Parametric Simulation of Headache Day Frequency Using A Negative Binomial Distribution: A Case Study of Erenumab in Episodic Migraine

Abstract

A361with the observed values. Results: The squared errors for the NB distributions were consistently lower than the Poisson and binomial fits. Across the full 64 weeks, the total errors (average errors) were 0.1506 (0.009), 0.5493 (0.034) and 0.7003 (0.044) for the NB, Poisson and binomial distributions, respectively. NB distributions also consistently provided a better visual fit to the observed values. ConClusions: The use of NB distributions with estimated dispersion parameters provided estimates with lower errors in comparison with Poisson or binomial fits to the distribution of patients by headache day frequency. This approach provides a solution where distribution parameters cannot be observed directly, and facilitates the modeling of comparisons for which patient level data are not available.

Estimating the clinical effectiveness and value-based price range of erenumab for the prevention of migraine in patients with prior treatment failures: a US societal perspective

“…31 The risks of death are assumed to be unaffected by MMD or treatment, and life expectancy is parametric models (C). 33,34 As shown in hypothetical time points  and , the shape of the distribution of individual patients by MMD changes to account for both the mean MMD and the asymmetric spread of individual patients.…”

“…These distributions have previously been shown to accurately approximate the distributions of migraine day count data from the erenumab clinical studies. 33,34 In addition to the mean migraine day counts over 28 days (28 Bernoulli trials), the negative binomial and beta binomial distributions are characterized by additional parameters which account for the spread of individuals by migraine day frequency (the dispersion parameter and intraclass correlation coefficient, respectively). The longitudinal regressions provide estimates of these parameters, which are assumed constant across the patient population, irrespective of treatment and time.…”

Migraine day frequency in migraine prevention: longitudinal modelling approaches

“…The distributions in the erenumab cohorts of the studies were estimated and compared to the observed distribution across the double-blind period. Zero-inflated negative binomial models with robust standard errors clustering at patient level (presented in Additional file 1: Table S1 and Additional file 2: Figure S1) were also fitted, but only the non-zero-inflated were considered here because zero-inflated models did not improve the model’s fit, and there was no substantial inflation of zeros due to the eligibility criteria for the study [20].…”

“…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].…”

“…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].…”