Left-hand & right-hand side nomenclature in regression models

$$y_i = \beta_{0} + \beta_{1}x_{1i} + \varepsilon_{0i}$$

The language to describe regression models, such as the very simple linear regression specified above often varies and such variations often carry subtle shifts in meanings. For example, the part of the model on the left-hand side of the equation may be termed (among others I am ignorant of) with connotations and denotations in parentheses:

• Dependent variable (hints at causal dependence)
• Predicted variable (implies the model forecasts/makes predictions)
• Response variable (implies causality, or at least temporal sequencing)
• Outcome variable (implies causality)

The variation in nomenclature is also true on the right-hand side of the equation (same disclaimer that I am an ignoramus about other terms):

• Independent variable (implies causal priority, hints at experimental design)
• Predictor variable (implies forecasts, implies that the variable has a non-zero parameter estimate associated with it)

In the course of proposing vetting, or communicating research I have had occasion to not only be called on the use of one term or another, but to subsequently be called on the term I chose to replace it with. While the people calling were of course being pedantic (NB: I am a professional pedant, so I sympathize), because of course we all understood what was being communicated, I still wonder:

Are there commonly used terms for the left-hand and right hand variables in regression models that are agnostic with respect to (a) the external uses of the model, (b) causal relationships between the variables, and (c) aspects of the study designs used to produce the variables themselves?

NB: I am not asking about the important issues of proper modeling and proper interpretation (i.e. I care very much about causality, study design, etc.), but am more interested in a language for talking about such models generally.

(I realize that "left-hand variables" and "right-hand variables" might, I suppose, be construed as a credible answer, but these terms seem clunky... maybe this is a clunky question. :)

– Carl
Nov 18, 2016 at 6:55
• I think the short answer is no. This is for good reason in my opinion. In formal instances, language used to identify variables should be nuanced enough to imply a distinguished interpretation within a models intended application/domain (i.e. it is very important to know whether or not causality is implied in a regression model and proper use of nomenclature will help with this). Nov 18, 2016 at 18:09
• @ZacharyBlumenfeld (a) Don't answer in comments. :) (b) And yet, we talk about "regression" itself in general terms, without recourse to study design, disciplinary knowledge domains, etc. (e.g. plenty of folks speak and write about the least squares estimator without invoking study design, causality, etc.). If we have an application-agnostic language to describe a broad class of statistical endeavor, why is there not a similarly agnostic language for the components of such endeavors? Nov 18, 2016 at 18:18
• Not sure if this is worth to be considered an answer, so I am posting it as a comment: Maybe there is some terminology coming from projections (as $X(X'X)^{-1}X'y$ is a projection of $y$ onto the $X$ space)? Like projectors and projectand (I am making these up now as I don't remember the relevant terms). That terminology should be free of your (a), (b) and (c). Nov 18, 2016 at 20:09
• @Kenji I agree wholeheartedly with most of your perspective. However, I do not agree that one can/ought only talk about regression equations in an applied case: we should have a language that can talk about left and right hand variables of all regression models, for example, when examining the application of such methods on a meta level across disciplines. Feb 4, 2017 at 23:02

This is an excellent question. Actually, it is so good that there are no answer to it. To the best of my knowledge, there are no true "agnostic" term for describing Y.

In my experience and readings, I found that the semantic is domain-specific and also model-objective-specific.

Econometricians will use the Dependent variable terms when building a model that is explanatory. They may use the terms Predicted or Fitted or Estimated variable when they are building a forecasting model that is more focused on accurate estimation/prediction than on theoretical explanatory power.

The Big Data/Deep Learning crowd uses a completely different language. And, they will typically use the terms Response variable or Target variable. Their models are such black boxes that they typically do not attempt to explain a phenomenon as rather to predict it and estimate it accurately. But, somehow they wouldn't be caught using the term Predicted. They far prefer the terms Response or Target.

I am less familiar with the term Outcome variable. It may be prevalent in other areas I am less exposed to such as social sciences including psychology, medicine, clinical trials, epidemiology.

In view of the above, I could not provide you with any "agnostic" semantic for describing Y. Instead, I provided a bit of information on what semantic to use when catering to different audience and also reflecting the objective of your model. In summary, I don't think anyone gets hurt if you talk about Dependent variable with econometricians and Response or Target variable with Deep Learning types. Hopefully, you can separate those crowds apart otherwise you could have a verbal food fight on your hand.

• I wish I could give you an extra up vote for "verbal food fight" :D Feb 6, 2018 at 21:06
• +1 "The Big Data/Deep Learning crowd" use features for the right hand side and to me as a non-native English speaker that sounds pretty agnostic as well. Mar 14 at 14:39
• @Bernhard Thank you for that illuminating comment! I am not in the BD/DL crowd so that is new to me. As an English-native speaker, "features" does not speak to me because it is not at all clear why left hand variables would not also be labeled "features" and also because I have never encountered the word in a regression context. But I will start to look for it now. :D Mar 14 at 17:08