A study measures blood pressure at 6 time points for each of 200 patients after initiating antihypertensive treatment. Why would a standard linear regression of blood pressure on time be inappropriate?
ALinear regression cannot handle time as a predictor
BThe 6 observations from each patient are correlated (not independent), violating the independence assumption of standard regression, which produces incorrect standard errors
CLinear regression requires at least 10 observations per patient
DBlood pressure is not normally distributed
Standard linear regression assumes all observations are independent, but repeated measures on the same patient are correlated — Patient A's blood pressures cluster together because of shared genetics, lifestyle, and physiology. Ignoring this correlation produces standard errors that are too small (because the effective sample size is not 1,200 independent observations but something closer to 200 independent patients), leading to inflated Type I error rates. A mixed-effects model with random intercepts (and potentially random slopes) explicitly accounts for within-patient correlation.
Question 2 Short Answer
In a mixed-effects model with a random intercept for patient, the random intercept captures between-patient variability in baseline levels. What additional structure does a random slope for time add?
Think about your answer, then reveal below.
Model answer: A random slope for time allows each patient to have a different rate of change over time, not just a different starting point. Some patients' blood pressure may decline rapidly while others decline slowly or not at all. Without a random slope, the model assumes all patients change at the same rate (the fixed slope) and differ only in their starting level. The random slope captures heterogeneity in treatment response across patients.
The random slope introduces a correlation between intercept and slope — patients who start higher might decline more (or less), and this correlation is estimated from the data. The covariance matrix of the random effects (intercept variance, slope variance, and their correlation) characterizes the population-level heterogeneity in trajectories, which is often of primary scientific interest.
Question 3 True / False
Mixed-effects model predictions for individual clusters are 'shrunk' toward the population mean. This shrinkage is a defect of the method that should be corrected.
TTrue
FFalse
Answer: False
Shrinkage is a feature, not a defect. Subject-specific estimates from a mixed-effects model are empirical Bayes predictions that blend the individual's data with the population average. Subjects with more observations or less noisy data are shrunk less (their data are more informative), while subjects with few observations are shrunk more toward the population mean. This regularization reduces the mean squared error of individual predictions by trading a small bias for a large reduction in variance — the same principle that makes Bayesian and ridge regression estimators more accurate than unbiased estimates in many settings.
Question 4 Multiple Choice
A researcher has data on 50 hospitals, each contributing 10-100 patients. She runs a standard regression with hospital as a fixed effect (49 dummy variables) instead of a random effect. What are the consequences?
ANo consequences — the two approaches give identical results
BFixed hospital effects use 49 degrees of freedom, cannot predict for new hospitals, cannot estimate between-hospital variance, and provide unreliable estimates for hospitals with few patients
CRandom effects are only valid when there are at least 100 groups
DFixed effects are always preferred because they make fewer assumptions
Fixed hospital effects consume many degrees of freedom, especially problematic with many hospitals. They provide no shrinkage, so hospitals with 10 patients get the same weight as hospitals with 100, producing noisy estimates. They cannot generalize to new hospitals (the estimates are specific to observed hospitals only) and do not estimate the between-hospital variance component, which is often of scientific interest. Random effects treat hospitals as a sample from a larger population, estimate the between-hospital variance, shrink small-sample hospitals toward the mean, and use far fewer parameters (variance components instead of dummy variables).