This article proposes a nonlinear complementarity problem (NCP) formulation for the risk-aversive stochastic transit assignment problem in which in-vehicle travel time, waiting time, capacity, and the effect of congestion are considered as stochastic variables simultaneously and both their means and variances are incorporated into the formulation. A new congestion model is developed and captured in the proposed NCP formulation to account for different effects of on-board passengers and passengers waiting at stops. A reliability-based user equilibrium condition is also defined based on the proposed generalized concept of travel time budget referred to as effective travel cost, and is captured in the formulation. A column generation based algorithm is proposed to solve the NCP formulation. A survey was conducted to validate that the degree of risk aversion of transit passengers affects their route choices. Numerical studies were performed to demonstrate the problem and the effectiveness of the proposed algorithm. The results obtained show that underestimating the congestion effect and ignoring the risk aversion behavior can overestimate the patronage of transit service, which have profound implications on the profit of the operators involved and the development of transit network design models.