© 2006 Lincoln Technologies Inc. A Phase Forward Company

Data Mining for Drug SafetyStatistical Analyses of Spontaneous Reports, Clinical Safety Data, and Longitudinal Medical Records

William DuMouchel, PhDChief Statistical Scientist

Rutgers University Biostatistics DayFebruary 22, 2008

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Legal Statement

This presentation contains forward looking statements within the meaning of the Private Securities Litigation Reform Act of 1995 relating to Phase Forward. These statements include, without limitation, statements related to the features and availability of one or more unreleased products. These statements are subject to a variety of risks and uncertainties including, but not limited to, technical difficulties encountered in the development of the planned product or changes in Phase Forward’s product plans which might result in the failure of Phase Forward to release the product as scheduled, as described, or at all, the failure of customers to purchase the product, or the availability of competitive products or services, as well as the risks set forth in Phase Forward's public filings with the Securities and Exchange Commission, including without limitation, its latest quarterly report on Form 10-Q. Phase Forward’s development plans are subject to change or withdrawal without further notice. We do not assume any obligation to update the forward-looking statements contained in this presentation.

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Data Mining for Drug Safety

Spontaneous reports• Disproportionality and Bayesian smoothing• Computer environment—Drilldown, signal management• Drug interaction signals

Clinical trial safety data analyses• CDISC standard—pooling events from many trials• Multiple comparisons—Many events, many subgroups• Bayesian smoothing of rare-event probabilities• Searching for unexpected syndromes due to treatment• Subgroup analyses

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Data Mining for Drug Safety (cont.)

Longitudinal Medical Records• FDA using and licensing databases• PDUFA IV – contains strong funded mandate for mining healthcare data for

safety– ‘‘…to provide for active adverse event surveillance– “…a postmarket risk identification and analysis system to link and analyze safety

data from multiple sources, with the goals of including, in aggregate—– (I) at least 25,000,000 patients by July 1, 2010; and– (II) at least 100,000,000 patients by July 1, 2012…”

Statistical Methodologies Common to All Data Types• Search for expected and unexpected associations• Bayesian shrinkage to resist biases due to post-hoc selection• Two-by-two tables with adjustment for covariates• Logistic regression and related methods

– Additive vs. multiplicative models– Screening for interactions

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Databases of Spontaneous Adverse Drug Reaction Reports

US FDA Spontaneous Report System (SRS/AERS)• Post-Marketing Surveillance of all Drugs since 1969• Renamed AERS (Adverse Event Reporting System) in 1997

– New ADR Coding System (COSTART vs. MedDRA)

• Version without Identifiers Available Publicly

US FDA/CDC Vaccine Adverse Events (VAERS)• Stricter US Laws for Vaccine Adverse Event Reporting

Other Databases for Medical Devices, etc.

World Health Organization VIGIBASE• Includes Data from many Countries• ADR Coding System WHOART

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Constructing a Denominator for N

For every DiEj pair = (Drug of Interest, Event of Interest)• Use the database to tabulate a 2 x 2 table of report counts• Compute an expected or baseline count e from (a, b, c, d)

– Based on assumption of no association between Drug and Event– e = b(a + c)/(b + d) [Proportional Reporting Ratio method]– e = bc/d [Reporting Odds Ratio method]– e = (a + b)(a + c)/(a+b+c+d) [Relative Report Rate: MGPS method]

• This method works best when adjusting for trend or demographic covariates in computation of e

• n/e = Measure of Disproportionality for this Drug and Event

Reports With Drug i

ReportsW/O Drug i Total

Reports With Event j nij = a b a + b

Reports W/O Event j c d c + d

Total a + c b + d a+b+c+d

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Disproportionality Analyses

Although the idea of computing n/e ratios for all or some drug-event combinations is simple, its widespread use is very recent

• Computer and database advances enabled ease of use and evaluation

Biostatisticians were uncomfortable with performing formal analyses on tabulations of spontaneous reports

• Unknown reporting mechanism can lead to reporting biases• Frequent noncausal associations with indications and comorbidities• All large values of n/e require follow-up for medical validity

Small values of n and/or e require statistical sophistication• PRR requires threshold values of n and 2 x 2 table chi-squared value• Bayesian statistical methods produce “shrinkage” values of n/e

– Help avoid the “multiple comparisons” fallacy• US FDA, UK MHRA and WHO UMC have each adopted Bayesian

disproportionality methods

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Combined Analysis of Drug-Event Counts in a Database

