# Logistic vs. linear regression for "inherently continous" variable - comparing probability

This is a situation that arises commonly in my area (medicine).

• Suppose there is an inherently continuous variable $$y$$
• Suppose there is some normal range for this variable, say 80 - 120
• Suppose there is a dichotomous categorization of $$y$$ as "within normal limits", and "outside normal limits" that are commonly used

This sort of setting is exceptionally common.

Now, let's say you wanted to build a prediction model for the patient's categorical status of $$y$$:

$$y_{status} = f(x_1, x_2, ..., x_k)$$, where $$x_i$$ are various measures like age, body weight, smoking status, ...

The usual approach seems to be to use logistic regression. However, it seems you could also predict the blood pressure as a continuous variable, say using a Bayesian approach producing a posterior distirbution, then estimate how much of the posterior is within or beyond the normal limits.

The two approaches yield (at least slightly) different answers, in my experience. Also, I basically never see the latter (Bayesian) approach - which suggests to me there is something fundamentally wrong with this idea. However, I can't understand where the logic is incorrect.

Any guidance on this would be greatly appreciated.

Turning a continuous variable into a binary variable is almost always a bad idea. It has been termed dichotomization. Frank Harrell (@FrankHarrell)explains and gives some examples. Steven Senn gave it the name dichotomania: Dichotomania: an obsessive-compulsive disorder that is badly affecting the quality of analysis of pharmaceutical trials

Basically, the problem is that information is lost when you convert a continuous reading into a binary normal vs. not normal. Worse, it can encourage investigators to try different cutoffs for defining the two groups, and then only publish the results with the cutoff value that gives the results closest to what they were hoping for.

• Thank you for the reply. It seems to support my notion that examining the Bayesian posterior is a legitimate (possibly superior?) approach. Nevertheless, the overwhelming preponderance of examples in the literature exhibit the dichotomania you refer to. If this Bayesian/Posterior approach is legitimate (maybe even preferable) - I wonder why virtually every stats book I have seen - including relatively advanced ones from Bayesians (e.g., Regression and other stories by Gelman et al.) never mention this - but, instead, stick to the logistic regression model. Feb 28 at 18:05
• Although I agree regarding dichotomisation of continuous variables, I believe the true issues are deeper than just information loss. Statisticians discard information all the time: they perform PCA, discard variables etc. I believe to have read somewhere that statistics is essentially a (lossy) data compression technique: Summarise large amounts of data into a handful of numbers easy to interpret. So the problem of dichotomisation is not information loss by itself; it is about which information is discarded, why, and how. Mar 1 at 8:13

Medical doctor and researcher here. I have often thought about this as well. I believe there are 2 main reasons, but ofcourse these are personal opinions and open to discussion:

First, the medical world, including doctors and patients, are mainly interested in whether or not the tests are normal. As a basic example, doctors are not usually interested if the fasting blood sugar (FBS) of a patient is 80 or 85. While this 5 units of change may be statistically significant in the appropriate context, this will in no way change how the doctor treats the said patient. Thus, statistical analyses that predict an average of 5 units of increase in FBS are of secondary importance to doctors and policy makers, the actual audience of medical papers, compared to a predicted 1.5 odds of higher than normal FBS values which may have implication for the diagnosis of diabetes.

As a side-note, I believe this is mainly because of the dichotomized nature of medical education and guidelines on how to treat patients. Although this is gradually changing and doctors are paying more attention to the spectrum of diseases rather than absolute states, we still have a long way to go.

Second, as the established medical literature is accustomed to dichotomized outcomes, future studies are more likely to go that way as well. Its not that the Bayesian and continuous approaches are wrong, in fact, in my experience, they provide more reasonable results. However, as researchers read the previous literature, they are more likely to use the already well established analyses as well, creating an unending cycle of dichotomized outcomes. Nevertheless, I am seeing more and more of Bayesian analyses recently. While it maybe a confirmation bias on my part, I am hopeful.

• Your second reason is probably correct, but it suggests that medical research is kind of a cargo cult: Everyone else is doing it, they get their papers published, so we need to do it, too. But I don't think this attitude is limited to medicine. It took computer security some 40 years or so to stop forcing users to periodically change their passwords. Feb 28 at 19:16