In this thesis, we develop statistical methodology to find solutions to contemporary problems in renal research. These problems include 1) assessing the association of the underlying kidney function and the risk of survival events, 2) early detection of progression towards renal failure amongst primary care patients, and 3) long-term influences of acute kidney injury occurrences on the subsequent kidney function. Joint modelling of longitudinal and time-to-event outcome and Cox model with time-varying covariate are considered to answer the first problem. Whilst parameters are estimated by maximum likelihood (ML) using an expectation-maximisation (EM) algorithm for the former model, by partial likelihood for the latter. Results show that Cox model underestimates the association parameter between the longitudinal and survival processes, and joint models correct this. A longitudinal model with a non-stationary stochastic process is developed for the second problem. Parameters are estimated by ML using a Fisher-Scoring algorithm. Based on the results of this model, we obtain the predictive distribution of meeting the clinical guideline for detecting progression. Results show that there are patients with very high probability and emerging behaviour of progression. By these probabilities, we aim to inform clinical decision-making. Another longitudinal model with a class of stationary stochastic processes and heavy tailed response distribution is developed for the third problem. Parameters are estimated by ML using an EM algorithm, and random effects are predicted using the conditional distribution of random effects given data. Results show that AKI might have serious impacts on kidney function such that on average the loss of kidney function doubles after having an AKI. Nonetheless, there are substantial between patient heterogeneity in terms of this influence. The R package lmenssp which enables inference for a range of mixed models with non-stationary stochastic processes is developed and its core features are presented.