Marginal modeling in community randomized trials with rare events: Utilization of the negative binomial regression model

Background/aims This work is motivated by the HEALing Communities Study, which is a post-test only cluster randomized trial in which communities are randomized to two different trial arms. The primary interest is in reducing opioid overdose fatalities, which will be collected as a count outcome at the community level. Communities range in size from thousands to over one million residents, and fatalities are expected to be rare. Traditional marginal modeling approaches in the cluster randomized trial literature include the use of generalized estimating equations with an exchangeable correlation structure when utilizing subject-level data, or analogously quasi-likelihood based on an over-dispersed binomial variance when utilizing community-level data. These approaches account for and estimate the intra-cluster correlation coefficient, which should be provided in the results from a cluster randomized trial. Alternatively, the coefficient of variation or R coefficient could be reported. In this article, we show that negative binomial regression can also be utilized when communities are large and events are rare. The objectives of this article are (1) to show that the negative binomial regression approach targets the same marginal regression parameter(s) as an over-dispersed binomial model and to explain why the estimates may differ; (2) to derive formulas relating the negative binomial overdispersion parameter k with the intra-cluster correlation coefficient, coefficient of variation, and R coefficient; and (3) analyze pre-intervention data from the HEALing Communities Study to demonstrate and contrast models and to show how to report the intra-cluster correlation coefficient, coefficient of variation, and R coefficient when utilizing negative binomial regression. Methods Negative binomial and over-dispersed binomial regression modeling are contrasted in terms of model setup, regression parameter estimation, and formulation of the overdispersion parameter. Three specific models are used to illustrate concepts and address the third objective. Results The negative binomial regression approach targets the same marginal regression parameter(s) as an over-dispersed binomial model, although estimates may differ. Practical differences arise in regard to how overdispersion, and hence the intra-cluster correlation coefficient is modeled. The negative binomial overdispersion parameter is approximately equal to the ratio of the intra-cluster correlation coefficient and marginal probability, the square of the coefficient of variation, and the R coefficient minus 1. As a result, estimates corresponding to all four of these different types of overdispersion parameterizations can be reported when utilizing negative binomial regression. Conclusion Negative binomial regression provides a valid, practical, alternative approach to the analysis of count data, and corresponding reporting of overdispersion parameters, from community randomized trials in which communities are large and events are rare.

Studying the Third Cumulant of the Mixture of Dirichlet-multinomial Distributions

Traditionally, the overdispersion parameter ϕ is estimated by using Pearson’s lack of fit statistic X2or the Deviance statistic D, which do not perform well in the case of sparse data. This paper particularly focuses on an estimator ϕnew of overdispersion parameter which was proposed for sparse multinomial data. The estimator was derived on the basis of an assumption on the 3rd cumulant of the response variable.When the data comes from the Dirichlet-multinomial distribution ϕnew is known to have the lowest root mean squared error comparing to the other three estimators. In this paper the 1st to 3rd order raw moments of the finite mixture of Dirichlet-multinomial distributions are derived, which results in complicated mathematical expressions. Furthermore, it is found that the 3rd cumulant of this mixture does not satisfy the assumption which is considered in the derivation of ϕnew . Dhaka Univ. J. Sci. 69(2): 96-100, 2021 (July)

Modelling menstrual cycle length in athletes using state-space models

The ability to predict an individual’s menstrual cycle length to a high degree of precision could help female athletes to track their period and tailor their training and nutrition correspondingly. Such individualisation is possible and necessary, given the known inter-individual variation in cycle length. To achieve this, a hybrid predictive model was built using data on 16,524 cycles collected from a sample of 2125 women (mean age 34.38 years, range 18.00–47.10, number of menstrual cycles ranging from 4 to 53). A mixed-effect state-space model was fitted to capture the within-subject temporal correlation, incorporating a Bayesian approach for process forecasting to predict the duration (in days) of the next menstrual cycle. The modelling procedure was split into three steps (1) a time trend component using a random walk with an overdispersion parameter, (2) an autocorrelation component using an autoregressive moving-average model, and (3) a linear predictor to account for covariates (e.g. injury, stomach cramps, training intensity). The inclusion of an overdispersion parameter suggested that $$26.36\%$$ 26.36 % $$[23.68\%,29.17\%]$$ [ 23.68 % , 29.17 % ] of cycles in the sample were overdispersed. The random walk standard deviation for a non-overdispersed cycle is $$27.41 \pm 1.05$$ 27.41 ± 1.05 [1.00, 1.09] days while under an overdispersed cycle, the menstrual cycle variance increase in 4.78 [4.57, 5.00] days. To assess the performance and prediction accuracy of the model, each woman’s last observation was used as test data. The root mean square error (RMSE), concordance correlation coefficient and Pearson correlation coefficient (r) between the observed and predicted values were calculated. The model had an RMSE of 1.6412 days, a precision of 0.7361 and overall accuracy of 0.9871. In conclusion, the hybrid model presented here is a helpful approach for predicting menstrual cycle length, which in turn can be used to support female athlete wellness.

