  We derive information bounds for the regression parameters in Cox models when
data are missing at random. These calculations are of interest for
understanding the behavior of efficient estimation in case-cohort designs, a
type of two-phase design often used in cohort studies. The derivations make use
of key lemmas appearing in Robins, Rotnitzky and Zhao [J. Amer. Statist. Assoc.
89 (1994) 846-866] and Robins, Hsieh and Newey [J. Roy. Statist. Soc. Ser. B 57
(1995) 409-424], but in a form suited for our purposes here. We begin by
summarizing the results of Robins, Rotnitzky and Zhao in a form that leads
directly to the projection method which will be of use for our model of
interest. We then proceed to derive new information bounds for the regression
parameters of the Cox model with data Missing At Random (MAR). In the final
section we exemplify our calculations with several models of interest in cohort
studies, including an i.i.d. version of the classical case-cohort design of
Prentice [Biometrika 73 (1986) 1-11]
