Ankenman et al. introduced stochastic kriging as a metamodeling tool for representing stochastic simulation response surfaces, and employed a very simple example to suggest that the use of Common Random Numbers (CRN) degrades the capability of stochastic kriging to predict the true response surface. In this article we undertake an in-depth analysis of the interaction between CRN and stochastic kriging by analyzing a richer collection of models; in particular, we consider stochastic kriging models with a linear trend term. We also perform an empirical study of the effect of CRN on stochastic kriging. We also consider the effect of CRN on metamodel parameter estimation and response-surface gradient estimation, as well as response-surface prediction. In brief, we confirm that CRN is detrimental to prediction, but show that it leads to better estimation of slope parameters and superior gradient estimation compared to independent simulation.