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After reading this answer https://stats.stackexchange.com/a/239583/353716 and the article cited, I wonder if there are any situations where one would want to carry out an analysis where we try to explain a variable $y$ with only one independent variable $x_1$ ($y = \alpha x_1 + c$ ) if they have the possibility* to carry out a multivariable analysis $y = \alpha x_1 + \beta x_2 + ... + c$. I know everything isn't linear but for the sake of simplicity, let's only consider linear regressions for this question.

* by possibility I mean, because in the data available, we have $y, x_1, x_2, ...$ for all patients.

For example predicting the number of infections of a patient. Or the value of the concentration of a certain molecule in the patient's bloodstream. And the independent variables could be age, how long they exercise a day, etc ... And the question my analysis could try to answer is: does sex have an impact on the molecule's concentration (looking at the p_value of the coefficient could be useful)? If yes, what's the impact (looking at the value of the slope $\alpha$ could be useful)?

If not, why are articles still full of boxplots and stuff, which are univariate (can I say univariate here?) in essence?

Scientific papers studying this or intuitions are welcome!

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  • $\begingroup$ Could you please explain what you mean by "multivariate analysis" and give a more specific scope of intended applications? The thread you reference refers to multiple regression which might or might not be viewed as "multivariate" because it has a univariate response. It would also help to understand how any analysis involving more than one variable could be viewed as being either univariate or multivariate -- any good analysis will involve many different considerations and techniques, some of which might be univariate and others multivariate. $\endgroup$
    – whuber ♦
    Commented Jan 16, 2023 at 14:33
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    $\begingroup$ See for example B. Hildago and M. Goodman, "Multivariate or multivariable regression?" Am. J. Public Health 103:39-40 (2013) for a brief explanation of the terminology. When you edit the question to clarify that, please also say something about the type of investigation that you are referring to. For example, the answer might be a good deal different between a controlled laboratory experiment, a randomized clinical trial, and an observational study. Please edit the question directly, as comments are easy to overlook and can be deleted. $\endgroup$
    – EdM
    Commented Jan 16, 2023 at 14:40
  • $\begingroup$ I actually meant multivariable regression, sorry I misused the term multivariate, I wasn't aware of the difference. $\endgroup$
    – FluidMechanics Potential Flows
    Commented Jan 17, 2023 at 10:29
  • $\begingroup$ My setting would be something like $y = \alpha x_1 + \beta x_2 + ... +c$ where $y$ is a real number and so are $x_1, x_2$. For example predicting the number of infections of a patient. Or the value of the concentration of a certain molecule in the patient's bloodstream. And the independent variables could be age, how long they exercise a day, etc ... And my question was: is there any situation where making a linear regression of the form $y = \alpha x_1 + c$ (without $x_2$, $x_3$, etc ...) would be relevant? $\endgroup$
    – FluidMechanics Potential Flows
    Commented Jan 17, 2023 at 10:30
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    $\begingroup$ Terminology has shifted. Wide sense and often a few decades ago, multivariate meant looking at many variables, although in practice looking at say histograms of several variables wouldn't be called multivariate. Narrow sense and usually now multivariate means multiple responses. In some texts say 50 years ago, multivariate meant (almost) more difficult and requiring a computer, when computer access was not routine. So single-predictor regressions could be done with a calculator, but that was much harder for multiple regressions. $\endgroup$
    – Nick Cox
    Commented Jan 17, 2023 at 10:53

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  1. If you want to predict, there may be situations where for future observations you won't have all predictors that are in your current data set, and there may even be only one of them, in which case you'd need a univariate model. Observing certain variables may also be costly, so there may be good reason to not observe them even if in principle you could. (You may also be interested in a potential one-variable situation for interpretative reasons, even if when predicting in the future you won't be in that situation.)

  2. You may be in an essentially univariate situation in which either only one variable is really informative, or where all variables are so strongly correlated that every single variable is good enough to predict. (Variable selection may lead to a univariate model.)

  3. If you have very many variables, you could use forward selection, which starts from fitting every model with one of the variables, then takes the best and goes on adding variables - but univariate models are used in the first step. (This will often not be the best method for variable selection, but very occasionally it can be.)

  4. "Why are articles still full of boxplots and stuff?" Because the distributions of the individual variables are often interesting in their own right, to check model assumptions, diagnose outliers, make statements like 99% percent of babies weigh more than 2 kilo etc. Note by the way that, as opposed to boxplots, univariate regression actually looks at two variables, x and y. Papers and screen show 2-d information, so 2-d displays such as residuals vs. fitted are essential for diagnosing model assumptions and all kinds of exploratory data analysis also for higher dimensional data.

That said, item 1-3 refer to particular situations, and if you have several predictors, it is indeed not normally recommended to fit univariate regressions. (At least not if you intend to take them seriously... just out of curiosity, why not?)

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