# Effect of a Unit Increase in \( x \) on \( y \) Given: - Logistic regression model \[ \text{logit}(p) = \beta_ + \beta_1 x \] - Estimated coefficient: \[ \beta_1 = .5 \] where: - \( p = P(y=1 \mid x) \) --- ## Interpretation of \( \beta_1 \) - \( \beta_1 \) represents the **change in the log-odds** of \( y=1 \) for a one-unit increase in \( x \). - Specifically, for a one-unit increase in \( x \): \[ \text{Change in log-odds} = \beta_1 = .5 \] ## Impact on the Odds - The **odds ratio (OR)** for a one-unit increase in \( x \): \[ OR = e^{\beta_1} = e^{.5} \approx 1.65 \] - **Interpretation**: The odds of \( y=1 \) **increase by approximately 65%** with each additional unit of \( x \). ## Effect on the Probability \( p \) - The probability \( p \) changes in a non-linear fashion as \( x \) increases. - **General trend**: As \( x \) increases, \( p \) increases, but the rate of change depends on the current value of \( p \). --- ## Summary | Aspect | Effect of 1-unit increase in \( x \) | |---------|----------------------------------------| | **Log-odds** | Increases by \( .5 \) | | **Odds** | Multiplies by approximately \( 1.65 \) | | **Probability \( p \)** | Increases, but the exact change depends on the current \( p \) (non-linear) | **In essence:** > A one-unit increase in \( x \) raises the odds of \( y=1 \) by about 65%.