Interpreting Negative Coefficients in Logistic Regression

In logistic regression, which is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome, the logit link is utilized to establish a relationship between the odds and the explanatory variables through a logarithmic link.

The odds are defined as the ratio of the probability of success (p) to the probability of failure (1-p), representing the proportion of successes in a given number of trials.

The formula for the odds can be expressed as:

Odds p / (1-p)

Where p represents the probability of success. In the logistic regression model, the log of odds (logit) is linearly related to the explanatory variables. The model is typically expressed as:

log(odds) logit intercept β1X1 β2X2 … βnXn

Here, intercept and β1, β2, βn are coefficients. The intercept represents the log odds of the outcome when all independent variables are zero.

Interpreting the Intercept

To interpret the intercept, we calculate the exponential of the intercept (exp(intercept)).

For example, if exp(intercept) 0.368, then the probability (p) can be calculated as:

p 0.368 / (1 0.368) 0.269 or 26.9%

Here, we have shown three significant figures. If our data were only accurate to two significant figures, we would round this to 27%.

Negative Coefficients in Logistic Regression

A negative coefficient in logistic regression means that as the value of the corresponding explanatory variable (X) increases, the log odds of the outcome decrease. This translates to the odds of the outcome decreasing, indicating a negative relationship between the variable and the likelihood of the event.

For example, if β1 is negative, it means that an increase in X1 will result in a decrease in the log odds of the outcome. The odds ratio (OR) associated with β1 is exp(β1), and if β1 is negative, then the odds ratio is less than 1, indicating a negative slope on the log-logit scale.

Negative Coefficients in Continuous Data

When applying logistic regression to continuous data, particularly in the range [0,1], a negative coefficient suggests a negative slope on the log-logit scale. This is logical because the logit function (log(odds)) transforms the probability scale (which is constrained between 0 and 1) into a linear scale.

Conclusion

The negative coefficients in logistic regression indicate an inverse relationship between the explanatory variable and the log odds of the outcome. Understanding these coefficients is crucial for correctly interpreting the results and making accurate predictions based on the logistic regression model.