Logistic Regression Part 2: A Worked Clinical Example

decrease in the odds of response per one-unit increase in severity.

The Full Odds-Ratio Table

A typical clinical regression table might therefore look like this:

For a clinical report, the treatment effect would usually be the primary parameter of interest, while the baseline covariates provide adjustment and additional clinical context.

From Odds Ratios to Predicted Probabilities

Odds ratios are useful, but clinicians often find probabilities easier to interpret.

The inverse-logit transformation converts a linear predictor back to a probability:

where:

Predicted Probability for a Control Patient

Consider a hypothetical control patient with:

• Age = 55 years
• Severity = 3.0
• Treatment = 0

The linear predictor is:

Therefore:

The predicted probability is:

So the predicted probability of response for this representative control patient is approximately 18.6%.

Predicted Probability for an Experimental Patient

Now change only treatment from control to experimental.

The linear predictor becomes:

Thus:

The predicted probability is:

The representative experimental patient therefore has an estimated probability of response of approximately 58.1%.

Adjusted vs. Unadjusted Treatment Effects

The crude odds ratio was approximately:

The adjusted odds ratio from the multivariable model is approximately:

These estimates are not identical.

The crude estimate compares treatment groups without accounting for age or severity. The adjusted estimate compares treatment groups after accounting for the covariates in the specified model.

R Implementation

A logistic regression can be fit in R using glm() with the binomial family.

model <- glm(
  response ~ treatment + age + severity,
  data = clinical_data,
  family = binomial(link = "logit")
)

summary(model)

The coefficient estimates are on the log-odds scale.

To obtain odds ratios:

exp(coef(model))

To obtain 95% Wald confidence intervals for the odds ratios:

results <- cbind(
  OR = exp(coef(model)),
  exp(confint.default(model))
)

results

Obtaining Predicted Probabilities in R

Predicted probabilities can be generated using type = "response".

new_patient_control <- data.frame(
  treatment = 0,
  age = 55,
  severity = 3.0
)

new_patient_treatment <- data.frame(
  treatment = 1,
  age = 55,
  severity = 3.0
)

predict(
  model,
  newdata = new_patient_control,
  type = "response"
)

predict(
  model,
  newdata = new_patient_treatment,
  type = "response"
)

This produces probabilities on the original 0-to-1 scale rather than on the log-odds scale.

SAS Implementation

In SAS, the corresponding analysis can be performed with PROC LOGISTIC.

proc logistic data=clinical_data;
    class treatment(ref='0') / param=ref;

    model response(event='1') =
        treatment
        age
        severity;

    oddsratio treatment;
    oddsratio age;
    oddsratio severity;
run;

The event='1' option makes the modeled event explicit: the probability of being a responder.

Why the EVENT Definition Matters

Suppose response is coded:

• 1 = responder
• 0 = non-responder

Then the model should estimate:

If the event is accidentally defined as the non-response category, the model will estimate:

The resulting odds ratios will be interpreted in the opposite direction.

Checking the Model

Fitting the model is not the end of the analysis.

Important questions include:

• Are the covariates clinically appropriate?
• Are continuous covariates modeled appropriately?
• Are there sparse cells or separation?
• Are influential observations present?
• Is the model excessively complex for the available number of events?
• Does the model discriminate between responders and non-responders?
• Is calibration adequate?

Linearity on the Logit Scale

A particularly important assumption concerns continuous predictors.

For example, the model above assumes that:

This means the relationship between age and the log odds is linear.

It does not mean that the probability itself changes linearly with age.

Interactions

A basic logistic model assumes that the treatment effect is the same across the levels of the included covariates unless interaction terms are included.

For example, a treatment-by-age interaction could be written as:

In that model, the treatment effect depends on age.

The treatment odds ratio at a particular age is no longer simply \(e^{\beta_1}\).

Instead, it depends on the interaction term.

Odds Ratio vs. Risk Ratio

One of the most important distinctions in clinical interpretation is the difference between odds and risk.

Suppose two groups have response probabilities:

The risk ratio is:

But the odds ratio is:

Thus an odds ratio of 3.5 does not mean the probability is 3.5 times as high.

Model Discrimination

A logistic model can also be evaluated for its ability to distinguish patients with and without the event.

A common measure is the area under the receiver operating characteristic curve, or AUC.

An AUC near 0.5 indicates little discrimination, while larger values indicate better discrimination.

However, a strong AUC does not prove that the model is well calibrated.

Model Calibration

Calibration asks whether predicted probabilities agree with observed outcome frequencies.

