Background
I have a colleague interested in a particular disease and specifically if the continuous variable $X$ is different between controls and patients. Preliminary results suggest that patients have higher values of $X$ than controls.
The straightforward approach would be a unpaired t-test to test if there is a difference in mean values between patients and controls.
However, the literature suggests that age and sex are correlated with $X$ so my colleague wanted to control for this when testing. Hence, she decided to use regression to solve the problem.
Problem
My colleague's supervisor reasons that since they do not know if it is the disease that causes raised $X$ or if raised $X$ causes the disease they could use either linear regression or logistic regression to solve the problem. The supervisor also argued that logistic regression was the preferable approach.
If $d$ is a categorical variable with levels {control, disease} then the first model could be written
$$X = \beta_0 + \beta_1 d + \beta_2 \text{age} + \beta_2\text{sex} + \epsilon$$
Where the interpretation was that there is a correlation between disease and $X$ if $\beta_1$ was found significant.
and the second model
$$\log(\text{Odds}(d=\text{disease})) = \beta_0 + \beta_1 X + \beta_2\text{age} + \beta_2\text{sex} + \epsilon$$
where the interpretation was that there is a correlation between disease and $X$ if $\beta_1$ was found significant.
What is the opinion among the experts on cross validated? Are there other methods?
comments
The patient and control groups were not perfectly matched and my colleague wants to make sure that she has controlled for both sex and age to avoid upsetting the reviewers. I do not know if there were any significant differences between the sex and age distributions between patients and controls.
There were also indications that the variance in $X$ was different in the two groups and questions were asked if this would influence the regression models.
My personal opinion is that because my colleague is interested in the conditional expected value $E(X|d,\text{age},\text{sex})$ she should use the ordinary regression model or just a t-test (if there are no age or sex differences between groups).