Polynomial vs. Multinomial Regression in Clinical Research

🔍 Polynomial vs. Multinomial Regression in Clinical Research

1. Polynomial Regression

Purpose: Captures nonlinear trends in continuous outcomes using powers of a predictor (e.g., age, dosage).

Example Use: Modeling birth weight across gestational age — expecting a curved (non-linear) trend.

Structure:

• Outcome: Continuous (e.g., blood pressure)

• Predictor: Continuous (e.g., age, age², age³)

Interpretation:

• The curve's shape is determined by the sign and size of the polynomial terms (e.g., positive age² = U-shape).

Stata Tip:

gen age2 = age^2
regress bp age age2

2. Multinomial Regression

Purpose: Models categorical outcomes with 3 or more unordered groups.

Example Use: Delivery mode (vaginal, elective CS, emergency CS).

Structure:

• Outcome: Nominal (e.g., delivery mode)

• Predictor: Categorical or continuous

Interpretation:

• Each category is compared to a base (reference) category using relative risk ratios (RRR).

Stata Tip:

mlogit delivery_mode i.maternal_age i.gestational_age, baseoutcome(1)

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🧮 Polynomial Logistic vs. Multinomial Logistic Regression

3. Polynomial Logistic Regression

Purpose: Extends logistic regression to capture non-linear effects of continuous predictors on a binary outcome.

Example Use: Risk of preeclampsia modeled against maternal age and age².

Structure:

• Outcome: Binary (e.g., disease vs. no disease)

• Predictor: Continuous (plus polynomial terms)

Interpretation:

• Odds of the outcome may increase then decrease with age (if age² is negative).

Stata Tip:

gen age2 = age^2
logistic preeclampsia age age2

4. Multinomial Logistic Regression

Purpose: Logistic analogue of multinomial regression. Predicts probabilities of multiple non-ordered categories.

Same as #2, but uses logistic framework to derive log-odds or RRR.

Stata Tip (same):

mlogit outcome i.exposure i.predictors, rrr

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🔹 Without "logistic" = Regression for continuous outcomes

• Polynomial regression → Outcome is a number (e.g., birthweight, blood pressure)

• Multinomial regression → Outcome is a category (3+ groups), but not binary

🧠 You're predicting a value or group.

🔸 With "logistic" = Regression for categorical outcomes (esp. binary)

• Polynomial logistic regression → Outcome is yes/no (e.g., disease vs. no disease), but predictor is non-linear (e.g., age + age²)

• Multinomial logistic regression → Outcome is 3+ categories (unordered), modeled with log-odds

🧠 You're predicting the chance (odds or probability) of being in a category.

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📊 Summary Table

Model	Outcome Type	Predictor	Output Type	Clinical Use Case
Polynomial Regression	Continuous	Continuous (X²)	Beta coefficients	Curved dose-response (e.g., age vs. BP)
Multinomial Regression	Categorical (≥3)	Any	RRRs	Mode of delivery, infection types
Polynomial Logistic	Binary	Continuous (X²)	Odds ratios	U-shaped risk curves (e.g., age vs. disease risk)
Multinomial Logistic	Categorical (≥3)	Any	RRRs	Type of referral diagnosis, treatment decisions