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Presented By: Chris Gravel

Data Collection Methods -Spontaneous Reporting Systems (SRS) gather data passively from health care professionals and consumers -Some limitations of this method include -under/over reporting -multiple drugs/adverse drug reactions (ADR) -limited background information -dose, exposure time etc.

Assumptions for this discussion -to discuss the simplest form of these statistical methods, we will assume -single g drugg and ADR -no knowledge of the reporting probability and background information

The data -The data can be g gathered into a large g dimensional contingency g y table where the i-th row represents p a particular drug, Di , and the j-th column represents a particular ADR, A j . -The data can then be condensed into a 2 x 2 table by summing over the columns and rows: E t Event

Oth Events Other E t

Aj

ACj

nij

ni. − nij

ni.

DiC

n. j − nij

n.. − ni. − n. j + nij

n.. − n. j

Total

n. j

n.. − n. j

n..

Drug

Di Oth Drugs Other D

ni. = ∑ nij j

n. j = ∑ nij i

Total

n.. = ∑∑ nij j

i

-the random variable nij is an integer based, counting variable that records a count when an interaction between drug Di and ADR A j is observed.

Defining a Signal -aa signal, signal or a ‘suspiciously’ large interaction between drug and ADR ADR, is identified when the observed interaction appears larger than would be expected due to chance alone.

Proportional P ti l Reporting R ti Ratio R ti (PRR) : A ratio ti comparing i the th conditional diti l probability b bilit off observing b i ADR A j given i drug, Di relative to the conditional probability of observing A j with any other drug.

Reporting R ti Odds Odd R Ratio ti (ROR) : A ratio ti comparing i the th odds dd (probability ( b bilit off success/probability / b bilit off failure) f il ) of observing drug Di and ADR A j versus the odds of not observing the interaction.

Di Relative R l i Reporting R i Ratio R i (RRR) : A ratio i comparing i the h observed b d number b off occurrences off drug d and ADR A j to the expected number of occurrences of Di and A j under independence.

Sequential Probability Ratio Test (SPRT) : A comparison of the log likelihood ratio statistic to upper and lower boundaries which are functions of type 1 and type 2 error. The statistic is updated at regular time intervals until the point in which the statistic crosses a threshold.

Estimator

Signal Detection Criteria

nij

Proportional Reporting Ratio (PRR)

PRR =

⎡ 1 1 1 1 ⎤ exp ⎢ln( PRR) − 1.96 − + − ⎥ >1 nij ni. n. j − nij n.. − ni. ⎥⎦ ⎢⎣

ni . n. j − nij n.. − ni.

OR

PRR ≥ 2, χ Y2 ≥ 4, nij ≥ 3 Chi-Square with Yate’s Correction

Reporting Odds Ratio (ROR)

Yule’s Q

2

2

χ = ∑∑ 2 Y

i =1 j =1

ROR =

( nij − Eij − 0.5) 2 Eij

nij (n.. − n. j − ni. + nij ) (n. j − nij )(ni. − nij )

Q=

ROR − 1 ROR + 1

Eij =

ni.n. j n..

⎡ 1 1 1 1 ⎤ exp ⎢ln( ROR) − 1.96 + + + ⎥ >1 nij ni. n. j − nij n.. − ni. ⎥⎦ ⎢⎣

ROR − 1 1 1 1 ⎛1 ⎞ 1 − 1.96⎜ (1 − Q 2 ) ⎟ + + + >0 ROR + 1 ⎝2 ⎠ nij n. j − nij ni. − nij n.. − n. j − ni. + nij

Bayesian Methods and the Relative Reporting Ratio (RRR) The WHO method and the Information Component (IC) P( A j | Di ) =

If

IC = log 2

P( A j , Di ) P( Di )

P ( A j , Di ) P ( A j ) P ( Di )

=

P( A j , Di ) P( A j ) P( Di )

P( A j )

> 1 ⇒ P( A j | Di ) > P( A j )

nij n.. P ( A, D) which can be estimated by ICij = log 2 ni.n. j P ( A) P( D)

Signal detection criteria Iff the h 22.5-th h percentile il off the h distribution di ib i off IC C is i above b zero, a signal i l is i concluded. l d d

E ( IC ) − 1.96 * SD ( IC ) > 0

IC.025 > 0

Calculating the 2.5-th percentile of IC

The distribution of IC is difficult to determine analytically however, the distributions of the components of IC are easily attained given that the marginal counts are considered to be binomially distributed with parameters n.. and the respective probabilities. probabilities

These probabilities are then assigned prior distributions as

P ( A j ) ~ Beta (α 1 , α 0 )

