How do random intercept and random slope differ?

How do random intercept and random slope differ?

• A. Random slope means the intercept cannot vary.
• B. They are both fixed effects across groups.
• C. Random intercept allows the intercept to vary across groups, while random slope allows the slope of a predictor to vary across groups.
• D. Random intercept affects the residual variance only.

Answer: (C)

In mixed-effects models, you can let different parts of the relationship vary across groups. A random intercept means each group has its own baseline level—the intercept can vary from one group to another. A random slope means the impact of a predictor on the outcome can differ by group—the slope is not the same for all groups. For example, predicting test scores from hours studied with students nested in schools: a random intercept lets each school have its own average score when hours studied is zero, while a random slope lets the effect of one more hour of study differ from school to school. You can include both, allowing both the starting point and the strength of the predictor’s effect to vary across groups. The other statements mischaracterize these concepts. Random slope does not imply the intercept cannot vary, and both random intercepts and random slopes are about variability across groups rather than being fixed across groups. The random intercept is not about residual variance alone.

In mixed-effects models, you can let different parts of the relationship vary across groups. A random intercept means each group has its own baseline level—the intercept can vary from one group to another. A random slope means the impact of a predictor on the outcome can differ by group—the slope is not the same for all groups.

For example, predicting test scores from hours studied with students nested in schools: a random intercept lets each school have its own average score when hours studied is zero, while a random slope lets the effect of one more hour of study differ from school to school. You can include both, allowing both the starting point and the strength of the predictor’s effect to vary across groups.

The other statements mischaracterize these concepts. Random slope does not imply the intercept cannot vary, and both random intercepts and random slopes are about variability across groups rather than being fixed across groups. The random intercept is not about residual variance alone.