Migraine day frequency in migraine prevention: longitudinal modelling approaches

Migraine is a common neurological disorder characterised by debilitating, recurrent headaches, often divided into episodic (EM) and chronic (CM) forms based on month headache days (MHD) and monthly migraine days (MMD) (EM, 4–14 MMD and < 15 MHD, or CM, ≥15 MHD and ≥ 8 MMD) [1‐3]. Migraine pain is typically unilateral, pulsating in quality, of moderate or severe intensity and aggravated by routine physical activity, such as walking or climbing stairs. In addition, diagnosis depends on the presence of associated symptoms of nausea, vomiting, photophobia or phonophobia in various combinations [1, 2, 4]. The burden of migraine is considerable, both in terms of the physical and emotional effects on the individual, and the economic impact of lost productivity and healthcare resource use [5]. It is ranked as the leading cause of neurological disability worldwide and is one of the five leading causes of long-term disability [6, 7].

Preventive treatment intended to reduce the frequency and severity of headaches is an important aspect of management; all patients with CM would benefit from preventative treatment. Among patients with EM, experiencing 4 or more headache days per month is a leading reason for considering preventative therapy [8]. MMD and MHD are counts that have values that include zero as well as positive integers; count data typically have skewed distributions [9]. Reductions in the frequencies of migraine days (MDs) and headache days are key measures of the efficacy of migraine prophylaxis.

Clinical studies typically examine the mean change in MMD frequency; patient-level data are not widely published. However, examining the mean change in MMD frequency across a cohort of patients may not capture the clinically meaningful effects of migraine prevention, such as the improvement in an individual’s ability to perform daily activities or health-related quality of life. Furthermore, examining the mean change in the MMD frequency for a population in clinical studies may not be applicable in the real-world, as treatments may shift the frequency distribution.

A higher frequency of MMD per 28 days is associated with lower health-related quality of life, increased use of medical resources, acute medication use and increased productivity losses, with the impact of each additional MD increasing with overall frequency. As such, the average outcomes across a patient cohort may not be the same as the outcomes of a patient with the average MMD frequency. The frequency distribution of MHD and MMD is important when it comes to modelling the effectiveness and cost-effectiveness of prophylaxis [10]. Previous analyses examining cost-effectiveness models for migraine have approached this issue by defining health states as categorical event frequency (transition from ≥15 MHD to < 15 MHD) or as response status (≥50% reduction in MHD) [10‐13] which may not adequately account for inter-patient variability. These models group together a heterogeneous set of patient outcomes, rendering the models less precise; for example Markov models tend to categorise patients into broad categories, which can be challenging when assessing benefits. In general, categorising count/continuous variables can lead to several problems including loss of information and may also increase the risk of false positives [14]. Furthermore, use of a data-derived ‘optimal’ cutpoint may lead to bias [15]. Migraine is a disease with considerable variability in the frequency, duration and severity of migraine attacks [16]. Therefore, there is a need for an approach that estimates the change in mean frequency of MMD but also the distribution of individual patients by MMD frequency within a cohort at subsequent time points.

Selection of the most appropriate model is important when fitting MHD or MMD data [17]. There are several approaches to modelling these data. Reports on modelling MMD frequency in the literature are limited but previous analyses have used Poisson and negative binomial to model headache day frequency [17‐21]. Zero-inflated variants of these distributions have also been used to improve goodness-of-fit [17, 22]. The Poisson distribution belongs to the family of discrete probability distributions traditionally used to model count data. In general, the model assumes that the mean and variance of the count data are equal [23]. It is considered appropriate for unrestricted count data [24], and because MMD frequency is a count variable, Poisson distribution may be considered an eligible model. However, its ability to model the variation seen in the patient-level data has proved limited [20, 25] due to insufficient accounting for overdispersion (where a single parameter is insufficient to characterise the mean and variance) [26]. More recently, thanks to Shmueli et al. there has been a resurgence in interest in Conway-Maxwell-Poisson distributions, originally proposed by Conway and Maxwell to handle queuing systems [27, 28]. The main characteristic of these distributions, which are an extension of the Poisson distribution, is the ability to handle both overdispersion and underdispersion. These distributions are limited, however, due to the lack of a hierarchical model to assess repeated measurements and parameterization that is not made directly via the mean of counts, making these distributions not easily comparable to other count regression models. By contrast, the negative binomial distribution, which uses an additional dispersion parameter to represent the additional variation seen in the data, has provided superior fits when modelling migraine populations [17, 20].

A preliminary analysis, based on cross-sections of the data, has indicated that the beta-binomial is an alternative distribution that could be used to model MMD frequency data and has been shown to provide comparable fits to the negative binomial models [25]. The beta-binomial model is commonly used to account for intraclass correlation coefficients (ICC) among dichotomous outcomes in cluster sampling [29]. The use of the beta-binomial model may offer some advantages because the outcome can be restricted to a maximum number of possible successes (i.e. a maximum of 28 MMD per 4-week period).

In order to assess the feasibility of fitting MMD data using negative binomial or beta-binomial models, longitudinal data from two erenumab studies were examined [30, 31]. Erenumab is a fully human monoclonal antibody that specifically binds to and blocks the calcitonin gene-related peptide (CGRP) receptor [32]. Erenumab has been evaluated as a prophylactic treatment for migraine in 2 pivotal clinical trials in patients with EM and CM [30, 31, 33].

