# [R] Modeling Dependent Bernoulli Trials?

Spencer Graves spencer.graves at pdf.com
Fri Jul 23 04:55:09 CEST 2004

```      Is there any software to estimate dependence between Bernoulli
trials?

I have N observations on k-vectors of 0s and 1s, and I need to
model the dependence.  Consider the simple case with k = 2.  There are 4
possible outcomes:  (0, 0), (0, 1), (1, 0), and (1, 1).  I assume 0 =
"failure" and 1 = "success".  I assume further that there are 3
independent failure mechanisms, one that makes the first outcome 0 with
probability (1-q1), another that makes the second outcome 0 with
probability (1-q2), and a third that make them both 0 with probability
(1-q12).  I compute the probabilities of these 4 outcomes as follows:

P00 = Pr(0, 0) = q1*q2*q12 = probability of no failures.

P01 = Pr(0, 1) = q1*(1-q2)*q12 = probability of a failure from the
second mechanism and none from the other 2.

P10 = Pr(1, 0) = (1-q1)*q2*q12 = probability of a failure from the
first mechanism and none from the other 2.

P11 = Pr(1, 1) = 1-Pr(0,0)-Pr(0,1)-Pr(1,0).

With this setup, I can write a likelihood function, parameterize
it appropriately, and estimate what I need.  However, I wonder if