If the coefficient of 'Income' in a logistic regression is 0.005, and the Odds Ratio is 1.005, what happens when income increases by 1 unit?

Question

BlackTom AI · Accepted Answer

A logistic-regression coefficient of 0.005 implies the odds multiply by exp(0.005) ≈ 1.005, so a one-unit increase in income raises the odds by about 0.5%.

1) Relevant concepts:
- In logistic regression the linear predictor beta relates to odds by Odds Ratio (OR) = exp(beta).
- The percent change in odds for a one-unit increase is (OR − 1) × 100%.
- Odds and probability are related but not identical: probability = odds / (1 + odds).

2) Step-by-step reasoning and calculations:
- Given beta = 0.005. Compute OR = exp(0.005).
- Use series or calculator: exp(0.005) ≈ 1 + 0.005 + (0.005^2)/2 ≈ 1 + 0.005 + 0.0000125 = 1.0050125 ≈ 1.005 (as stated).
- Percent change in odds = (1.005 − 1) × 100% = 0.005 × 100% = 0.5%.

3) Intermediate numeric example to illustrate difference from probability:
- Suppose baseline odds = 1 (probability 0.5). After increasing income by 1 unit, new odds = 1 × 1.005 = 1.005. New probability = 1.005 / (1 + 1.005) ≈ 0.5012469, an absolute probability increase ≈ 0.001247 (0.1247 percentage points), not 5%.

4) Why the correct answer is correct:
- The coefficient maps to multiplicative change in odds via the exponential function. Since OR ≈ 1.005, the multiplicative change is a 0.5% increase in odds per unit increase in income.

5) Why other options are incorrect:
- "The probability of the outcome increases by 5%." Wrong because the coefficient/OR refers to odds, not direct probability, and the percent (5%) is ten times too large.
- "The odds of the outcome decrease by 0.5%." Wrong sign; OR > 1 implies an increase, not decrease.
- "Income has no effect." Wrong because a nonzero coefficient (0.005) implies a measurable effect.

Conclusion: The correct interpretation is that the odds of the outcome increase by 0.5%, matching the provided answer.

If the coefficient of 'Income' in a logistic regression is 0.005, and the Odds Ratio is 1.005, what happens when income increases by 1 unit?单项选择题

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