A Flexible Bayesian Method to Model Adverse Event Hazards

1
A Flexible Bayesian Method to Model Adverse Event
Hazards

• Quan Hong, Scott Andersen, Dave DeBrota
• Midwest Biopharmaceutical Statistics Workshop
• May 19, 2009

2
Value Proposition

• Model Adverse Event Hazard Rate
• Identify risk factors
• such as patient gender, ethnicity, or age
• Or rule them out
• E.g., not because of dosing duration.
• Applicable even when N is small (N 30 or 40),
  or n is small (n 3 or 5)

3
Motivational Example

• Preclinical animal toxicology studies
• Animals were given different dosages 50mg/kg,
  100mg/kg , 300 mg/kg
• Animals were dosed once a day for 3 months, 6
  months or 1 year.
• 5 10 animals per dose group
• Some animals had an emisis in higher dose group,
  while others did not through the duration of the
  studies

4
A Red dot indicates an animal having an emesis
2 dogs (out of 10) on 300mg/kg had events on day
2 and 10
2 dogs (out of 8) on 200mg/kg had events on day
30 and 60
1 dog (out of 14) on 100mg/kg had an event on day
200 and 300
5
Motivational Example (Contd)

• Was emesis due to increased doses?
• Was emesis due to longer dosing duration?

6
Modeling
7

• Poisson distribution expresses the probability
  of a number of events occurring in a fixed period
  of time if these events occur with a known
  average rate and independently of the time since
  the last event.

Average event rate
Number of events
8
How many emeses did the dog have on day 1?
on day d?
Day 1
Day 2
Day d
How many emeses did the dog had on day 2?
9
Model
is the number of adverse events a
patient i had on day d, then follows Poisson
distribution ,
where
is the average rate of adverse events rate of
patient i on day d
If AE rate is related to dosing duration (days),
If AE rate is also related to daily dosage (mg),
a and b are model parameters
Put Bayesian priors on a and b,
10
a, the dose coefficient
If a lt 0

• If a gt 0

If a 0
Dose
Dose
More AEs at higher doses
AEs evenly across doses
Not likely!
11
b, the time coefficient

• If b gt 0

Increased AE frequency with time
Time
If b 0
AEs evenly across time
Time
If b lt 0
Decreasing AE frequency with time
Time
12
The data look like
Dog 1 had an emesis on day 7, Dog 2 never had an
emesis during the 30 day study
Dog Dose Day Event
1 10 1 0
1 10 2 0
1 10 3 0
1 10 4 0
1 10 5 0
1 10 6 0
1 10 7 1
2 10 1 0
2 10 2 0
2 10 3 0
..... ..... ..... .....
..... ..... ..... .....
..... ..... ..... .....
2 10 29 0
2 10 30 0
Dog 1 had event on day 7
Dog 2 did not have event
13
Apply the Model in WinBUGS or R/BRugs
model for ( i in 1 N) responsei
dpois(lambdai) lambdai lt- a dosesi b
timesi a dnorm(0,4.0E4) b
dnorm(0,4.0E4)
14
Simulation Flow Diagram

• Step 1 Select pre-specified a and b, N ( of
  dogs), and T (duration of study)
• - e.g., a 0.0005, b 0.00001
• Step 2 Simulate a dataset according to selected
  and
• Step 3 Apply the model to get a and b
  estimators
• Step 4 Repeat steps 2 and 3 for M times, e.g (M
  1000), and get M sets of estimators
  and
• Step 5 Assess the simulation results of 1000
  sets of estimators for precision and bias

15
Simulation
16
Simulation Scenario I N is small

• a 0.0005, b 0.0001
• 3 dose groups 10, 20 and 50 mg/kg
• N 10 / group
• Typical simulated dataset shown below

17
(No Transcript)
18
Simulation Scenario II n is small

• a 0.0001, b 0
• 3 dose groups 10, 20 and 50 mg/kg
• N 10 / group
• Typical simulated dataset is shown below

19
Simulation Scenario II
20
Case Examples
21
Example I (Preclinical) Motivational Example
22
Example I - Results
23
Example I - conclusion

• Dog emesis rate increases with dose
• Dog emesis rate does not change with time

24
Example II Olanzapine Long-Acting Injection

• Post-injection syndrome (29 events in 41,193
  injections)
• Previously identified risk factors
• Logistic regression
• Dose
• Age
• BMI
• Also interested in question of constant hazard

25
Example II - WinBUGS Model
model for ( i in 1 41193) eventyni
dpois(lambda2i) lambdai lt- b0 b1 bmii
b2 agei b3 sdydosei b4
injnoi lambda2ilt-max(lambdai,0.00001)
b0 dnorm(0,6.25E6) b1 dnorm(0,6.25E6) b2
dnorm(0,6.25E6) b3 dnorm(0,6.25E6) b4
dnorm(0,6.25E6)
26
Example II - Results
27
Example II - conclusion

• Previously identified risk factors were
  significant
• Hazard rate does not increase over time
• Hazard for a specific patient
• age 50
• BMI 22
• dose 405mg
• 0.0019 (logistic regression calculated as 0.0022)

28
Flexibility

• The Poisson probabilistic part of the model
• Can be replaced with other distributions and
  models as determined by different problems, for
  example

29
Concluding Remarks

• AE hazard rate modeling
• Works well even when N is small or n is small
• Applies to a wide variety of data preclinical,
  clinical or market data
• Risk factors
• Identify risk factors, or rule them out.
• Address toxicity or safety concerns early and
  effectively.
• Flexibility
• May use different distributions, different priors
  or different hazard rate functions depending on
  situation
• Easy to implement using WinBUGS or R/BRugs