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1. Including cluster(ID) does not change the point estimates of the parameters. It does change the way that the standard errors are computed however.

More details can be found in Therneau & Grambsch's book Extending the Cox Model, chapter 8.2.

2. With the cluster(ID) option you force a "robust" estimate of the standard errors. This is because if you (wrongly) assume that the observations are independent (as done in the Cox model) the inverse of the information matrix would underestimate the true standard errors. In more technical terms, the score function for the parameters does not change, but the variance of this score does. A more intuitive argument is that 100 observations on 100 individuals provide more information than 100 observations on 10 individuals (or clusters).

3. Vague indeed. In short, +frailty(ID) in coxph() fits standard frailty models with gamma or log-normal random effects and with non-parametric baseline hazard / intensity. frailtypack uses parametric baseline (also flexible versions with splines or piecewise constant functions) and also fits more complicated models, such as correlated frailty, nested frailty, etc.

Finally, +cluster() is somewhat in the spirit of GEE, in that you take the score equations from a likelihood with independent observations, and use a different "robust" estimator for the standard errors.