The name is absent

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individual and collective interests (Pocock, 1992; Palmer and Rosenberger, 1999) is
necessary. The individual interest suggests to put the patient on what the physician
considers the “best treatment” available. If we only followed this goal, then many
trials would not proceed. The collective interest requires to treat some individuals
not under this “best treatment.” This is done in order to gather data to prove that
there is a superior treatment. The disagreement among physicians about the “best
treatment” makes randomization ethical (Freedman, 1987). The objective of clinical
trials is to create consensus about the best treatment among physicians. Finally, it
is worth to mention that adaptive random allocation designs for phase II and III
clinical trials are being considered. The idea of these designs is that the probability
of assigning a patient to a group changes each time the information is updated and
increases the probability of a new patient entering the study to be assigned to what
at that point of the study is thought to be the “best treatment”.

2.2.2 Binary Data Model

Suppose that we have a vector of binary outcomes z = (zɪ,..., z_n) with z_i ∈ {0,1}.
Assume that each observation is associated with a vector of measurements (covariates)
x_i of dimension p. We are interested in estimating the probability of observing the
response к for a future observation with covariate vector xj, к = 0 or 1.

A common approach to this problem is the use of generalized linear models. In
particular, the probit and the logit models. They assume that z_i ∣ π_i are independent
and Bernoulli^) distributed. The value of π_i is related to the covariate x_i by means
of a known link function τ∕>^¹ mapping the interval (0,1) into the real line. The
variable x_i is linearly related with ≠~¹(7Γj), that is '0^-1(τΓj) = xj β where β is a p-
dimensional unknown parameter vector. Commonly ψ is chosen to be a cdf. When
ψ is the normal cdf we have a probit model and when it is a logistic cdf we are
working with a logistic model. An important property of the probit model is that
we can choose prior distributions for β that makes posterior inference tractable. See

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