Accounting for overdispersed count data What could possibly go wrong? PSI Conference | 17th May 2017 Dan Lythgoe and Audrone Aksomaityte

Contents Count data in clinical trials Modelling count data Overdispersion Modelling excess variance An illustrative example Simulation study Bladder Tumour example Conclusions References

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

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

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 OVERDISPERSION ONLY APPARENT (Hilbe page 157) Add appropriate predictor Construct required interactions Transform predictor(s) Transform response Adjust for outliers Use correct link function Consequence of not sufficiently accounting for overdispersion is that standard errors may be underestimated.

Modelling excess variance Expectation Variance Poisson 𝜇 𝒙 𝜇 𝒙 Quasi-Poisson 𝜇 𝒙 𝜙𝜇 𝒙 Negative Binomial (NB-2) 𝜇 𝒙 𝜇 𝒙 +𝑘𝜇 𝒙 2 Heterogenerous Negative Binomial (NB-H) 𝜇 𝒙 𝜇 𝒙 +𝑘(𝒙)𝜇 𝒙 2 𝒙 is a vector of covariates. 𝜙 > 0, estimated using Pearson or Deviance statistic. 𝑘 Yee - require(VGAM) Hilbe - require(msme) SAS parameterisation of NB-2: https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_genmod_sect030.htm#statug.genmod.genmodrpd We are talking about SAS parameterisation. Large values of k represent large overdispersion (SAS), but this is not the case in R. > 0. Hilbe (2011), Yee (2015)

Research questions: How robust are NB-2 results when the dispersion parameters differs across treatment groups? Does NB-H offer any improvement? Random variables 𝑌 𝑖 , 𝑖=1,…,𝑛, represent counts with means 𝜇 𝑖 and 𝑉𝑎𝑟(𝑌 𝑖 )= 𝜇 𝑖 + 𝑘 𝑖 𝜇 𝑖 2 Models fitted ln( 𝜇 i )= 𝛽 0 + 𝛽 1 𝑥 𝑖 rnegbin() - The function uses the representation of the Negative Binomial distribution as a continuous mixture of Poisson distributions with Gamma distributed means.  Dispersion parameter NB-H: ln 𝑘 𝑖 = 𝛾 0 + 𝛾 1 𝑥 𝑖 NB-2: ln 𝑘 𝑖 = 𝛾 0

An illustrative example Two treatment arms, 𝑥𝑖 (Control = 0, Experimental = 1) Control: 𝑌 𝑖 ~𝑁𝐵(1,1), Experimental: 𝑌 𝑖 ~𝑁𝐵 1,5 50,000 patients per arm Results: Treatment group Observed NB-2 NB-H Control (0) Mean 1.0 Variance 2.0 3.4 Experimental (1) 6.0 3.5 NB2 k is 2.46

An illustrative example

Simulation study Two treatment arms, 𝑥 𝑖 (Control = 0, Experimental = 1) 100 patients per arm. Mean exacerbation rate for Controls = 1 exacerbation per patient-year (Sharafkhaneh et al. 2012). Dispersion parameter ( 𝑘 0 ) in Control group set to 1. Simulated data: 𝑌 𝑖 ~𝑁𝐵(1,1) if 𝑥 𝑖 =0 [Control arm], 𝑌 𝑖 ~ 𝑁𝐵(𝜃, 𝑘 1 ) if 𝑥 𝑖 =1 [Experimental arm]. 𝜃 ∈ 0.5, 1, 1.5, 2 𝑘 1 ∈{0.5, 1, 2, 3, 4, 5} 10,000 simulations per scenario. Fit NB-2 and NB-H models to each data set. Formoterol arm from Sharafkhaneh

Goodness of fit – low values good, high values bad CI Coverage - % of times that the CI includes true value. It should be ~95%. Values less that 95% indicate SE too small and type I error rate will be too high. Bias – measure of how much the estimated treat effect deviates from the true trt effect

The worst case scenario The worst case scenario. Mean on exp arm is lower but the dispersion parameter is higher

More realistic scenarion – mean in experimental arm is higher and so is the variance. NB2 in some scenarios performs poorly, whilst in some it performs surprisingly well. It appears to be a pretty robust model.

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) Andrews DF, Hertzberg AM (1985), DATA: A Collection of Problems from Many Fields for the Student and Research Worker, New York: Springer-Verlag. R survival package: http://127.0.0.1:12467/library/survival/html/bladder.html Size: min 1, max 8, median 1 Number: ranges from 1 to 8 Stop: range 1 to 64 months, median = 32 months.

Bladder tumour trial example Poisson model fit is poor ( 𝜒 2 = 195.41, 74 df, p < 0.0001) NB-2 model fit is good ( 𝑘 = 1.17, 𝜒 2 = 64.27, 74 df, p = 0.78) NB-H:   Estimate SE p exp(Estimate) 95% Lower 95% Upper 𝛽 0 Intercept -3.090 0.398 0.000 0.045 0.021 0.099 𝛽 1 Trt: Pyridoxine 0.165 0.363 0.650 1.179 0.578 2.403 𝛽 2 Number 0.085 0.095 0.371 1.088 0.904 1.310 𝛽 3 Size 0.025 0.098 0.802 1.025 0.846 1.242 𝛾 0 0.441 0.843 0.601 1.555 0.298 8.108 𝛾 1 1.281 0.642 0.046 3.602 1.023 12.678 𝛾 2 -0.337 0.312 0.279 0.714 0.388 1.315 𝛾 3 -0.097 0.184 0.597 0.907 0.633 1.301 1/0.056 = 1 recurrence every 17.86 months 1/0.066 = 1 recurrence every 15.15 months NB-2 AIC = 278.8145 NBH AIC = 280.6236 NBH model used to look closely for the source of overdispersion. The results made no difference on trt effect.   Estimated rate (per patient-month) 𝒌 𝒊 Estimated variance Placebo 0.056 0.672 0.058 Pyridoxine 0.066 2.422 0.077 RR: 1.179 (0.578, 2.403)

Conclusions Overdispersion is an important consideration in recurrent event modelling. There are 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.

References Agresti A, Categorical data analysis, Wiley 2002. Hilbe JM, Negative binomial regression, 2nd 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 pMDI 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.

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