Interpretation of a logistic regression coefficient

Question

I always find it difficult to interpret the coefficients of a logistic regression, especially with transformed variables, so I would like a confirmation of my conclusion or a correction if necessary.

I'm trying to predict tumor penetration of the prostate capsule. Here is the description of the variables:

• Tumour penetration of Prostatic Capsule (0 = No penetration, 1 = Penetration) (Capsule)

• Results of the ‘Digital Rectal Exam (1 = No Nodule, 2 = Unilobar Nodule (Left), 3 = Unilobar Nodule (Right), 4 = Bilobar Nodule) (Dpros)

• Prostatic Specific Antigen Value’ in mg/ml (PSA)

• Tumour volume obtained from ultrasound in cm3 (Vol)

• Total gleason score (Gleason)

Here are the results of the logistic regression I chose:

                 (Intercept)  **DprosUnilobar Nodule (Left)** 
                  -8.1264570                    0.7169127 
**DprosUnilobar Nodule (Right)**          **DprosBilobar Nodule** 
                   1.6227430                    1.5086911 
                     **log.PSA**                   **square.Vol** 
                   0.5077599                   -0.1087285 
                     **Gleason** 
                   0.9273380

I've transformed PSA with log and Vol with sqrt.

Here is the interpretation:

• The log-odds of having a unilobular nodule (left) and having a penetration compared to a no nodule is 0.7169, holding the other variables constant. If we exponentiate this we get

exp(0.7169) [1] 2.049

and it is the odds ratio of penetration for the unilobular nodule (left) compared to no nodule - i.e. the odds of penetration for the unilobular nodule (left) is 104.9 % higher than the odds of penetration in the capsule for no nodule, if the rest remains constant. Or the probability of having penetration into the capsule with a unilateral nodule (left) compared to no nodule is 2.049/(1+2.049)=0.672 so 67.2%, if all else remains constant.

• Every 1 unit increase in Gleason score is associated with a 0.9273380 increase in log-odds of survival holding the other variables constant. If we exponentiate this:

exp(0.927338) [1] 2.528

Thus, each one-unit increase in the Gleason score is associated with a 152.8% increase in the odds of penetration into the capsule, with the other variables remaining constant. We can also say that for every 1% increase in the Gleason score, the probability increases by about 0.39561/(1+0.39561)=0.2835 28.4% of penetration into the capsule, with the other variables remaining constant.

• Every 1 unit increase in PSA score is associated with a exp((log(1.01)*0.51))= 1.0020 increase in log-odds of penetration holding the other variables constant. If we exponentiate this:

exp(1.0020) 1 2.724

Thus, each one-unit increase in the PSA score is associated with a 172.4% increase in the odds of penetration into the capsule, with the other variables remaining constant.

• 10% increase in volume, increases the odds by exp(sqrt(10)*-0.11)= 0.71 and thus the probability by 4.15%, if all the other variables are kept fixed

Answer 1

The main problem with your interpretation is that you cannot transform an odds ratio into a probability change. You can make no claim about how the probability changes simply by transforming the odds ratio. You can change an odds into a probability, but the coefficients (other than the intercept) correspond not to odds but to odds ratios (when exponentiated).

Here is how I would change your interpretation:

The difference in the log odds of penetration between having a unilobular nodule (left) and having no nodule is 0.7169, holding other variables constant. If we exponentiate this we get

exp(0.7169) 1 2.049

and it is the odds ratio of penetration for the unilobular nodule (left) compared to no nodule - i.e. the odds of penetration for the unilobular nodule (left) is 104.9 % higher than the odds of penetration in the capsule for no nodule if the rest remains constant. (Equivalently, the odds of penetration for those with unilobular nodule (left) is 2.049 times that for those with no nodule, holding the other variables constant).

Every 1 unit increase in Gleason score is associated with a 0.9273380 increase in log-odds of penetration holding the other variables constant. If we exponentiate this:

exp(0.927338) 1 2.528

Thus, each one-unit increase in the Gleason score is associated with a 152.8% increase in the odds of penetration into the capsule, with the other variables remaining constant.

There are procedures for estimating changes in probability rather than odds ratios, but they are fairly complicated and cannot be done simply by reading coefficients from a logistic regression. Note that logistic regression is not the only appropriate model for binary outcomes, and other models may give similar predictive ability with more interpretable coefficients. See Huang (2019) for an example.