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.