Interpretation of linear regression where the outcome can only possibly cause the value of predictors and not viceversa

Question

We want the verify the association between artificial pregnancy (artificial insemination) versus natural pregnancy and a series of pregnancy conditions (hypertensions, diabetes, etc).

We first verified the effect of the reproduction type adjusted for subject confounders like age, BMI on each single pregnancy condition, producing a series of regressions.

Regression 1:

$$condition_1\ \tilde\ \ preg\_type + age + BMI\\ condition_2\ \tilde\ \ preg\_type + age + BMI\\ ...\\ condition_n\ \tilde\ \ preg\_type + age + BMI\\$$

We thought that we could actually do the opposite. That is, using a logistic model we assess the independent effect of having one those conditions (modeled as covariates, together with the aforementioned confounders) on the type of pregnancy.

Regression 2:

$$preg\_type\ \tilde\ \ age + BMI + condition_1 +\ ...\ + condition_n$$ This would allow to identify the independent effect of each condition. Obviously we know that these conditions cannot "cause" the use of an artificial reproduction techniques. But regression methods assesses only association, not causation, so conceptually the model should be valid.

Our questions are:

• Which is the difference in interpretation between the two kind of analysis? Which questions do they answer?
• Is regression 2 a proper analysis design or it's basically flawed?
• Is it a valid interpretation stating that using the conditions as covariates would provide insight regarding the independent association between each condition and the type of pregnancy?

Answer 1

preg_type  ~ age+BMI+condition1+ ... +conditionn

The second regression that you have can be used for cases where you don’t have any information about whether the pregnancy was natural or artificial. You are trying to make guess about the pregnancy using the conditions. It’s totally a different question here. You would build the model with the data in hand, and then you would predict the type of pregnancy in cases where you only have info on conditions (not pregnancy type).

Let’s say dependent variable takes the value of 1 when the pregnancy is natural and zero otherwise. Here is how I would interpret the coefficients in the second regression

Coefficient on Condition 1 is the odds (log odds) that the pregnancy was natural given someone have Condition 1.

Similarly, coefficient on Condition 2 is the odds that the pregnancy was normal given someone have condition 2.

Answer 2

While it is certainly true that regression does not assess causation, I see two problems with your second model.

First, switching the types of variables (dependent and independent; outcome and predictor; whatever names you want to give them) leads to conceptual confusion along the causation line. To take a simple linear regression example, we regress weight~height and not the other way round, because height "causes" weight (I put cause in quotations because it's not exactly causal, but I don't know of a good word for what is going on here).

Second, your second model is clearly an attempt to make one equation instead of several. But ... the model is now controlling for the "effect" of the other variables. I don't see how this makes sense in this situation.