Dimension reduction methods have come to the forefront of many applications when the number of covariates, p, far exceeds the sample size, n. In this work, we focus on two dimension reduction methods. In particular, we proposed a variant of Partial Least Squares, denoted by Rank-based Modified Partial Least Squares (RMPLS). The method is insensitive to outlying values of both the response and the predictors, and takes into account the censoring information in the construction of its components. We also focus on the dimension reduction method of Random Projection (RP). The motivation for RP is the Johnson-Lindenstrauss (JL) Lemma, which allows for the projection of n points in p-dimensional Euclidean space onto a k-dimensional Euclidean space such that the pairwise distances between the points are preserved within a factor of 1 ± e. In this work, we revisit the JL Lemma, and provide an improvement on the lower bound for k by working directly with the distributions of the random distances rather than resorting to the moment generating function technique. We further provide a lower bound for k using pairwise L_2 distances in the space of the original points and pairwise L_1 distances in the space of projected points.