Poisson Mixture Models for Randomized Controlled Trials with Count Data

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Poisson Mixture Models for Randomized Controlled
Trials with Count Data

• Hideaki Uehara1 2, Kunihiko Takahashi1, Masako
  Nishikawa1, Toshiro Tango1
• 1 Department of Technology Assessment and
  Biostatistics,
• National Institute of Public Health, Japan
• 2 Tsumura Co. , Japan
• uehara_hideaki_at_mail.tsumura.co.jp

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Background

• We are interested in data from clinical studies
  for reducing risks of recurrent events (asthma,
  epilepsy, hot flash, migraine headache, etc.)
• Example of cross sectional data -gt
• To exclude possibly non-informative patient, we
  need screening.

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Study design
______
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xi0(Control) xi1(Test)

• Design Issues
• Allocation ratio (r) -gt Sample Size (for
  randomization)
• Patient selection criterion (c) -gt Sample Size
  (for screening) -gt Duration of study
• Lengths of study periods(t0,t1 ) -gt Patients
  burden

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Objective

• To build up appropriate and interpretable
  statistical models for the analysis of count data
  from pre-post comparative experiments with
  subject screening.

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Literature

• Truncated negative trinomial model
• McMahon et al(1994) Test Sample Size
• (frequency difference )
• Cook Wei (2002) Conditional model
• (frequency ratio )
• Cook Wei (2003) Sample Size

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Poisson-mixture model 1
ai Random effect for patient heterogeneity lt
Event frequency per unit time during period t
gt Covariate effect during period t zi
Covariate of i-th patient b treatment
effect d treatment by covariate interaction xi
Binary indicator of group for i-th patient
(0Control , 1Test)
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Dealing with patient heterogeneity

• Conditioning by total frequency
• Conditioning by the baseline frequency (ANCOVA
  type approach 23)

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Conditioning by total frequency

• Nit Poisson(µit), t0,1 Ni0Ni1ni? Ni1ni
  Binomial(ni, pi), piµi1/(µi0µi1)
• In other words, we can cancel out ai by
  considering the proportion of event counts
  observed in the treatment period for each
  patient.
• In term of interpretability, this patient-wise
  adjustment may be intuitively appealing.

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Conditioning by total number of events reflecting
screening
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Conditioning by the baseline frequency
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Decomposition of likelihood function
The first term is for truncated baseline
distribution. The second term is for conditional
distribution of response. Separation of c and b
in likelihood function means that b estimation
in this approach may be indifferent to c.
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Conditional Gamma-Poisson Model (Cook Wei, 2003)

• Ni0 Poisson(ui?l0t0) Ni1 Poisson(ui?l1t1ebxi)
  where uiGamma(a,a)
• gtNi1 ni0 Poisson(vi?yebxi(ani0)) where
  yl1t1/(al0t0),viGamma(ani0,ani0)

lt-screening
Linear relationship-gt
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Epilepsy study

• Leppik et al. (Neurology, 1987)
• A double-masked 2x2 crossover study for an
  anti-epileptic drug (Progabide)
• t0t18(weeks)
• 59 patients from 2 centers
• Age at entry 18 to 42 years
• Used as well Phenytoin / Carbamazepine with
  standard therapeutic dose monitoring

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Epilepsy study(2)

• Eligibility criteria
• Simple and/or complex partial seizures
• At least four seizures in one month of the two
  months prior to accrual and at least one in the
  other month prior to entry.

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Individual data from epilepsy study

• Thall and Vail (Biometrics, 1990)
• They showed only baseline and first treatment
  period data.
• For baseline period, only total seizure counts
  are reported. (Let us use c5.)

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Problems in Poisson-mixture model 1

• naïve to the outlier(?) or unable to deal with
  the unexpected extra-variation.

• Epilepsy data example
• (Fitted to the full dataset)
• Parameter Estimate se
• b -0.1016 0.06507
• y (/day) 1.1148 0.05230
• 1E-8 .
• (Fitted after deleting Outlier)
• Parameter Estimate se
• -0.2995 0.06976
• y 1.1148 0.05230
• a 1E-8 .

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Model by Diggle et al. (2002)(No consideration
for screening)

• Two types of heterogeneity
• Heterogeneity at baseline (background)
• Heterogeneity in pre-post change

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Poisson-mixture model 2
Log-normal mixture model
Finite mixture model
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(all assumes )
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Fitting results
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Summary of fitting results

• Diggle et al.s result suggests that the
  assumption of independence between ai and bi is
  reasonable.
• Treatment effect estimates given by models
  differed by distributional assumption about
  pre-post change heterogeneity.
• Model B1 is preferred according to AIC/BIC.

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Classification by Model B2 (K3)
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Concluding remark

• We developed four candidate models possibly
  suitable for our objective.
• Treatment effect estimate is dependent on the
  distributional assumption for heterogeneity in
  pre-post change.
• Superiority of our models over Diggle et al.s
  has not been clarified yet.

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Concluding remark (Cont.)

• Nevertheless, sensitivity analysis using our
  models can be useful to see the appropriateness
  of distributional assumptions or influence of
  screening, if any.
• Our model B2 can be useful for the interpretation
  of the pre-post data, if the necessary
  assumptions were acceptable.

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Thanks for your attention.