Logistic regression using a predictor that's part of the outcome Say I have 2 continuous variables measuring the same thing (e.g., at-home blood pressure monitor and in-office blood pressure cuff with in-office measurements being the gold standard). At a cut-off of BP > 130, we would consider a person hypertensive. However, because in-office measurements are more troublesome to do, we want to see how often at-home blood pressure monitors misclassify hypertension.
So, my outcome is if both measurements agree at the threshold of 130 (if one is <130 and the other is >130, then the outcome = 1 (disagree) and if both are > 130 or both are < 130, then the outcome = 0(agree)). I'm interested in seeing if there are certain values of at-home blood pressure measurements where the odds of disagreement are higher. Would it be appropriate to run a logistic regression where outcome is predicted by at-home blood pressure measurement? (i.e., logit(outcome) = at-home blood pressure measurement)
I'm a bit unsure since it seems like we'd be using a variable that is part of the outcome as a predictor.
 A: What you need to do is to evaluate the calibration of at-home versus in-office blood pressure (BP) monitoring. Forget the cutoff.
A systolic BP value > 130 mmHg has been used as a cutoff for defining "hypertension," because that's the lowest value at which the risk of cardiovascular events has historically been distinguished statistically from the risk when systolic BP is at a reference level below 100. But the risk rises continuously with BP.
You can see that in the recent study by Ji et al, Sex Differences in Blood Pressure Associations With Cardiovascular Outcomes, Circulation 2021;143:761–763. The authors display hazard ratios (HR) for 4 different cardiovascular events broken down both by sex and by 10-mmHg systolic BP bins from 100 through 159, with a final bin for $\ge$ 160 mmHg. Point estimates of hazard ratios and event incidence per person-year rise continuously with systolic BP.
Their results for "Cardiovascular disease" in men show that the hazard is statistically distinguishable from that of the baseline starting with the group between 130-139 mmHg. If you wish to use 130 mmHg as a cutoff for "hypertension" based on that, consider the following. For women, you can see a "statistically significant" difference from baseline for "Cardiovascular disease" even in the 100-109 mmHg group! So your cutoff should be 100 mmHg. For men, a cutoff chosen that way for Myocardial Infarction or Stroke would be at 150 mmHg instead.
Furthermore, systolic BP is only one of many criteria used to evaluate the risk of cardiovascular events. The American College of Cardiology has a Risk Estimator that includes age, sex, race, both systolic and diastolic BP, cholesterol (all of total, HDL, and LDL), histories of diabetes and smoking, and whether an individual is on therapy with aspirin, a statin, or an anti-hypertensive. A clinician basing therapy choices and patient counseling solely on systolic BP isn't serving his patients well.
So focus on the BP calibration curve itself, plotting for example values for home measurements against those for office measurements for individuals. You might also need to take into account things like time of day or even day of the week.* What you should care about are things like the mean-squared error between the types of determinations, and whether there are any systematic changes in error as a function of BP or other conditions of measurement.
What might be best would be to have patients bring their home devices with them to the clinic and do direct paired comparisons under identical conditions. Then do a Tukey mean-difference/Bland-Altman plot, which doesn't require either method to be a "gold standard," to evaluate overall agreement and any systematic differences as a function of BP.

*With my home monitor, I find the highest BP on afternoons when I am working.
