In assessing time to event endpoints, data are said to exhibit competing risks if subjects can fail from multiple mutually-exclusive causes. For competing risks data, the Fine--Gray proportional hazards model for subdistributions has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function. However, in many important applications, the requisite proportional hazards assumption may not be satisfied, including multi-center clinical trials, where the baseline subdistribution hazards may not be common due to varying patient populations. We consider a stratified competing risks regression which allows the baseline subdistribution hazard to vary across levels of the stratification covariate. According to the relative sizes of the number of strata and strata sizes, two stratification regimes are considered. Using partial likelihood and weighting techniques, we obtain consistent estimators of regression parameters. The corresponding asymptotic distributions are provided for the two regimes separately, along with various estimation techniques. Data from a breast cancer clinical trial and from a European bone marrow transplantation (EBMT) registry illustrate the potential utility of the stratified Fine--Gray model. We also extend the Fine--Gray model to clustered competing risks situations where the failure times are grouped in a manner that can lead to within-group correlation. Adapting the marginal model approach for classical survival analysis and the Fine--Gray model for unclustered data, we obtain consistent parameter estimators under an independence working assumption. The variance-covariance matrix and a consistent estimator are then acquired in a manner that accounts for the within-cluster correlation in the data. Comparisons of sizes and powers are conducted to show the utility of the proposed approach. The method is also illustrated by the EBMT registry data. The remaining topic of our research concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine--Gray model, similarly to the tests proposed by Grambsch and Therneau (1994) for independent censored data. We develop a score test for the time-varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non-proportionality. We also propose graphical diagnostics for identifying the functional form of the time-varying coefficients.