Large Two-Way Table with Possibly Millions of Cells• One Column for each Drug, One Row for each Event• Rows and Columns May Have Thousands of Categories• Most Cells Are Empty, even though N.. Is very Large

“Bayesian Data Mining in Large Frequency Tables”• The American Statistician (1999) (with Discussion)• SRS Database with 1398 Drugs and 952 AE Codes• Nij = Count of Reports Containing Drug i and Event j• Only 386 000 out of 1 331 000 Cells Have Nij > 0• 174 Drug-Event Combinations Have Nij > 1000• Develops and Illustrates Bayesian Estimation Method “GPS”

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Empirical Bayes Gamma-Poisson Shrinker (GPS Method)

Estimate λij = µij /Eij , where Νij ~ Poisson(µij )

Assume Superpopulation Model for λ• Prior Distribution Is Mixture of 2 Gamma Distributions• Estimate the 5-Parameter Prior from All the (Nij , Eij) Pairs

Posterior Distributions of each λij Are Used to Create “Shrinkage” Estimates

• EBGM = Empirical Bayes Geometric Mean of Posterior Dist.– Estimate of µij /Eij Has Smaller Variance than Nij /Eij

• Rank Cells by EB05ij = Lower 5% Point of Posterior Dist.• More “Interesting” than Ranking Cells Based on “P-Values”

– Compare (N = 10, E = 0.1) to (N = 2000, E = 1000)

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Plot of Classical Estimate with Conf. Int. and Bayesian “Shrinkage” Estimates [O]

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NISOLDIPINEACARBOSEORTHO-NOVUM1/8PLATECONCENTRHURSODIOLEMLAORTHO-NOVUMSQCOLFOSCEPALMITAVCHYDROCORTISONEGANCICLOVIRGLYCINESELENIUMSULFIDSUPROFENNONOXYNOLMINIZIDEPRILOCAINEFENFLURAMINEPENTAZOCINETROPICAMIDELEVONORGESTRELLEVONORGESTRELWARFARINNICOTINEDIATRIZOICACIDMINOXIDILPERMETHRININSULINHUMANESTRADIOLIOTHALAMICACID URTICARIA

APPLICASITREACNODRUGEFFECTNODRUGEFFECT

ALOPECIAURTICARIA

APPLICASITREACPROTHROMBINDEC

REACTUNEVALMETRORRHAGIA

MYDRIASISFIBROINJECTSIT

HYPERTENSPULMMETHEMOGLOBIN

HYPOKALEMPENISDIS

PAINKIDNEYSEBORRHEA

HYPONATREMRETINITIS

OTITISEXTBALANITIS

INTESTSMALLPERCARCINOMALARYN

HYPALGESIALIVEDAMAGAGGRALIVEDAMAGAGGRACARCINOMALIVERHEPATINONSPECIHEPATINONSPECI O

OOOOOO

OO

OOOOOOOOOOO

OOO

OO

OO

OOO

0.3 1 3 10 30 100 300 1000 10000 100000

Relative Risk

0.3 1 3 10 30 100 300 1000 10000 100000

Relative Risk

O Empirical Bayes Geometric Mean of RR ---|--- Observed RR with 99.9% Classical Conf. Int.

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Comparisons of NSAIDS in AERS

AERS to 3Q03 (Suspect drugs)

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Importance of Studying Drug Interactions

Drug interactions cause up to 2.8% of hospital admissions

(Grymonpre et al. J Am Geriatr Soc 1988)

50% of elderly take 5 drugs/week; 12% take 10 drugs/week

(Harvard Health Letter: March 2004)

The more medications you take, the greater your chance of a drug interaction

Brand Name Generic NameBaycol Cerivastatin

Propulsid CisaprideSeldane TerfenadineHismanal AstemizolePosicor Mibefradil

Drugs withdrawn due to severe drug interactions

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Simple Interaction Analysis Approach

Overview of Method• Yang, XM, Pharmacoepidemiology and Drug Safety 2004, 13, suppl 1: S247

• Introduce observed drug pairs as additional “pseudo-drugs”• Reports with 3 drugs can be treated as reports with 6 “drugs”

AbacavirCisaprideErythromycinAbacavir-CisaprideAbacavir-ErythromycinCisapride-Erythromycin

