Accounting for overdispersed count data What could possibly
Accounting for overdispersed count data What could possibly go wrong? PSI Conference | 17 th May 2017 Dan Lythgoe and Audrone Aksomaityte
Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 1 Count data in clinical trials Modelling count data Overdispersion Modelling excess variance An illustrative example Simulation study Bladder Tumour example Conclusions References This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Count data in clinical trials • Discrete counts, e. g. • Brain lesions in multiple sclerosis • Events that recur over time, e. g. • Incontinence episodes (bladder studies) • Exacerbations of asthma/COPD 2 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Modelling count data - Poisson regression – most common statistical model - Offset (ln(t)) used to account for different times of patient follow up. - Distributional assumptions: - Events are independent and occur randomly in time - Events have a constant rate per unit of time - Mean = variance 3 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Overdispersion • Often in practice the variance exceeds the mean • ‘Apparent’ versus ‘genuine’ dispersion • Causes of apparent overdispersion: • Outliers • Misspecified model (link function, missing predictors/terms, variable transformations, not accounting for excess zeros) • Causes of genuine overdispersion: • Events are correlated • Excess variation in counts Consequence of not sufficiently accounting for overdispersion is that standard errors may be underestimated. 4 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Modelling excess variance Expectation Poisson Quasi-Poisson Negative Binomial (NB-2) Heterogenerous Negative Binomial (NB-H) is a vector of covariates. > 0, estimated using Pearson or Deviance statistic. > 0. Hilbe (2011), Yee (2015) 5 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd Variance
Research questions: 1. How robust are NB-2 results when the dispersion parameters differs across treatment groups? 2. Does NB-H offer any improvement? Models fitted Dispersion parameter NB-H: NB-2: 6 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
An illustrative example • Results: Treatment group Control (0) Experimental (1) 7 Observed NB-2 NB-H Mean 1. 0 Variance 2. 0 3. 4 2. 0 Mean 1. 0 Variance 6. 0 3. 5 6. 0 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
An illustrative example 8 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Simulation study 9 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
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Bladder tumour trial example • Bladder tumour study by the Veteran Administration Co-operative Urological Research Group (Byar, 1980). • 74 patients (Placebo - 47 patients, Pyridoxine - 31 patients). • 144 events (Placebo – 87 events, Pyroxidine – 57 events). • Raw rates are approx. 0. 06 recurrences per patient-month in both arms. • Baseline variables: • Size: Size of largest initial tumour (cm) • Number: Initial number of tumours • Trt: Randomised treatment • Outcome variable: • Recur: Number of bladder cancer recurrences • Offset: • ln(Stop): ln(Time in months) 13 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Bladder tumour trial example Estimate SE p -3. 090 0. 398 0. 000 0. 045 0. 021 0. 099 Trt: Pyridoxine 0. 165 0. 363 0. 650 1. 179 0. 578 2. 403 Number 0. 085 0. 095 0. 371 1. 088 0. 904 1. 310 Size 0. 025 0. 098 0. 802 1. 025 0. 846 1. 242 Intercept 0. 441 0. 843 0. 601 1. 555 0. 298 8. 108 Trt: Pyridoxine 1. 281 0. 642 0. 046 3. 602 1. 023 12. 678 -0. 337 0. 312 0. 279 0. 714 0. 388 1. 315 Estimated 0. 907 0. 597 variance 0. 633 1. 301 Intercept Number Size Placebo Pyridoxine 14 Estimated rate 0. 184 -0. 097 (per patient-month) 0. 056 0. 066 0. 672 2. 422 exp(Estimate) 95% Lower 95% Upper 0. 058 0. 077 RR: 1. 179 (0. 578, 2. 403) This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Conclusions • Overdispersion is an important consideration in recurrent event modelling. • There a multiple options for incorporating overdispersion, including NB-2 and NB-H. • Simulation study: • NB-H: can offer improved standard errors compared with NB-2 if dispersion parameters differ considerably across treatment groups. • NB-2: quite robust to violation of the assumption of a single common dispersion parameter. • NB-H can be a useful alternative to NB-2 analysis for identifying sources contributing to heterogeneity. 15 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
References • Agresti A, Categorical data analysis, Wiley 2002. • Hilbe JM, Negative binomial regression, 2 nd Edition, Cambridge University Press 2011. • Law M. , et al. Misspecification of at‐risk periods and distributional assumptions in estimating COPD exacerbation rates: The resultant bias in treatment effect estimation. Pharmaceutical Statistics 2016, 1 -9. • Sharafkhaneh A, et al. Effect of budesonide/formoterol p. MDI on COPD exacerbations: A double-blind, randomized study. Respiratory Medicine 2012 106, 257 -268. • Yee TW, Vector generalized linear and additive models: with an implementation in R, Springer Series in Statistics 2015. 16 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
Thank you for listening. Questions? 17 This document is property of VIVOS Technology Ltd and may not be copied or communicated without the written consent of VIVOS Technology Ltd
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