  Most quantum information tasks based on Bell tests relie on the assumption of
measurement independence. However, it is difficult to ensure that the
assumption of measurement independence is always met in experimental
operations, so it is crucial to explore the effects of relaxing this assumption
on Bell tests. In this paper, we discuss the effects of relaxing the assumption
of measurement independence on 1-parameter family of Bell (1-PFB) tests. For
both general and factorizable input distributions, we establish the
relationship among measurement dependence, guessing probability, and the
maximum value of 1-PFB correlation function that Eve can fake. The
deterministic strategy when Eve fakes the maximum value is also given. We
compare the unknown information rate of Chain inequality and 1-PFB inequality,
and find the range of the parameter in which it is more difficult for Eve to
fake the maximum quantum violation in 1-PFB inequality than in Chain
inequality.
