We apply the distinction between parameter independence and outcome independence to the linear and nonlinear models of a recent nonrelativistic theory of continuous state vector reduction. We show that in the nonlinear model there is a set of realizations of the stochastic process that drives the state vector reduction for which parameter independence is violated for parallel spin components in the EPR-Bohm setup. Such a set has an appreciable probability of occurrence (≈ 1/2). On the other hand, the linear model (...) exhibits only extremely small parameter dependence effects. We investigate some specific features of the models and we recall that, as has been pointed out recently, if one wants to be able to speak of definite outcomes (or equivalently of possessed objective elements of reality) at finite times, one has to slightly change the criteria for their attribution to physical systems. The concluding section is devoted to a detailed discussion of the difficulties which one meets when one tries to take, as a starting point for the formulation of a relativistic theory, a nonrelativistic scheme which exhibits parameter dependence. Here we derive a theorem which identifies the precise sense in which the occurrence of parameter dependence forbids a genuinely relativistic generalization. Finally we show how the appreciable parameter dependence of the nonlinear model gives rise to problems with relativity, while the extremely weak parameter dependence of the linear model does not give rise to any difficulty, provided one takes into account the appropriate criteria for the attribution of definite outcomes. (shrink)