For example, among patients predicted to have approximately 70% probability of response, we would ideally observe a response rate close to 70%.

Separation and Sparse Data

Logistic regression can become unstable when a predictor or combination of predictors almost perfectly separates responders from non-responders.

For example, suppose every patient with a particular rare baseline characteristic responds and every patient without it fails.

Maximum likelihood estimates may become extremely large or fail to converge.

This problem is called separation.

Potential approaches include:

• Reviewing the clinical data for sparse cells
• Reducing unnecessary model complexity
• Combining clinically appropriate sparse categories
• Using penalized or exact methods when justified
• Prespecifying appropriate handling in the statistical analysis plan

Clinical Interpretation of the Worked Example

The main result from our example is:

The interpretation is:

The model also suggests that increasing age and greater baseline severity are associated with lower odds of response, although the scientific importance of these associations should be considered separately from the primary treatment comparison.

How to Report the Result

A concise clinical-trial report might state:

This wording identifies:

• The endpoint
• The statistical method
• The adjustment variables
• The treatment comparison
• The adjusted odds ratio
• The confidence interval
• The P-value

What Not to Report

Avoid statements such as:

• "Treatment increased response by 607%."
• "Patients were 6.07 times more likely to respond."
• "The probability of response was 6.07 times higher."
• "The treatment caused a six-fold increase in response probability."

These statements confuse odds ratios with probability or risk ratios.

A Practical Analysis Workflow

Common Mistakes

1. Calling an odds ratio a risk ratio. An odds ratio and a risk ratio are different measures.
2. Interpreting \(e^\beta\) as a probability change. Exponentiating a logistic coefficient produces an odds ratio.
3. Ignoring the reference category. Every categorical treatment or covariate effect is interpreted relative to its specified reference level.
4. Failing to specify the modeled event. Reversing the event definition reverses the direction of interpretation.
5. Adding many covariates without justification. An unnecessarily complex model can become unstable, especially with limited numbers of events.
6. Assuming continuous predictors are automatically linear. Logistic regression assumes linearity on the logit scale for ordinary continuous predictor terms.
7. Reporting only the P-value. The estimated effect and confidence interval are essential for understanding magnitude and precision.
8. Confusing adjusted and unadjusted estimates. The crude treatment comparison and the multivariable treatment effect answer different questions.
9. Assuming statistical significance implies clinical importance. A small P-value does not by itself establish that an effect is clinically meaningful.
10. Ignoring model diagnostics. Convergence, separation, influential observations, discrimination, and calibration should be considered.

When Logistic Regression Is Especially Useful

Logistic regression is particularly useful when:

• The primary endpoint is binary.
• Adjustment for baseline prognostic factors is desired.
• The treatment effect needs to be expressed as an odds ratio.
• Multiple predictors are clinically relevant.
• Predicted probabilities are useful for interpretation or prediction.
• Effect modification or interactions are scientifically important.

When Logistic Regression May Not Be the Best Choice

Logistic regression is not automatically the best model for every clinical endpoint.

The model should be selected based on the structure of the endpoint and the scientific question, not simply because logistic regression is familiar.

One More Important Distinction: Association vs. Prediction

The worked example is primarily an inferential clinical model. The principal question is whether treatment is associated with response after adjustment for prespecified covariates.

A predictive model has a different goal: accurately predicting outcomes for future patients.

Predictive modeling may require additional considerations such as:

• Internal validation
• Bootstrap validation
• Cross-validation
• Calibration assessment
• Optimism correction
• External validation
• Prediction intervals or uncertainty assessment

The Complete Worked Example in One View

The Most Important Concept

The most important lesson from a clinical logistic regression analysis is that the model connects three different scales:

The model is fit on the log-odds scale, coefficients are exponentiated to produce odds ratios, and the inverse-logit transformation converts the fitted model back into predicted probabilities.

References

Hosmer, D.W., Lemeshow, S. & Sturdivant, R.X. (2013). Applied Logistic Regression. 3rd ed. Wiley.

Agresti, A. (2018). An Introduction to Categorical Data Analysis. 3rd ed. Wiley.

Harrell, F.E. (2015). Regression Modeling Strategies. 2nd ed. Springer.

Vittinghoff, E., Glidden, D.V., Shiboski, S.C. & McCulloch, C.E. (2012). Regression Methods in Biostatistics. 2nd ed. Springer.

SAS Institute Inc. PROC LOGISTIC Documentation.

R Core Team. R: A Language and Environment for Statistical Computing.