# Two variables measure the same thing, are they confounding?

There is an unpublished paper on how upper trunk rotation strength at a base line date correlates with injuries obtained during a full year of play in a specific sport. The study population consist of teenagers of one gender. The linear regression is not given in the paper so I can't provide the algorithm used unfortunately.

They claim to test age, weight and posterior chain strength as confounding factors.

My question is what happens if you try something to be a confounder if that variable is already present? In this case I suspect trunk rotation strength and general strength has a near perfect positive correlation. It's also likely that age is closely correlated with trunk rotation strength, most people aged 19 will produce more force in the specified exercise than do people aged 13.

Would you see general strength as a confounder in this scenario if you tried them?

$$\newcommand{\op}{\operatorname{#1}}$$ Let's use some abbreviations as follows: \begin{align*} \op{UTRS}&=\text{Upper Trunk Rotation Strength}\\ \op{I}&=\text{injuries}\\ \op{A}&=\text{Age}\\ \op{W}&=\text{Weight}\\ \op{PCS}&=\text{Posterior Chain Strength}\\ \op{GS}&=\text{General Strength}. \end{align*} Now the basic model under consideration is $$\op{UTRS}\to\op{I}.$$ As in, via causal diagrams, we are considering $$\op{UTRS}$$ as the cause of $$\op{I}.$$ Suppose, like the paper, we are considering age as a confounder; that would require something like this: While I could certainly buy the arrow from $$\op{A}\to\op{UTRS},$$ it's not at all clear, given the age range in question of teenagers, why $$\op{A}\to\op{I},$$ unless you are considering that older teenagers are more experienced, and less likely to get an injury. That's a possibility. Similarly, we are right to question $$\op{W}$$ as a similar confounder. But here, the question would be, why does $$\op{W}$$ influence $$\op{UTRS}?$$ What would be the causal mechanism there?

The reason I bring up these questions is that, depending on your causal model, sometimes you should adjust or condition for variables because they confound the results - that is, they produce a backdoor path such as in my diagram above, where $$\op{UTRS}\leftarrow\op{A}\to\op{I}$$ is a backdoor path. That is a true confounder. But what if, e.g., with weight, the arrow goes the other way? Like this: If this is the case, and I could imagine a scenario where it would be, it's incorrect to condition on $$\op{W},$$ because it would obscure the total causal effect of $$\op{UTRS}$$ on $$\op{I}$$ - precisely that portion mediated through $$\op{W}.$$

So the answer to your first question, as to what happens when you try something to be a confounder if it is already present, depends on the directions of the arrows. What causes what? In the first diagram, you should condition for $$\op{A}$$ because it is a confounder (assuming the model is correct, which it might not be). In the second diagram, you should NOT condition for $$\op{W},$$ because it is not a confounder but a mediator (assuming the model is correct, which it might not be).

The answer to your second question, as to whether general strength is a confounder, also depends on the arrow directions, and whether, if you want to include general strength in your model, it sets up a backdoor path or not. That's always the question: can you shut down all the backdoor paths? If so, and the variables you need are all measured, then you can identify the causal effect of $$\op{UTRS}$$ on $$\op{I}.$$ Otherwise not.

My answer here, you must understand, is more about pointing you to a framework in which you can answer your own questions, more than answering them myself. There are clarifications necessary in your problem before you can give a final answer.