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

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

2. 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

3. 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

4. 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.

5. 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:
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)

6. 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

7. 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

8. An illustrative example

9. 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

10. 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

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

12. 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.

13. 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 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:
Size: min 1, max 8, median 1
Number: ranges from 1 to 8
Stop: range 1 to 64 months, median = 32 months.

14. Bladder tumour trial example
Poisson model fit is poor ( 𝜒 2 = , 74 df, p < )
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 months
1/0.066 = 1 recurrence every months
NB-2 AIC =
NBH AIC =
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: (0.578, 2.403)

15. 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.

16. References Agresti A, Categorical data analysis, Wiley 2002.
Hilbe JM, Negative binomial regression, 2nd Edition, Cambridge University Press
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 ,
Yee TW, Vector generalized linear and additive models: with an implementation in R, Springer Series in Statistics 2015.

17. Thank you for listening. Questions?