Use of linear regression or logistic regression when testing conditioned group differences

Question

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).

Answer 1

I say you are right. You should use OLS regression here, not logistic regression.

The question of causality is a red herring. Causality is not required for either linear regression or logistic regression, and it is fine to model a cause as a (e.g., linear) function of an effect. In fact, there are predictive models that do so. As an example, researchers studying the collapse of the Mayan civilization have hypothesized that drought may have initiated its decline. Predictive models have been built that allow researchers to make an educated guess about rainfall levels (i.e., causes) from traces that remain (i.e., effects; e.g., analyses of core samples from lake beds), to clarify this possibility.

Which variable should be made the response variable, and which the explanatory variable should be decided based on the question you want to answer. It is clear from your setup that you are wondering about possible differences in the level of $X$ given the disease state. Thus, $X$ should be the response variable, and disease state should be the explanatory variable.

Answer 2

I agree with your supervisor, logistic regression is the better choice. The continuous predictor allows you to make a more nuanced probabilistic statement about the disease relationship rather than the more blunt categorization statement of the relationship resulting from using disease as the predictor.

You did word your initial question about whether X was different between controls and patients but it isn't clear that future use of the information might better be served by being able to make statements about the probability of disease given the value of X. I'm trusting that your supervisor has better knowledge of the more general use of the findings and therefore would offer more appropriate advice than any of us can here.

It gets a bit tricky looking at your covariates of age and sex when they're primarily supposed to be related to X. My guess is they're related to both X and disease, in which case you really need to treat them not as covariates, that are generally ignored, but as fully analyzed predictors. You'll need to know the correlations among your predictors and how they influence your model.