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)

Improved Landmark Dynamic Prediction Model to Assess Cardiovascular Disease Risk in On-Treatment Blood Pressure Patients: A Simulation Study and Post Hoc Analysis on SPRINT Data

Landmark model (LM) is a dynamic prediction model that uses a longitudinal biomarker in time-to-event data to make prognosis prediction. This study was designed to improve this model and to apply it to assess the cardiovascular risk in on-treatment blood pressure patients. A frailty parameter was used in LM, landmark frailty model (LFM), to account the frailty of the patients and measure the correlation between different landmarks. The proposed model was compared with LM in different scenarios respecting data missing status, sample size (100, 200, and 400), landmarks (6, 12, 24, and 48), and failure percentage (30, 50, and 100%). Bias of parameter estimation and mean square error as well as deviance statistic between models were compared. Additionally, discrimination and calibration capability as the goodness of fit of the model were evaluated using dynamic concordance index (DCI), dynamic prediction error (DPE), and dynamic relative prediction error (DRPE). The proposed model was performed on blood pressure data, obtained from systolic blood pressure intervention trial (SPRINT), in order to calculate the cardiovascular risk. Dynpred, coxme, and coxphw packages in the R.3.4.3 software were used. It was proved that our proposed model, LFM, had a better performance than LM. Parameter estimation in LFM was closer to true values in comparison to that in LM. Deviance statistic showed that there was a statistically significant difference between the two models. In the landmark numbers 6, 12, and 24, the LFM had a higher DCI over time and the three landmarks showed better performance in discrimination. Both DPE and DRPE in LFM were lower in comparison to those in LM over time. It was indicated that LFM had better calibration in comparison to its peer. Moreover, real data showed that the structure of prognostic process was predicted better in LFM than in LM. Accordingly, it is recommended to use the LFM model for assessing cardiovascular risk due to its better performance.

7. Fixed-Effects Negative Binomial Regression Models

This paper demonstrates that the conditional negative binomial model for panel data, proposed by Hausman, Hall, and Griliches (1984), is not a true fixed-effects method. This method—which has been implemented in both Stata and LIMDEP—does not in fact control for all stable covariates. Three alternative methods are explored. A negative multinomial model yields the same estimator as the conditional Poisson estimator and hence does not provide any additional leverage for dealing with over-dispersion. On the other hand, a simulation study yields good results from applying an unconditional negative binomial regression estimator with dummy variables to represent the fixed effects. There is no evidence for any incidental parameters bias in the coefficients, and downward bias in the standard error estimates can be easily and effectively corrected using the deviance statistic. Finally, an approximate conditional method is found to perform at about the same level as the unconditional estimator.