Dimensionality reduction is a common unsupervised learning task. In this paper we propose novel deep autoencoder-based nonlinear dimensionality reduction methods by extending Kramer (1991). Our contributions are two-fold: (1) We introduce Deep Autoencoder-based Monotone (DAM) methods for promoting monotonicity between the reconstructed output and the latent components. While DAM methods do not completely solve the problem of model selection for each nonlinear component, they do reduce the model space considerably when the monotonicity assumption is reasonable. (2) We propose a new, multi-stage simultaneous (MSS) deep learning model for estimating multiple nonlinear components. This allows construction of loss curves to assess how the reconstruction error in the original input space decreases as new components are added. Using DAM and MSS together allows us to construct a loss curve that provides a more solid basis for choosing the number of nonlinear components. Our method is motivated by immune response datasets generated from vaccine studies, and will be evaluated in such datasets.