To the best of our knowledge, longitudinal negative binomial and beta-binomial regression models that accommodate over-dispersed data have not been used previously in the assessment of MMD frequency. Here, we describe an assessment of these models of the change in MMD frequency, using data from the placebo and erenumab 140 mg arms of two pivotal erenumab clinical trials.

This analysis is an assessment of the ability of longitudinal parametric models to capture intra- and inter-patient variability in MMD frequency over time, using data from two erenumab clinical trials as examples. Patients with migraine experience considerable day-to-day variability in the frequency, duration and severity of attacks [16]. This approach was used to estimate patient distribution accurately by the frequency of MMD using mean MMD values for the overall patient population. Modelling MMD with negative binomial and beta-binomial longitudinal regression models can be advantageous because they can accommodate overdispersed data (with a variance larger than the mean) and account for the variation in MMD both within and between individual patients.

The approaches described here allows the distribution of individual patients by MMD to be modelled using only the clinical endpoint of the studies - the mean change from baseline in MMD compared with placebo at a single time point. The beta-binomial regression method allows restriction of the maximum successes (i.e. maximum of 28 MDs), whereas the negative binomial does not. Despite this, the negative binomial showed a better goodness of fit to the MD distributions than beta-binomial. The modelled data from the negative and beta-binomial regressions show a closer fit with the observed values, compared with the Poisson reference model. The zero-inflated negative binomial regressions did not substantially improve the goodness of fit of the predicted distributions. In contrast to clinical trial populations that may have a lower bound of MDs per month, the zero-inflation model may be more useful in a real-world population where a greater proportion of people have zero MDs in a month.

The choice of distributions is important when measuring count data. The Poisson and negative binomial distributions have been used in previous studies to model count data [20, 23, 37] and have also been used to approximate headache day frequency data in published migraine studies [17, 38]. However, these distributions may be inappropriate when event counts are limited by a maximum possible frequency or measuring multimodal distributions. The Poisson and negative binomial distribution have indefinite support for positive integers and, therefore, have the potential to generate inappropriate values, especially with migraine cohorts of higher MD frequency.

Modelling data as continuous events rather than categorising data has many advantages, including the reduction of bias and more accurately estimating the extent of variation in outcomes between groups [14].This analysis takes the approach of modelling migraine frequency as a continuous outcome and addresses a key limitation of previous modelling approaches which define health states by categorical event frequency or response status. The proposed approach also provides a greater capability to model indirect comparisons than previous models, as the published endpoints of clinical studies (i.e. mean change in MMD) can be used to estimate the distributions of patients, assuming the patient-level variation is similar across cohorts. Using a count-based structure makes indirect comparisons straightforward because data can be linked to study primary endpoints. Estimating the distribution of patients by MMD also allows outcomes linked to MD frequency (such as health-related quality of life or pain medication use) to be quantified directly as a function of frequency. Furthermore, because clinical trials in migraine are commonly placebo-controlled, this approach could be used to parameterise indirect comparison in migraine prevention where patient-level frequency data are not available.

While this approach addresses key limitations of previous approaches, such as defining health states by categorical event frequency or response status, some potential improvements could be made to it. The implementation of a negative binomial regression with upper bound (28 MDs) could be considered and treatment-visit interactions could be included. Additionally, the data are required to fit to the smooth distributions of the model; however, this is not always the case. The predicted distributions observed in the CM study did not fit as well as the EM study owing to the greater spread in distribution in the CM study and may also be due to the differences in the patient populations between the EM and CM cohorts. Therefore, alternative approaches may be required to better model these cohorts.

The method described here has applications in economic evaluations of preventative medication and policy decisions in migraine. The parametric approach proposed can be used to perform extrapolations of treatment effects beyond trial observations. Extrapolation of data is particularly relevant when considering economic evaluations [39] as patient-level data collected within the duration of clinical studies are often too short to assess the long-term relationship between migraine frequency and health status. Further research may consider how such data should be extrapolated into the future, as whilst survival-modelled extrapolation has become well-established and standardised, the parametric approach is relatively novel, and the way in which the data can be best extrapolated is yet to be defined [40]. Furthermore, there is an inherent risk to extrapolation, as the clinical trajectory can be uncertain.

Modelling outcomes as continuous variables rather than health states has advantages when data are limited. Therefore, this approach has implications for use in various disease analyses which have simplified continuous outcomes associated with health states, which may result in loss of information or bias. This approach could be used to evaluate the disease progression of patients with HIV/AIDS, where multistate Markov models based on CD4 cell counts have previously been used [41] or modelling health assessment questionnaire (HAQ) scores in patients with psoriatic arthritis [42].

Modelling MMD with regression models that can accommodate overdispersion in a longitudinal framework is a statistically valid method to estimate the variation in MMD, both within and between individual patients. This approach, which estimates the distribution of patients by MMD, allows outcomes (such as health-related quality of life or pain medication use) to be directly quantified and linked to MD frequency. This has important applications in the evaluation of preventive medications for migraine and beyond.