AbstractComposite likelihood appears to be an appealing alternative to the full likelihood when the latter is too complex. For a given model, there may be different ways to formulate a composite likelihood, for example, pairwise marginal or conditional likelihood. To guide the choice, we develop a suggestion in Cox & Reid (2004) and we explore a combination of pairwise and independence likelihoods which leads to a new objective function that depends on a constant to be chosen. Exact and asymptotic properties are explored. The former allow to identify a range of admissible values for the constant, while efficiency considerations can be used to choose an optimal value. Two examples are analysed in detail, also through simulation studies.The Canadian Journal of Statistics42: 525–543; 2014 © 2014 Statistical Society of Canada