Using joint models to disentangle intervention effect types and baseline confounding: an application within an intervention study in prodromal Alzheimer’s disease with Fortasyn Connect

The LipiDiDiet trial is a 24-month randomised, controlled, double-blind, multi-centre trial, performed at 11 study sites across different countries. The goal of the LipiDiDiet trial was to investigate the effects of Fortasyn Connect on cognition and related measures in prodromal AD patients. For this purpose, several longitudinal measures of cognitive functioning were recorded. In this paper we include two of them: the Clinical Dementia Rating sum of boxes (CDR-SB) and memory domain from a neuropsychological test battery (NTB memory domain).

The CDR-SB score reflects global clinical impression and ranges from a score of 0 to 18, with a higher score indicating a worse status. It is obtained through a semi-structured interview of patients and informants, summing scores of cognitive functioning on each of the following domain box scores: memory, orientation, judgement and problem solving, community affairs, home and hobbies, and personal care.

NTB memory domain is a composite z-score based on Consortium to Establish a Registry for AD (CERAD) 10-word list learning immediate recall, CERAD 10-word delayed recall, and CERAD 10-word recognition. A higher z-score indicates a better memory.

Individual patients’ scores were measured at baseline, where randomisation to either the test or control group took place, as well as around months 12 and 24 with an additional visit around 6 months for NTB memory domain. At each visit it was recorded whether patients had received the diagnosis of dementia. Progression to dementia was diagnosed according to criteria defined by DSM-IV, the National Institute of Neurological and Communicative Disorders and Stroke, and the AD and Related Disorders Association criteria for AD.

In this article, we focus on AD dementia as a specific form of dementia. The study sample consisted of 311 patients (modified intention-to-treat population in the LipiDiDiet main paper [22]), of whom 57 (36%) patients in the control group and 62 (41%) in the test group had received the AD dementia diagnosis. The median follow-up times were respectively 1.96 years in the control, and 1.94 years in the test group. Despite the randomisation procedure, a statistically significant difference between the intervention groups was found in baseline Mini–Mental State Examination (baseline MMSE, p=0.039, two-sided t-test), reflecting baseline cognitive performance. The higher baseline MMSE score in the control group denotes better performance and suggests a lower risk of receiving the dementia diagnosis in this group at baseline. Figure 1 displays the histograms of baseline MMSE scores in the test and control group. For further information regarding the LipiDidiet trial, including information on the randomisation procedure, we refer to the LipiDiDiet main paper [22].

As the name suggests, a joint model for longitudinal and survival data consists of a longitudinal sub-model and a survival sub-model. The longitudinal sub-model is typically a mixed effects model aiming to describe the shapes of the patient-specific longitudinal profiles. For continuous longitudinal data, linear mixed models can take into account that repeated measurements from the same patient may be more correlated than measurements from other patients, by including not only fixed effects but also patient-specific random effects. For background information on mixed models, we refer to Verbeke & Molenberghs (1997) [25] and Fitzmaurice et al. (2008) [26].

In order to formulate our longitudinal sub-model for the longitudinal trajectories, as a first step we investigated the observed longitudinal profiles for six randomly selected patients. Figures 2 and 3 show the longitudinal profiles for respectively their CDR-SB and NTB memory domain observations; these figures show that there is a lot of variation between patients. Therefore, we allowed each patient to have its own trajectory, by incorporating patient-specific intercepts and slopes. For the average CDR-SB and NTB memory domain trajectories we used linear effects of time (β₁) but more complicated functions of time such as quadratic or higher order polynomials, e.g., using splines, are also possible [27]. We also tried trajectory functions for CDR-SB and NTB memory domain using quadratic time effects, but these were found to give similar results (results not shown). To model the effect of Fortasyn Connect, we included both a main effect of the intervention (β₂) and an interaction of intervention by time (β₃) in order to allow the trajectories of the intervention groups to be different over time. This is necessary, because the intervention is expected to have a gradual effect, possibly resulting in CDR-SB and NTB-memory domain levels for the test group that are worsening more slowly. Further, we included and intercept (β₀) and main effects for baseline MMSE (β₄) and site (β₅). This resulted in the following longitudinal sub-model for the CDR-SB observations, and similarly defined for NTB memory domain where CDR_i(t) are the observed values of CDR-SB for patient i at actual time points t, and the time points at which measurements take place may vary between patients. Further b_i0 and b_i1 denote respectively the patient-specific intercept and slope. The longitudinal profile of observed values CDR_i(t) is broken down in a trajectory function \(\tilde {\text {CDR}}_{i}(t)\) and a random error term ε_i(t), which is assumed to be normally distributed. The trajectory function is assumed to describe the true but unobserved trajectory of the longitudinal marker, and as will be seen later, is used in the survival sub-model ‘joining’ the two sub-models. The main effect of the intervention, β₂, denotes the difference between the intervention groups at baseline, while the interaction effect β₃, describes the intervention effect over time.

