## 1 Introduction

Growth models are often used in life course epidemiology to describe patterns of growth or change, to assess the influences of rates and timing of change on later health outcomes and to investigate the effects of exposures on patterns of change [1-6]. In some cases two or more change processes occur in parallel, such as gains in height and weight during childhood, and it may be of interest to assess relationships between such processes, or to adjust for the effects of one change process when assessing the associations of another with a later health outcome. It may also be hypothesised that change in one process causes a change in another, or that an unmeasured or latent construct causes change in both of the processes simultaneously. Bivariate multilevel models (MLMs) for repeated measurements with linear relationships of both outcome variables with time have been used to demonstrate the correlation between values of the two variables at baseline and rates of change in the variables over time [7-9]. However, this does not demonstrate whether change in one variable temporally precedes change in another and is thus unable to provide evidence towards a causal effect.

Linear spline models have recently been proposed as a method of representing growth [5] or change [10], which reduces the dimensionality of repeated measurements. The shape of the trajectory of change is assumed to be piecewise linear, with knot points defining changes in the magnitude or direction of association of the response variable with time. The selection of the number and location of knot points may be determined by the data or by prior knowledge and therefore the linear splines represent the shape of the change trajectory in a meaningful way, and the linear periods of change provide coefficients that are more interpretable than for polynomial curves. By first fitting splines to each change process and then modelling these processes in parallel, we are able to model associations between observed rates of change in one variable in one period of time and observed rates of change in another variable in a subsequent period of time, with time periods defined by the splines, and thus we may identify whether changes in one variable precede changes in another. By examining the temporal sequence of events there is the possibility of improving causal inferences for pathways between variables. Furthermore, prior knowledge or hypotheses about which pathways exist may be tested by constraining covariances between individual-level random effects for different periods of change to be zero and comparing the model fit to models where these covariances are freely estimated.

We will use the example of weight and mean arterial pressure (MAP) during pregnancy to demonstrate these models. In normal pregnancy there is a decrease in blood pressure in early pregnancy followed by a rise in late pregnancy [11]. Hypertensive disorders of pregnancy (HDP), defined by high blood pressure in late pregnancy (after 20 weeks' gestation), are associated with risk of adverse health outcomes for both the mother and offspring [12-14]. and gestational weight gain (GWG) has been found to be positively associated with the risk of developing an HDP [15-17]. However, it is not clear whether changes in weight during pregnancy precede changes in blood pressure, or whether increases in blood pressure precede GWG because of increases in oedema and plasma volume expansion [18]. MAP is the average pressure in an artery over a complete cycle of one heart beat. It is estimated by combining systolic and diastolic blood pressure and allowing for the lower pressure during the diastolic phase of the cardiac cycle (see below for description of how this is calculated). In this paper we have used MAP so that we have one blood pressure variable that we can relate to weight change in pregnancy and that takes account of blood pressure in both systole and diastole. We show how MLMs and structural equation models (SEMs) with linear splines for gestational age can be applied to weight and MAP to learn whether an increase in weight precedes a rise in MAP, and in which periods of pregnancy associations are strongest. The paper is structured as follows: in Section 2 we describe the data. In Section 3 we define the multilevel multivariate response model and the derivation of regression coefficients from the variances and covariances of random effects. In Section 4 we show that the model can be equivalently defined in SEM form. In Section 5 we provide an application of the model to weight and MAP changes in pregnancy, and compare results from fitting the model in MLM software (MLWIN) (Rasbash J, Browne W, Healy M, Cameron B, Charlton C, Centre for Multilevel Modelling, University of Bristol, UK) and SEM software (MPLUS) (Muthen and Muthen, Los Angeles, California). Section 6 is a discussion of the methods described.