Background: In many epidemiological scenarios, understanding change is crucial. Repeated measures (or longitudinal) data are one of the pillars of this understanding. Cohort studies and randomised controlled trails produce such data since a group of individuals are followed over time with repeated measurement of key exposure(s) or outcome(s). A plethora of methods exist for modelling such data, with mixed or multi-level models being at the forefront. However there is a lack of approaches capable of estimating features of trajectories borne out of such data. To accurately estimate a feature (such as the minimum or maximum of a trajectory) requires a descriptive approach not restricted by parametric assumptions. Furthermore, many features of a trend can only be extracted through derivative estimation. For example, derivative estimates are needed to obtain the maximum rate of change of a biological process or the time at first decline of a biomarker.

There is a strong positive association of systolic and diastolic blood pressure (SBP and DBP) across most of their distribution with increased cardiovascular disease risk. Higher SBP and DBP measured in adolescence and early adulthood are associated with increased coronary heart disease and stroke risk with magnitudes of association that are similar to those seen when blood pressure is measured in middle-life.

Blood pressure varies, by up to 20%, over the day (with lower levels during rest and in particular in deep sleep) and in response to different stimuli, one of which is physical activity. Variability in blood pressure across the day, and the magnitude of the difference between day- and night-time blood pressure have been proposed as independent cardiovascular risk factors over and above the mean level of blood pressure. Relatively little is known about these patterns and their correlates in healthy adolescents.

Aim: The aim of this proposal is to develop methods to accurately extract features of diurnal BP for individuals and for the sample. These may then be used to investigate associations with outcomes.

Hypothesis: Specific features of diurnal SBP and DBP trajectories in adolescents are associated with cardiovascular risk factors

Exposure: Extracted features of diurnal SBP and DBP. Linear mixed models are one commonly used approach for modelling continuous repeated measures. These allow separate modelling of the variability between members of the cohort and the variability within individuals. To handle non-linear trajectories over time, alterations to the linear mixed model are available, such as transforming the outcome, allowing for polynomial trends of the outcome over time (fractional polynomials) or allowing the outcome trend to change in different segments of time (regression splines). These can be useful if statistical inference is of primary concern. At the other end of the spectrum, when the question of interest is to describe the trend over time, these simple alterations can prevent goodness of fit because of the restrictions of a fully parametric model. In particular, when interest lies in identifying a feature of repeated measurements, a fully parametric model is not sufficiently flexible to obtain reliable estimates.

Flexible methods which borrow from the fields of mixed models and non-parametric smoothing come under the umbrella of functional data analysis (FDA). FDA encompasses the various modelling methods for these non-linear repeated measures data. Unfortunately, in the situation where measurement times are irregular across individuals, many methods under the umbrella of FDA become inefficient. In epidemiology such data are common since measurement often occurs within routine GP visits. Three approaches to modelling non-linear irregular repeated measures data have been identified, namely semiparametric mixed models, P-Spline mixed models and Principal components Analysis through Conditional Expectation (PACE). These have not been used to their full potential in the epidemiological literature.

In a range of disciplines it is often the case that the derivative, or rate of change, of observed data is of primary interest. In the situation where data are observed over time, the first derivative will correspond to velocity, the second to acceleration. In a LMM the rate of change is, by definition, constant for both the individual and cohort. Where the LMM contains a complex polynomial of time, this polynomial can be differentiated to give the rate of change of the biomarker at any timepoint, both on average and for each individual. When non-linear methods such as P-Splines are used, it can be difficult to obtain derivative estimates and their standard error analytically. Derivative estimation is a difficult problem for several reasons. Firstly, with no observations for derivatives, testing goodness of fit does not exist. Secondly, modelling the change of a variable over time is more sensitive to measurement error and outliers than modelling observed data. Third, standard errors can be difficult to obtain since many approaches use numerical methods to estimate the derivative as a by-product. However, derivative estimates can be very useful. For instance, when a first derivative estimate of a biomarker over time is below zero, the process is declining. Using the standard error to create variability bands, we obtain an analogous confidence interval comparison for evidence of decline. Derivative estimates allow us to obtain features of trend such as the maximum/minimum velocity or acceleration. These have been used in physiology as a marker for endurance and in child growth as a marker for puberty. Derivative estimates and their standard errors have yet to be obtained for several approaches to longitudinal data.

Outcome: Cardiovascular risk factors

Confounders: Age, sex, socio-economic status, smoking status, weight, height, fat mass, physical activity.