A common choice for the survival sub-model is a Cox model, which is used to model the hazard of experiencing the event, i.e., in this case receiving the dementia diagnosis. For background information on Cox models, see Cox (1972) [28], Klein & Moeschberger (1997) [29] and Therneau & Grambsch (2013) [30]. In case the proportional hazard assumption of the Cox model is violated, alternative modelling frameworks for the survival sub-model exist, such as the accelerated failure time model [31]. In this paper we formulated our joint model using a Cox model. Additionally, we fitted a joint model using an accelerated failure time model which gave similar findings (results not shown). In the Cox model we included the intervention as a time-independent effect, and the estimated true trajectory of the longitudinal marker as a time-varying effect. Since there can be variation across sites in how early a patient is diagnosed, we also corrected for site in the survival sub-model. The hazard λ_i(t) of dementia diagnosis at time t for patient i is therefore modeled using the following survival sub-model, where the parameter α links the longitudinal process, i.e., the trajectory function \(\tilde {\text {CDR}}_{i}(t)\), or similarly \(\tilde {\text {NTB}}_{i}(t)\), to the survival process. More specifically, the quantity exp(α) denotes the hazard ratio at time t for a one-unit increase in the trajectory of the longitudinal marker at the same time point. Further, λ₀(t) is the baseline hazard and γ₁ denotes a direct effect on the survival outcome. To gain a better understanding of how the intervention affects the risk of receiving the dementia diagnosis, and to explain what we mean by a ‘direct effect’, we distinguish three types of coefficients. These are schematically illustrated in Fig. 4. β describes the intervention effect on the longitudinal marker. As indicated before, there are two types of β’s here; β₂ denoting the difference in the longitudinal outcome between the intervention groups at baseline, and β₃ describing the intervention effect on the longitudinal outcome over time. Secondly, since the parameter α measures the effect of the longitudinal process on the survival outcome, together, β₃ and α, quantify the time-varying intervention effect on the risk of receiving the dementia diagnosis manifesting through the longitudinal marker. The third type of parameter involving the intervention is γ₁ and is directly related to the risk of receiving the dementia diagnosis. Within the joint model we can therefore distinguish the direct process (Fig. 4, bottom arm), capturing the direct effect on the survival outcome, and the indirect process (Fig. 4, upper arm), in which the coefficients quantify the indirect intervention effect on the survival outcome through the longitudinal marker.

As is the case for the Cox model, the intervention effect in the joint model is the hazard ratio of the test versus the control group. In particular, the total intervention effect is the hazard ratio between two generic patients, i in the test group (fortasyn_i=1) and i^′ in the control group (\(\text {\texttt {fortasyn}}_{i'} = 0\phantom {\dot {i}\!}\)) who do not further differ concerning other covariates. In the joint model, this hazard ratio is a combination of the indirect and direct process. For our formulated joint model the hazard ratio for the total intervention effect denotes exp{γ₁ +α(β₂+β₃×t)}, with the first part (i.e., γ₁), for the direct process and the latter (time-varying) for the indirect process.

The two processes of the joint model differ in how they handle the aspect of time. The indirect process can model how an intervention effect varies over time by modelling the intervention effect in the mixed model as a divergence of trajectories. In the direct process however, we are dealing with the proportional hazard assumption of the Cox regression model meaning that the direct effect on the survival outcome is assumed to be constant over the whole period. This arm is therefore likely to capture the effects already present right at the start of the intervention period. In a situation such as this one, where the effect of the intervention on the survival outcome - manifesting through the longitudinal marker - is expected to increase gradually over time, but an effect of any possible baseline confounding on the survival outcome is expected to be immediate, the baseline confounding will for a large extent end up in the direct arm of the model. This is a very appealing property of the joint model that makes it a very effective tool to investigate and control for the effect of potential baseline confounding.

