  In both criminal cases and civil cases there is an increasing demand for the
analysis of DNA mixtures involving relationships. The goal might be, for
example, to identify the contributors to a DNA mixture where the donors may be
related, or to infer the relationship between individuals based on a mixture.
  This paper introduces an approach to modelling and computation for DNA
mixtures involving contributors with arbitrarily complex relationships. It
builds on an extension of Jacquard's condensed coefficients of identity, to
specify and compute with joint relationships, not only pairwise ones, including
the possibility of inbreeding. The methodology developed is applied to two
casework examples involving a missing person, and simulation studies of
performance, in which the ability of the methodology to recover complex
relationship information from synthetic data with known `true' family structure
is examined.
  The methods used to analyse the examples are implemented in the new KinMix R
package, that extends the DNAmixtures package to allow for modelling DNA
mixtures with related contributors. KinMix inherits from DNAmixtures the
capacity to deal with mixtures with many contributors, in a time- and
space-efficient way.