P( Di ) ~ Beta( β1 , β 0 )

P ( A j , Di ) ~ Beta(δ1 , δ 0 )

Which results in the posteriors

P ( A j | ni. , n.. − ni. ) ~ Beta (α1 + ni. , α 0 + n.. − ni. )

P ( D j | n. j , n.. − n. j ) ~ Beta ( β1 + n. j , β 0 + n.. − n. j )

P ( A j , Di | ni. , n.. − ni.. , n. j , n.. − n. j ) ~ Beta B ( nij + δ1 , n.. − nij + δ 0 )

Approximations for the expectation and variance were derived using the delta method and exact derivations can be computed using the moment generating function technique. Approximate Expectation and Variance E ( ICij ) ≈ log 2

V ( ICij ) ≈

δ1 + nij . β1 + n. j α1 + ni.. − log 2 − log 2 δ1 + δ 0 + n.. α1 + α 0 + n.. β1 + β 0 + n..

⎞ β 0 + n.. − n. j δ 0 + n.. − nij α 0 + N − ni. 1 ⎛⎜ ⎟ + + log 22 ⎜⎝ (α1 + ni. )(α 0 + α1 + n.. + 1) ( β1 + n. j )( β 0 + β1 + n.. + 1) (δ1 + nij )(δ 0 + δ1 + n.. + 1) ⎟⎠

Exact Expectation and Variance E ( ICij ) =

1 (Ψ (δ1 + nij ) − Ψ (δ 0 + δ1 + n.. ) − [Ψ (α1 + ni. ) − Ψ (α 0 + α1 + n.. ) + Ψ ( β1 + n. j ) − Ψ ( β1 + β 0 + n.. )] ln(2)

V ( ICij ) =

1 (Ψ ' (δ1 + nij ) − Ψ ' (δ 0 + δ1 + n.. ) + [Ψ ' (α1 + ni . ) − Ψ ' (α 0 + α1 + n.. ) + Ψ ' ( β1 + n. j ) − Ψ ' ( β1 + β 0 + n.. )] ln (2) 2

where Ψ ( x ) = d ln(Γ ( x )) / dx Ψ ' ( x) = dΨ ( x ) / dx

Multiple authors have proposed different starting values for the parameters of the posterior distributions More recently a method for estimating the percentiles of the distribution of IC has been proposed using Monte Carlo simulation

The Gamma Poisson Shinker (GPS) and the Empirical Bayes Geometric Mean (EBGM) Let nij ~ Poisson ( μ ij ≡ λij Eij ) where

Eij =

ni.n. j n..

then if λ ij > 1 ⇒ μ ij > E ij (expectation under independence)

To ‘shrink’ shrink the estimates of λ ij we must first consider its prior distribution f (λij ) = P * Γ(λij ; α1 , β1 ) + (1 − P) * Γ(λij ; α 2 , β 2 )

where Γ represents the pdf of the gamma distribution

g (nij ) = ∫ h(nij , λij )dλij =P * NB(nij ;α1 , β1 , Eij ) + (1 − P) * NB(nij ; α 2 , β 2 , Eij ) where NB represents the pmf of the negative binomial distribution

and P is the prior mixing parameter To determine the values of the prior parameters, parameters max θ

∏ g (n ) ij

where θ = {α1 , α 2 , β1 , β 2 , P}

i, j

Posterior Mixing Parameter

Qnij =

P * NB(nij ; α1 , β1 , E ) P * NB(nij ; α1 , β1 , E ) + (1 − P) * NB(nij ; α 2 , β 2 , E )

The Posterior Distribution

p(λij | nij ) = Qnij Γ(λij ; α1 + nij , β1 + Eij ) + (1 − Qnij )Γ(λij ; α 2 + nij , β 2 + Eij ) The Posterior Expectation E (λij | nij ) = Qnij

α1 + nij α 2 + nij + (1 − Qn ) β1 + Eij β1 + Eij ij

E (log(λij ) | nij ) = Qnij [Ψ (α1 + nij ) − log(β1 + Eij )] + (1 − Qnij )[Ψ (α 2 + nij ) − log(β 2 + Eij )]

The Emprical Bayes Geometric Mean (EBGM)

Λ ij = exp( E (log(λij ) | nij )) Signal detection criteria The h 5-th h percentile il off the h distribution di ib i off Λ , denoted d d as EB05, can be b comparedd to 2 Shrinkage estimates

log(

n α +n ) ⎯n⎯→ log( ) ,E β +E E

however, for small values, Ψ ( x) ≤ log( x ) for x ≥ 0

⇒ E (log(λ ) | n) ≤ E (log(λ ))

