With larger data sizes and often unobservable variables, statistical models have become progressively larger and structurally complex and even computationally intractable. In applications, work-arounds have arisen that add say log-likelihoods and adjust for breakdown in Bartlett relations, but usually with the loss of first order accuracy. Meanwhile, likelihood asymptotics has sought progressively higher accuracy for p-value functions and now provides definitive third-order p-values and related likelihood functions. We apply the likelihood asymptotic approach to this combining problem and are able to convert composite likelihood to a fully first-order accurate procedure. The methods can then be extended to the combining of statistically dependent p-values and scores.