Temporal changes in the risk of superspreading events of coronavirus disease 2019

In order to identify the temporal change in the possible risk of superspreading events (SSE), we estimated the overdispersion parameter in two different periods of COVID-19 pandemic. We identified the possible risk of SSE was reduced 90% during the second epidemic period in South Korea.

Temporal changes in the risk of superspreading events of coronavirus disease 2019

In order to identify the temporal change in the possible risk of superspreading events (SSE), we estimated the overdispersion parameter in two different periods of COVID-19 pandemic. We identified the possible risk of SSE was reduced 34% during the second epidemic period in South Korea.

Modelling Menstrual Cycle Length in Athletes Using State-space Models

The ability to predict menstrual cycle length to a high degree of precision enables female athletes to track their period and tailor their training and nutrition correspondingly knowing when to push harder when to prioritise recovery and how to minimise the impact of menstrual symptoms on performance. Such individualisation is possible if cycle length can be predicted to a high degree of accuracy. To achieve this, a hybrid predictive model was built using data on 16,990 cycles collected from a sample of 2,178women (mean age 33.89 years, range 14.98 - 47.10, number of menstrual cycles ranging from 4 - 53). To capture the within-subject temporal correlation, a mixed-effect state-space model was fitted incorporating a Bayesian approach for process forecasting to predict the duration (in days) of the next menstrual cycle. The modelling procedure was split into three steps(i) a time trend component using a random walk with an overdispersion parameter, (ii) an autocorrelation component using an autoregressive moving-average (ARMA) model, and (iii) a linear predictor to account for covariates (e.g. injury, stomach cramps, training intensity). The inclusion of an overdispersion parameter suggested that 26.81% [24.14%,29.58%] of cycles in the sample were overdispersed where the random walk standard deviation under a non-overdispersed cycle is 1.0530 [1.0060,1.0526] days while under an overdispersed cycle it increased to 4.7386 [4.5379,4.9492] days. To assess the performance and prediction accuracy of the model, each woman’s last observation was used as test data. The Root Mean Square Error (RMSE), Concordance Correlation Coefficient (CCC) and Pearson correlation coefficient (r) between the observed and predicted values were calculated. The model had an RMSE of 1.6126 days, a precision of 0.7501 and overall accuracy of 0.9855. In the absence of hormonal measurements, knowing how aspects of physiology and psychology are changing across the menstrual cycle has the potential to help female athletes personalise their training, nutrition and recovery tailored to their cycle to sustain peak performance at the highest level and gain a competitive edge. In conclusion, the hybrid model presented here is a useful approach for predicting menstrual cycle length which in turn can be used to support female athlete wellness to optimise performance

Superspreading in early transmissions of COVID-19 in Indonesia

This paper presents a study of early epidemiological assessment of COVID-19 transmission dynamics in Indonesia. The aim is to quantify heterogeneity in the numbers of secondary infections. To this end, we estimate the basic reproduction number $$\mathscr {R}_0$$ R 0 and the overdispersion parameter $$\mathscr {K}$$ K at two regions in Indonesia: Jakarta–Depok and Batam. The method to estimate $$\mathscr {R}_0$$ R 0 is based on a sequential Bayesian method, while the parameter $$\mathscr {K}$$ K is estimated by fitting the secondary case data with a negative binomial distribution. Based on the first 1288 confirmed cases collected from both regions, we find a high degree of individual-level variation in the transmission. The basic reproduction number $$\mathscr {R}_0$$ R 0 is estimated at 6.79 and 2.47, while the overdispersion parameter $$\mathscr {K}$$ K of a negative-binomial distribution is estimated at 0.06 and 0.2 for Jakarta–Depok and Batam, respectively. This suggests that superspreading events played a key role in the early stage of the outbreak, i.e., a small number of infected individuals are responsible for large numbers of COVID-19 transmission. This finding can be used to determine effective public measures, such as rapid isolation and identification, which are critical since delay of diagnosis is the most common cause of superspreading events.

