Given a dataset with binary outcomes for a response variable, y, and a numeric predictor, x, we fit a logistic regression model ẞ0 + ẞ1x where
beta_{1} = 0.5
How does y change with a unit increase in

Question

Given a dataset with binary outcomes for a response variable, y, and a numeric predictor, x, we fit a logistic regression model ẞ0 + ẞ1x where
beta_{1} = 0.5
How does y change with a unit increase in x ?

VIPSolutions · Accepted Answer

# Effect of a Unit Increase in $ x $ on $ y $ in Logistic Regression

Given:
- Logistic model: $\text{log}(p) = \beta_ + \beta_1 x$
- $\beta_1 = .5$

where:
- $ p $ is the probability that $ y = 1 $.

---

## Interpretation of $\beta_1$

- $\beta_1$ represents the change in the **log-odds** of $ y = 1 $ for a one-unit increase in $ x $.

## How $ y $ changes with $ x $

- For a **unit increase** in $ x $:
  
  $$
  \text{logit}(p) \text{ increases by } .5
  $$

- The **odds ratio** associated with a one-unit increase in $ x $:

$$
  OR = e^{\beta_1} = e^{.5} \approx 1.65
  $$

- **Interpretation**:

For each additional unit of $ x $, the **odds** that $ y = 1 $ increase by approximately **65%**.

---

## Summary

| Change in $ x $ | Effect on $ y $                                              |
|-------------------|--------------------------------------------------------------|
| 1 unit increase   | Odds of $ y=1 $ multiply by $ e^{.5} \approx 1.65 $   |
| Probability $ p $ | Increases, but not by a fixed amount; depends on current $ p $ |

**Note:** The actual change in probability $ p $ depends on the current value of $ x $ and $ p $, since logistic functions are nonlinear.

✨ VIPSolutions

Given a dataset with binary outcomes for a response variable, y, and a numeric predictor, x, we fit a logistic regression model ẞ0 + ẞ1x where beta_{1} = 0.5 How does y change with a unit increase in x ?

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