the response variable.
The fundamental issue here comes down to a choice of how to do regression analysis.  Regression analysis is used in a context where the object of interest is the conditional distribution of the response variable, conditional on the explanatory variables.  In many cases the variables have a fixed status (i.e., there is a particular response variable of interest) and so the goal is to infer the conditional distribution of the variable that is the designated response variable.  So, in this problem, do we only care about making good inferences of that object, or do we want some broader consistency properties to hold (e.g., that the analysis remains consistent if we change the status of the variables)?  Answering that question goes a long way to determining whether OLS or RMA is a preferable estimation method.