Superspreading in Early Transmissions of COVID-19 in Indonesia

We estimate the basic reproduction number ℛ0 and the overdispersion parameter K at two COVID-19 clusters in Indonesia: Jakarta-Depok and Batam. Based on the first 397 confirmed cases in both clusters, we find a high degree of individual-level variation in the transmission. The basic reproduction number ℛ0 is estimated at 6.79 and 2.47, while the overdispersion parameter K of a negative-binomial distribution is estimated at 0.08 and 0.2 for Jakarta-Depok and Batam, respectively. This suggests that superspreading events played a key role in the early stage of the outbreak, i.e., a small number of infected individuals are responsible for large amounts of COVID-19 transmission.

Unifying the U–Pb and Th–Pb methods: joint isochron regression and common Pb correction

The actinide elements U and Th undergo radioactive decay to three isotopes of Pb, forming the basis of three coupled geochronometers. The 206Pb ∕238U and 207Pb ∕235U decay systems are routinely combined to improve accuracy. Joint consideration with the 208Pb ∕232Th decay system is less common. This paper aims to change this. Co-measured 208Pb ∕232Th is particularly useful for discordant samples containing variable amounts of non-radiogenic (“common”) Pb. The paper presents a maximum likelihood algorithm for joint isochron regression of the 206Pb ∕238Pb, 207Pb ∕235Pb and 208Pb ∕232Th chronometers. Given a set of cogenetic samples, this total-Pb/U-Th algorithm estimates the common Pb composition and concordia intercept age. U–Th–Pb data can be visualised on a conventional Wetherill or Tera–Wasserburg concordia diagram, or on a 208Pb ∕232Th vs. 206Pb ∕238U plot. Alternatively, the results of the new discordia regression algorithm can also be visualised as a 208Pbc ∕206Pb vs. 238U ∕206Pb or 208Pbc ∕207Pb vs. 235U ∕206Pb isochron, where 208Pbc represents the common 208Pb component. In its most general form, the total-Pb/U-Th algorithm accounts for the uncertainties of all isotopic ratios involved, including the 232Th ∕238U ratio, as well as the systematic uncertainties associated with the decay constants and the 238U ∕235U ratio. However, numerical stability is greatly improved when the dependency on the 232Th ∕238U-ratio uncertainty is dropped. For detrital minerals, it is generally not safe to assume a shared common Pb composition and concordia intercept age. In this case, the total-Pb/U-Th regression method must be modified by tying it to a terrestrial Pb evolution model. Thus, also detrital common Pb correction can be formulated in a maximum likelihood sense. The new method was applied to three published datasets, including low Th∕U carbonates, high Th∕U allanites and overdispersed monazites. The carbonate example illustrates how the total-Pb/U-Th method achieves a more precise common Pb correction than a conventional 207Pb-based approach does. The allanite sample shows the significant gain in both precision and accuracy that is made when the Th–Pb decay system is jointly considered with the U–Pb system. Finally, the monazite example is used to illustrate how the total-Pb/U-Th regression algorithm can be modified to include an overdispersion parameter. All the parameters in the discordia regression method (including the age and the overdispersion parameter) are strictly positive quantities that exhibit skewed error distributions near zero. This skewness can be accounted for using the profile log-likelihood method or by recasting the regression algorithm in terms of logarithmic quantities. Both approaches yield realistic asymmetric confidence intervals for the model parameters. The new algorithm is flexible enough that it can accommodate disequilibrium corrections and intersample error correlations when these are provided by the user. All the methods presented in this paper have been added to the IsoplotR software package. This will hopefully encourage geochronologists to take full advantage of the entire U–Th–Pb decay system.

Comparison Between Two Multinomial Overdispersion Models Through Simulation

A key assumption when using the multinomial distribution is that the observations are independent. In many practical situations, the observations could be correlated or clustered and the probabilities within each cluster might vary, which may lead to overdispersion. In this paper we discuss two well-known approaches to model overdispersed multinomial data, the Dirichlet-multinomial model and the finite-mixture model. The difference between these two models has been illustrated via simulation study. The forest pollen data is considered as a practical example of overdisperse multinomial data. The overdispersion parameter,φ, has been estimated using two classical estimators. Dhaka Univ. J. Sci. 68(1): 45-48, 2020 (January)