Statistical notes for clinical researchers: logistic regression

Abstract

Logistic regression is a regression model where the dependent variable is categorical and corresponding independent variables can be categorical or continuous. This article covers the case of a binary dependent variable such as an event occurring coded 1 = 'event' and 0 = 'no event'. Frequent outcomes are pass/fail, win/lose, disease/no disease, etc. The logistic regression model estimates the probability that an event occurs versus the probability that the event does not occur. An example: score and pass data Let's say that an institution performed an assessment procedure to determine pass and fail of the participants considering exam scores, interview result, and reputation among colleagues. Table 1 shows a data with 2 variables, exam scores and pass state (1 = pass, 0 = fail). We can notice that there is a trend that persons with lower scores are more likely to fail, while persons with higher scores tend to pass. When we plot the data as Figure 1A, we can see persons with value 1 (pass) have scores that shift to the right side, while persons with value 0 (fail) have those that shift to the left side. Persons with same score may not have the same outcome (e.g., cases of score = 799) because the assessment procedure comprises other factors. At least we can postulate that the probability of pass may be higher if the score is higher. What is the best-fit line for this data? A usual straight regression line ranging from minus infinity to infinity does not make sense for this case. Instead of ordinal regression the logistic regression can fit the probability more adequately. In Figure 1B, the probability estimated by logistic regression is presented. The estimated probability by the logistic regression model (red dot and line) seems reasonable because it reflects the observed reality that the probability of pass decreases close to zero with very low scores, while the probability increases close to one with very high scores.