Which are examples of mixed models with random effects on an original scale?

• A. Linear, beta-binomial, negative binomial (extension of Poisson)
• B. Logistic, Poisson with log link
• C. Cox proportional hazards model
• D. Multinomial logistic regression

Answer: (A)

The main idea is about where the random effects operate relative to the observed data scale. Mixed models with random effects on the original scale are those where the outcomes themselves are observed on their natural measurement scale (such as counts or proportions) and the random effect enters through a mixture or dispersion mechanism that keeps the data on that same scale. Linear mixed models fit continuous outcomes directly on the original scale and include random effects to account for clustering. Beta-binomial and negative binomial models extend this idea to overdispersed data: the beta-binomial handles clustered binomial data by letting the probability of success vary across clusters, and the negative binomial arises as a Poisson-Gamma mixture, giving extra-Poisson variation. In both cases, the observed data—counts or successes—remain on their natural scale.

Other approaches use link functions that transform the mean to a different scale. Logistic regression and Poisson models with a log link apply a transformation (logistic or log) to connect the linear predictor to the mean, so the random effects act on that transformed scale rather than the original response scale. The Cox proportional hazards model is a survival model where random effects (frailty) affect the hazard function, not the observed event counts or proportions on their natural scale. Therefore these involve random effects on transformed or different scales, not the original data scale.

So the examples that fit the idea of random effects on the original scale include linear, beta-binomial, and negative binomial models.