# Interpreting the estimated coefficients in binary logistic regression

The interpretation of the estimated coefficients depends on: the link function, reference event, and reference factor levels. The estimated coefficient associated with a predictor (factor or covariate) represents the change in the link function for each unit change in the predictor, while all other predictors are held constant. A unit change in a factor refers to a comparison of a certain level to the reference level. For more information on changing the reference level for categorical predictors, go to Specify the coding scheme for Fit Binary Logistic Model. For more information on changing the reference event for the response, go to Enter your data for Fit Binary Logistic Model.

The logit link provides the most natural interpretation of the estimated coefficients and is therefore the default link in Minitab. The interpretation uses the fact that the odds of a reference event are P(event)/P(not event) and assumes that the other predictors remain constant. For the logit link function, the natural log of the odds is a function of the estimated coefficients.

ln [P(event)/P(not event)] = β0 + β1x1 + β2x2 + ... + βnxn

Where:
• ln = natural log function
• P = probability of
• β0 = The intercept
• βi = The coefficient for xi
• xi = The predictors
The greater the log odds, the more likely the reference event is. Therefore, positive coefficients indicate that the event becomes more likely and negative coefficients indicate that the event becomes less likely. A summary of interpretations for different types of predictors follows.
Continuous predictors

The coefficient of a continuous predictor is the estimated change in the natural log of the odds for the reference event for each unit increase in the predictor. For example, if the coefficient for time in seconds is 1.4, then the natural log of the odds increase by 1.4 for each additional second.

Estimated coefficients can also be used to calculate the odds ratios, or the ratio between two odds. Exponentiate the coefficient for a predictor. The result is the odds ratio for when the predictor is x+1 compared to when the predictor is x. For example, if the odds ratio for mass in kilograms is 0.95 than for each additional kilogram the probability of the event decreases by about 5%.

For continuous predictors, the interpretation of the odds can be more meaningful than the interpretation of the odds ratio.

Categorical predictors with 1, 0 coding

The coefficient is the estimated change in the natural log of the odds when you change from the reference level to the level of the coefficient. For example, a categorical variable has the levels Fast and Slow and the reference level is Slow. If the coefficient for Fast is 1.3 then a change in the variable from Slow to Fast increases the natural log of the odds of the event by 1.3.

Estimated coefficients can also be used to calculate the odds ratio, or the ratio between two odds. Exponentiate the coefficient for a level. The result is the odds ratio for the level compared to the reference level. For example, a categorical variable has the levels Hard and Soft and Soft is the reference level. If the odds ratio for Hard is 0.5, then the change from Soft to Hard decreases the odds of the event by 50%.

Categorical predictors with 1, 0, -1 coding

The coefficient is the estimated change in the natural log of the odds when you change from the mean of the natural log of the odds to the level of the coefficient. For example, a categorical variable has the levels Before Change and After Change. If the coefficient for After Change is -2.1 then the natural log of the odds of the event decrease by 2.1 from average when the variable equals After Change.

Estimated coefficients can also be used to calculate the odds ratios. To find the value to exponentiate, subtract the coefficients that you want to compare. For example, a categorical variable has the levels Red, Yellow, and Green. To calculate the odds ratio for Red and Yellow, subtract the coefficient for Red from the coefficient for Yellow. Exponentiate the result. If the odds ratio is 1.02, then the change from Red to Yellow increases the odds of the event by 2%.

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