Introduction to Logistic Regression in Health Care Applications
By: Miguel Sotelo
Tags: Clinical Research, Logistic Regression, Healthcare, Mpirik Research, Machine Learning

Photo by Franki Chamaki on Unsplash

In clinical research, a common predictive model is Logistic Regression. This model has a wide range of applicability such as variable selection, patient classification and response prediction. Because of its prevalence in healthcare literature and use at Mpirik, we wanted to provide an introduction to Logistic Regression and share a few examples of its utility in healthcare.

Let’s start with the math. Logistic Regression takes a set of variables and assesses the relationship they have with a categorical response variable. The output is a coefficient weight for every input predictor. Below is the Logistic Regression equation.

The following is a description of how each variable relates with each other. Coefficients (β) are estimated and the weighted sum of the coefficients and the input variables (x) yield a representation of the probability of an event occurring (y = 1), for any given number of inputs (n). Thus, the log-odds of an event occurring over the event not occurring is equal to the weighted sum of the input predictors with weights equal to the estimated coefficients.

Let’s use an example to bring this model to life. In healthcare, response prediction is one of the main goals for logistic regression. It is useful to describe how the unit increase of an input predictor increases the odds of an event occurring. For instance,

What are the odds of re-hospitalization for heart failure in patients with Aortic Stenosis (AS) given that the aortic peak velocity (aopv) has increased by 1 m/s?

We can reorder the above logistic regression equation to answer this question. The resulting equation (below) reveals that the increase in the odds of hospitalization given an increase in aopv (represented as a ratio to the baseline odds) is equal to the exponential of the beta coefficient for aopv. For more detail, there are many blog posts or instructional guides regarding this formulation.

The exponential of the beta coefficient is the “odds ratio” (OR), a number that is typically reported in published manuscripts as a measure of the relative effect a predictor variable has on the odds of a response variable. For instance, if aopv has an OR of 2, then the odds of hospitalization is twice as high for a person with an aopv of 4 m/s compared to that of a person with an aopv of 3 m/s.

OR > 1 indicates an increased occurrence of the response variable given an increase in the predictor variable OR < 1 indicates a decreased occurrence of the response variable given an increase in the predictor variable.

Thus, the OR for aopv being greater than 1 indicates that as jet velocity increases, the odds of hospitalization also increases.

Photo by Enayet Raheem on Unsplash

There are other examples of logistic regression in recent literature that can help demonstrate the utility of this model. For instance, logistic regression was recently used to assess the predictive power of the ratio of the transmitral doppler early filling velocity to tissue doppler early diastolic mitral annular velocity (E/e’) on short term myocardial infarction (STEMI). Tai et al., (2020) found that elevated E/e’ was associated with a higher risk of STEMI (OR = 3.7).

Another example was a study in determining the severity of aortic stenosis in patients with low-gradient aortic stenosis. Sato et al., (2019) found that the time between the left ventricular and aortic systolic pressure peaks was associated with severe aortic valve calcification (OR = 1.02), suggesting that using this variable could add value in diagnosing severity of aortic stenosis.

Although there are many other modeling techniques that can predict a response variable (such as Neural Networks and clustering techniques), logistic regression has been widely adopted in clinical applications due to its interpretability and relatively high accuracy. At Mpirik, we have used logistic regression for a number of studies and applications. For example, we used logistic regression to estimate recommended thresholds of hemodynamic measurements, which was then incorporated into a decision tree. We also used logistic regression to inform multivariate models through variable selection and have used these models to predict hospitalization, death, and disease progression in patients with valvular heart disease.

Overall, logistic regression is a fantastic tool for understanding the relationship between predictor variables and a response variable. In my upcoming blog post, I will walk through a technical example on how we have used logistic regression to find an optimal threshold for a continuous variable to predict a categorical response.