If you turn a covariate into a outcome variable what would happen to the study?

Question

I have a question; I have mood as a covariate in one of my projects. It's an important covariate. But I had to remove covariates due to insufficient numbers, so I proposed turning mood into an outcome measure, and I was told that that would change the study completely. Can someone explain how and why?

The idea is IV is the testing condition, let's say condition a( Desktop) vs. condition b( VR) vs condition c( control), and were use mood as one covariate and memory recall as a DV. In this scenario, if I added the mood as a DV - would it not translate to observing the impact of each condition on Mood and memory?

Answer 1

One study proposes to use $p$ features, including mood, to predict memory. Another study proposes to use $p-1$ features, excluding mood, to predict mood and memory.

These are different questions being answered in different ways. Neither question is a bad question to ask, but they’re totally different.

Answer 2

A covariate is not, in common language, anything that "covaries" with another variable, although that might well be the case. In the context of linear modeling, a covariate means something similar to regressor or predictor or input variable.

Putting your analysis aside, here's a toy example: Suppose you measured smoking status versus IQ, a bivariate model might be written as:

$$ E[\text{IQ}] = \alpha_1 + \beta_1 \text{Smoking} $$

In this case the null hypothesis that Smoking and IQ are not associated is written $H_0: \beta_1 = 0$. This would be a bad analysis, as there are obvious confounders, and a randomized study is out of the question. Suppose we take "age" as a covariate. My model is then written:

$$ E[\text{IQ}] = \alpha_2 + \gamma_2 \text{Age} + \beta_2 \text{Smoking} $$.

The hypothesis is now written: $H_0: \beta_2 = 0$ that "IQ, adjusting for age, is not associated with smoking status". You see, it's a fundamentally different model, and a fundamentally different scientific question.

If you believed that "mood" stratified$^1$ recall, then removing mood from the model changes the primary hypothesis, regardless of whether you separately analyze it as an outcome.

Another situation where this arises has to do with mediation. In a mediation model, a mediator is a variable that is caused by the condition, and in turns causes the outcome. The condition may have residual causation of the outcome as well. Again the question is fundamentally different, if the mediator M is omitted from the model, then your bivariate association represents the "total effect" of an exposure X, directly, and indirectly through M. If M is included in the model, then the X,Y association measures only the direct effect, because adjusting for M "blocks" its effect on the Y.

$1$: PS - Stratify may in some cases be used to describe covariate that are *not associated with the main exposure or predictor in the model, in this case your variable: condition. If condition is, for instance, randomly assigned, the choice to adjust for mood as a stratification variable may not change the hypothesis, but rather the inference, although there are untestable and unnecessary assumptions about, for instance, the homogeneity of effect and lack of interaction that are best put aside to favor the "strict" interpretation of the hypothesis, such as in the case 2 I presented.