Sequential Probability Ratio Test Let X = ( x1 ,,...,, xn ) '

For a general case

if xi ∈ R0 ⇒ H 0 accepted

Subdivide sample space into three regions (R) such that:

if xi ∈ R1 ⇒ H 1 accepted otherwise keep sampling

Define:

p 0 m & p1m

α

Are the probability distribution functions associated with the posterior probability that Ho or H1 are true respectively is the p probabilityy of rejecting j g Ho g given that it is true (type ( yp 1 error))

β

is the probability of rejecting H1 given that it is true (type 2 error)

β p1m ≤ ⇒ Accept H 0 p0 m 1 − α p1m 1 − β ≥ ⇒ Accept H1 p0 m α

General Decision Rule:

Otherwise, continue to sample

In Pharmacovigilance for example, H 0 : μ = φ ( RR ) If

H 1 : μ = φ ( 2 RR )

pim ~ Poisson ( μ ) i = 0,1

then

⎛1− β ⎞ l ( 2) ⋅ nij − Eij ≥ ln ln( l ⎜ ⎟ ⎝ α ⎠

Discussion: Frequentist methods versus Bayesian methods F Frequentist ti t Poor choice of the complement set of drugs and ADRs will bias the results These estimators heavily rely on the data and as a result the distributions of these estimators can be highly skewed The calculation of the estimators at a specific cell is subject to the ‘complement’ set which may contain extremely large observations influencing their value Small expected frequencies may tend to produce false positives (Type 1 error)

Bayesian The distribution Th di ib i off the h Relative R l i Reporting R i Ratio R i will ill be b pulled ll d towards d the h chosen h prior i with iha minimal data requirement Over time, the updating of the parameters will cause the posterior to move towards the ‘true’ distribution In other words, in the short term, the results of signal detection algorithms will be dependent on the i off the h prior i distribution di ib i appropriateness Bayesian methods lower impact of random fluctuations of the relative reporting ratio (shrinkage) This may cause false negatives (Type 2 error) Empirical Bayes is a computationally intensive algorithm

Further topics in Passive Pharmacovigilance Research

Higher Dimensional Associations Multiple Drug/ADR associations

Confoundingg Factors Covariate Stratification: Using pre-existing knowledge (eg. stratify on high risk demographic groups) to lower the effect of confounding variables Multiple l i l Logistic i i Regression: i Modelling d lli a predictor di variable i bl with i h respect to all ll possible ibl covariates (ie. the set of all relevant drugs) Unsupervised Pattern Recognition: Using data visualization techniques to identify how the data is clustered (organized)

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Gould, A.L. “Practical Pharmacovigilance analysis strategies,” Pharmacoepidemiology and Drug Safety, vol. 12, pp.559-574,, 2003. pp Orre, R. et al. “Bayesian neural networks with confidence estimations applied to data mining,” Comput. Stat. Data Anal. 34, 2000. van Puijenbrook, E.P. et al. “Detecting drug-drug interactions using a database for spontaneous adverse drug reactions: an example with diuretics and non-steroidal non steroidal anti-inflammatory anti inflammatory drugs. drugs Eur. Eur J. J Clin. Clin Pharmacol. Pharmacol 56, 56 2000 2000. Hocine, M.N, et al. “Sequential case series analysis for pharmacovigilance,” J. R. Statist. Soc. 172 part 1 pp. 213236, 2009. Ahmed, I et al. “Bayesian pharmacovigilance signal detection methods revisited in a multiple comparison setting,” Statist. Med., 28(13), 2009. Orre, R, et al. “A Bayesian recurrent neural network for unsupervised pattern recognition in large incomplete data sets,” International Journal of Neural Systems. Vol. 15 No. 3, 2005. Norén, G Norén G. et al al. “Case Case based imprecision estimates for Bayes classifiers with the Bayesian Bootstrap, Bootstrap ” Machine Learning. 58, 2005. Norén, G. et al. “Extending the methods used to Screen the WHO drug safety database towards analysis of complex associations and improved accuracy for rare events,” Statistics in Medicine. vol. 25, 2006. DuMouchel, W. and Pregibon D. “Empirical Bayes screening for multi-item associations,” Proc. of the Seventh AACM SIGKDD International conference on Knowledge Discovery and Data Mining, pp. 67-76, 2001. Gould, A.L. “Accounting for multiplicity in the evaluation of ‘signals’ obtained by data mining from p report p adverse event databases,” , Biometrical Journal, vol. 49,, pp. pp 151-165,, 2007. spontaneous

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