2026-03-24 17:03 Tags:

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1. Start from the core equation

$$[\log \left(\frac{P}{1-P}\right)={\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p]$$

Focus on one variable (x_j):

$$[\log (\text{odds})=\cdots +{\beta }_j{x}_j]$$

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2. What does (\beta_j) mean?

Think: increase (x_j) by 1

What happens?

$$[\log ({\text{odds}}_{new})-\log ({\text{odds}}_{old})={\beta }_j]$$

So:

A 1-unit increase in (x_j) changes log-odds by (\beta_j)

But log-odds is not intuitive.

So we exponentiate.

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3. Convert to something interpretable

$$[e^{{\beta }_j}=\text{odds ratio}]$$

This is the key.

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4. Final interpretation (memorize this)

A 1-unit increase in (x_j) multiplies the odds by (e^{\beta_j})

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5. Concrete example

Suppose:

$$[{\beta }_{age}=-0.7]$$

Then:

$$[e^{-0.7}\approx 0.5]$$

Interpretation:

Each 1-year increase in age → odds are multiplied by 0.5
→ odds are cut in half

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Another example:

$$[{\beta }_{score}=0.7]$$ $$[e^{0.7}\approx 2]$$

Interpretation:

Each +1 in score → odds double

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6. Important: this is NOT probability

This is where most people get confused.

You are NOT saying:

probability increases by 0.2

You are saying:

odds are multiplied by 2

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7. Why odds instead of probability?

Because the relationship is nonlinear.

Example:

If baseline probability = 0.1
odds = 0.11

Multiply odds by 2 → 0.22
new probability ≈ 0.18

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If baseline probability = 0.8
odds = 4

Multiply odds by 2 → 8
new probability ≈ 0.89

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Same coefficient, very different probability change.

That’s why we use odds.

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8. Direction vs magnitude

Sign of coefficient

• (\beta > 0): increases probability

• (\beta < 0): decreases probability

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Magnitude

• larger (|\beta|) → stronger effect

• but better to compare using (e^{\beta})

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9. Special case: binary variable

If:

$$[x_j=0\text{ or }1]$$

Then:

$$[e^{{\beta }_j}]$$

means:

odds for group 1 vs group 0

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Example:

• gender (female = 1, male = 0)

• (e^{\beta} = 1.5)

Interpretation:

females have 1.5× the odds compared to males

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10. Intercept (\beta_0)

$$[{\beta }_0=\log (\text{odds when all }x=0)]$$

Often not meaningful unless variables are centered.

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12. Common mistakes

Mistake 1

“β = 0.7 means probability increases by 0.7”

Wrong.

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Mistake 2

Comparing coefficients directly across scaled vs unscaled data

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Mistake 3

Ignoring that effect depends on baseline probability

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13. Mental model

Think of logistic regression as:

• linear model → log-odds

• exponent → multiplicative effect on odds

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14. One-line summary

$$[{\beta }_j\Rightarrow \text{log-odds change}\to e^{{\beta }_j}\Rightarrow \text{odds multiplier}]$$

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Example: Interpreting Logistic Regression Coefficients

Model

$$[\log \left(\frac{P}{1-P}\right)=-2+0.7\cdot \text{tachycardia}]$$

Where:

• tachycardia = 1 → patient has tachycardia

• tachycardia = 0 → no tachycardia

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Step 1: Baseline (no tachycardia)

$$[\text{tachycardia}=0]$$ $$[\log (\text{odds})=-2]$$

Convert to odds:

$$[\text{odds}=e^{-2}\approx 0.135]$$

Convert to probability:

$$[P=\frac{0.135}{1+0.135}\approx 0.12]$$

Interpretation

Without tachycardia, the probability of hospitalization is about 12%

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Step 2: With tachycardia

$$[\text{tachycardia}=1]$$ $$[\log (\text{odds})=-2+0.7=-1.3]$$ $$[\text{odds}=e^{-1.3}\approx 0.27]$$ $$[P=\frac{0.27}{1+0.27}\approx 0.21]$$

Interpretation

With tachycardia, the probability of hospitalization is about 21%

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Step 3: What does the coefficient mean?

We focus on:

$$[\beta =0.7]$$

Exponentiate:

$$[e^{0.7}\approx 2]$$

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Key Interpretation

Having tachycardia doubles the odds of hospitalization

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Step 4: Verify numerically

• Original odds = 0.135

• New odds = 0.27

$$[0.27=2\times 0.135]$$

So the interpretation is exact.

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Step 5: Important insight

The coefficient affects:

Odds (multiplicative change)

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Not:

Probability (additive change)

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Step 6: Why probability change is not fixed

From the example:

• 12% → 21% (increase of 9%)

But if baseline were different:

Suppose baseline = 50%

• odds = 1

• multiply by 2 → odds = 2

$$[P=\frac{2}{1+2}=0.67]$$

Now:

• 50% → 67% (increase of 17%)

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Key Insight

Same coefficient:

• always multiplies odds by 2

• but probability change depends on baseline

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Step 7: Proper professional wording

You can write:

The presence of tachycardia is associated with a 2-fold increase in the odds of hospitalization.

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