AbacavirCisaprideErythromycin pseudo

drugs

Drug Event N EBGM PRRCisapride Torsade de pointes 92 19.525 69.919 Erythromycin Torsade de pointes 58 20.425 13.227

Cisapride-Erythromycin Torsade de pointes 18 228.733 755.355

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Acetaminophen – Ethanol Interactions

Ethanol

Acetaminophen

Acetaminophen-Ethanol

1 3 10 30 70

EBGM (EB05, EB95)

Ethanol

Acetaminophen

Acetaminophen-Ethanol

1 3 10 30 70

EBGM (EB05, EB95)

Ethanol

Acetaminophen

Acetaminophen-Ethanol

1 3 10 30 70

EBGM (EB05, EB95)

Ethanol

Acetaminophen

Acetaminophen-Ethanol

1 3 10 30 70

EBGM (EB05, EB95)

Encephalopathy INTSS=5.11Encephalopathy INTSS=5.11

Hepatic failure INTSS=4.68Hepatic failure INTSS=4.68

Hepatic necrosis INTSS=4.97Hepatic necrosis INTSS=4.97

Hepatocellular damage INTSS=5.00Hepatocellular damage INTSS=5.00

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Masking – Effect of Background Rate

A relative reporting rate needs a denominator• This background or “noise” rate should ideally exclude effects

of predictors having very large signals

MGPS, PRR and similar methods naively assume that all reports excluding the one drug being focused on are background noise

• The “control group” may include other drugs with very high signals for the event of interest

• Analysis should estimate the effects of more than one drug at a time

– 2 x 2 Table analysis is too simple

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Confounding

Unless multiple predictors are themselves uncorrelated, one-predictor-at-a-time analyses can be biased

• GPS, PRR and similar methods don’t account for effect of Drug-Drug associations on Drug-Event associations

• Drugs that are often prescribed together can be confounded– Co-prescribed drugs partially inherit each other’s associations

• Synonymous terms– Signal leakage– Innocent bystander effect

Need a multivariate methodology

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Logistic RegressionFocus on specific events and drugs

• Medically related events may be fit separately or pooled as a combined response event

• Presence/absence of specific drugs of interest are primary predictors

Add covariates to the model as additional predictors• Dummy variables for age, gender, report year, etc.

Add frequent concomitant drugs as more predictors• E.g., Drugs corresponding to top 200 Counts for response event

or top 200 values of EB05 from MGPS analysis

Fit regression and convert to Odds Ratios and conf. limits• Non-overlapping confidence intervals worth investigating• Note patterns of agreement across events

More on this later

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Contrast Media and Nephropathy

90% Confidence Intervals based on 2x2 Tables (O) and LR (X)

Larger Disproportionalities from LR Probably Due to Masking• Many other drugs in the database also associated with renal problems• See Solomon and DuMouchel, Investigative Radiology (2006) 41:651-660

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Szarfman (2005 FDA Science Forum)

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Clinical Trial Safety DataSimilarities with spontaneous report data

• Could use 2 x 2 table analyses or logistic regression• Must cope with multiple comparisons

Differences• Smaller sample sizes• Usually just two treatments vs. thousands in database• Prospective study• Randomized allocation of treatment• More valid comparison group

Active surveillance studies• Large longitudinal database of medical records• Attempt to match users of two drugs with propensity scores• Maybe closer to clinical trial data than to spontaneous reports

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Pooling Data Across Trials

Combined analysis of multiple trials comparing the same treatment

• May be the best option for studying rare adverse events• Analyses can adjust for varying background rate per trial• Similar to pooled-data meta-analyses

CDISC data format standard for clinical trials• Consortium of FDA, drug companies, and software firms• FDA has announced eventual requirement for all NDAs• Several such NDAs have been submitted already• Some FDA reviewers are now doing partial conversions to CDISC

SDTM format to allow combined safety reviews• Software to take advantage of the data standard available or under

development by various vendors

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Bayesian Shrinkage ModelsStatistical validity of searching for extreme differences

• Most significant adverse event or patient subgroup

Classical approach to post-hoc interval estimates• Maintain centers of CI at observed differences• Expand widths of every CI • Expansion is greater the more differences you look at• If you look at too many, the CI’s are too wide to be useful

Bayesian approach • Requires a prior distribution for differences

– Can estimate it from the multiple observed differences available• Centers of CI’s are “shrunk” toward average or null difference

– High-variance differences shrink the most• Widths of CI’s usually shrink a little too• The more you look at, the better you can model the prior dist.