MMSE, found to be significantly different at baseline, is noted to be an important predictor for outcome parameters [22]. This suggests that, before the start of the intervention, the test group might on average have been more likely to receive the dementia diagnosis than the control group due to an imbalance of baseline characteristics. This hampers the interpretability of the results as post baseline outcomes are a combination of the intervention effect and the effect of differences already present at baseline. To investigate this, we examined whether the lower baseline MMSE scores in the test group were related to higher risks of receiving the dementia diagnosis at the start of the trial, therefore possibly counteracting the intervention effect. This required fitting an additional joint model in which we corrected for the effect of the baseline MMSE score on dementia diagnosis by including its value in the survival sub-model, given by

We will illustrate how the coefficients of this extended joint model can be an effective tool to investigate and control for the effect of potential baseline confounding using the CDR-SB data from the LipiDiDiet trial.

Naturally, apart from being a useful property in investigating possible baseline confounding, the combination of an immediate (direct) and a progressive (indirect) effect helps us to understand the process by which the intervention affects the risk of dementia diagnosis.

Another aspect of the process by which the intervention affects the timing of dementia diagnosis is determined by the type of the association between the longitudinal and the survival process. The joint model defined in the previous section is the standard joint model and assumes that the value of the longitudinal outcome at any time t is related to the risk of an event at the same time point. However, the underlying relationship between the two processes could have a more complex nature. Examples of longitudinal characteristics possibly related to dementia diagnosis, are the current value, the stability at the current moment, the history of the longitudinal profile up to now or combinations of these characteristics [9]. For demonstration purposes we compared joint models that vary with respect to the type of association that is assumed between the longitudinal data i.e, NTB memory domain, and the survival process i.e., timing of dementia diagnosis. We investigated whether, given the current value of NTB memory domain, the rate of change (i.e., the slope) contains any additional information on the risk of receiving a dementia diagnosis. More specifically, the slope indicates by how much the NTB memory domain for a particular patient is increasing or decreasing at a specific time point. This required fitting a joint model in which we included the slope of NTB memory domain as an additional term in the survival sub-model, given by

where the slope of NTB memory domain is obtained by taking the derivative of the trajectory function, consisting of the fixed and random effects, that is, The parameter α₁ has the same interpretation as the parameter α before, and the parameter α₂ measures the association between the slope of the NTB memory domain trajectory and the risk of an event at the same time point, holding \(\tilde {\text {NTB}}_{i}(t)\) constant. Using this joint model, two patients with the same level of NTB memory domain at the current moment do not necessarily have to be at equal risk of receiving the dementia diagnosis. For example, if one patient’s NTB memory domain level is decreasing very rapidly while another patient’s NTB memory domain level is remaining constant, it might be more realistic to assume that the first patient has a higher risk of receiving the dementia diagnosis than the latter - although they have the same value at the current moment.

A similarity between this joint model and the standard joint model, is that the risk of an event at the current moment is related to characteristics of the trajectory at that same time point only. However, the risk of receiving the dementia diagnosis may not depend solely on the level of NTB memory domain or its rate of change at the current moment, but it might also be related to the history of the NTB memory domain levels. That is, two patients with the same characteristics at the current moment are not necessarily at the same risk of receiving a dementia diagnosis if their history of NTB memory domain levels were very different. One approach to take the history of NTB memory domain levels into account is by summarising its cumulative effect i.e., the area under the curve (AUC). The area under the curve indicates the cumulative effect of NTB memory domain values for a particular patient up to the current time point. We also investigated this type of association by fitting a joint model with the following survival sub-model

where α₃ measures how strongly the risk of an event at time t is related to the cumulative effect of NTB memory domain for patient i by time point t. A possible limitation of this joint model is that it gives all past values of NTB memory domain the same weight in terms of their impact on the risk of receiving the dementia diagnosis at the current time point. This may not always be a reasonable assumption. As an alternative, a weight function can be used that places different weights at different time points, for example to give more weight to more recent values of the longitudinal marker. For information on how to use this weight function we refer to [9].

Figure 5 gives a graphical representation of different ways of modelling the association, respectively using the current value, the current value plus the rate of change and the cumulative effect of the longitudinal trajectory.

All the statistical analyses in this paper were performed with statistical software package R, using R-package JM [32]. The package uses maximum likelihood for the parameter estimation and assumes right-censoring. The R code to fit the joint models can be found in the web appendix (Additional file 1).