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Searching for Event ClustersAn event cluster (associated with treatment) is a set of at least three AEs for which all pairs of said AEs tend to show up in Treatment patients more often than Comparator patients and also more than expected if the AEs are independent within each arm of the study

• Defining potential syndromes by event frequency distributions rather than by theoretical medical mechanisms

We declare a potential syndrome if all pairs within a cluster meet some distributional threshold

• Syndromic Odds Ratio for 2 events (Treatment vs. Comparator)– SOR(E1,E2) = OR(E1*E2)/max[OR(E1), OR(E2), 1]

• Bayesian statistical methods estimate smoothed probabilities for AEs and pairs of AEs for each arm of the studies

– EB versions of Beta-binomial model seem to work well

• Clustering algorithms find groups of events having high SORs

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Potential SyndromesExample from a Trial with n = 902, nt = 676

• Most frequent 50 AEs are analyzed• Example cluster cut height < 2/3 [SOR > 1.5]

Six Clusters returned (Each cell corresponds to a PAIR of events)Row Syndrome

1 1 HPRKALPT 2 1 6/0 RENALIMPPT 3 1 2/0 3/0 HYPOKALAEMPT 4 1 3/0 3/0 2/0 ANEMIAPT

5 2 THIRSTPT 6 2 52/0 URINFREQPT 7 2 28/0 21/2 DRYMOUTHPT 8 2 5/0 5/0 7/0 HYPRNATRAEMPT

9 3 AGGRFTGPT 10 3 6/1 WGTINCRPT 11 3 4/1 3/0 ORTHOPNPT

12 4 HEADACHEPT 13 4 7/0 UTINFPT 14 4 5/0 3/0 ARTHRALGIAPT

15 5 NAUSEAPT 16 5 8/0 DIARRHPT 17 5 15/2 7/1 VOMITPT

18 6 CCFAGGPT 19 6 12/4 CONSTIPT 20 6 16/2 8/0 INSOMPT

Counts of Event Pairs (Treatment/Comparator) for Events Belonging to Potential Syndromes.

4 < SOR < 82 < SOR < 41 < SOR < 2

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Logistic Regression for Subgroup Analyses of Multiple Events

Start from a set of Medically Related events to study• Set of events from potential signal• Set of events from SOR clusters (potential syndromes)• Set of ad-hoc events, or all events within a MedDRA SOC

Fit Logistic Regressions to each AE as a response• Use exactly the same predictor model for each AE

– Age, gender, concomitant medication, medical history, etc.

• Include treatment and interactions with treatment as predictors• Generate parameter estimates for predictors and interactions

Empirical Bayes shrinkage of estimated coefficients• Coefficients of each predictor borrow strength across AEs• Overall treatment and interaction effects shrink toward 0

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Rationale for EB Model Across Events

Coping with fine grain of adverse event data• Compare T vs. C on 20 varieties of hepatic issues• Approach 1—separate analyses of all 20 events

– Small counts lead to non significant comparisons– Adjustment for multiple comparisons further reduces sensitivity

• Approach 2—define a single event as union of the 20 events– Significant differences may be washed out by the pooling– Even if significant, little information about original 20 differences

Compromise approach—EB hierarchical model• 20 individual estimates that “borrow strength” from each other• Let Bjk = coefficient of jth treatment effect/interaction on kth AE

– Bjk ~ N(µj, σj2) [prior distribution shrinks AEs toward each other]

– µj ~ N(0, τ2) [prior for overall treatment effects shrinks toward 0]

• Estimated prior variances σj2 and τ2 control amount of shrinkage

– Appropriate amount of shrinkage avoids multiple comparisons fallacy

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Display of Subgroup Effects

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Comparing Different AEs and Subgroups

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Safety Analyses of Clinical DataData mining of clinical trial safety data has many of the same challenges as analysis of spontaneous reports

• Although the data will be cleaner, there will be less of it and the multiple comparisons issues are just as significant

Combined analyses of multiple trials is important• CDISC data standards make pooling data easier• This is a form of pooled-data meta-analysis

Bayesian models can be useful here too• Multivariate estimation of many possibly related AEs• Searching for potential syndromes (different AEs in the same patients) that are

associated with treatment• Searching for subgroup effects• Borrowing strength across medically related AEs

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3 types of data: Spontaneous Reports

Strengths Weaknesses

Data already pre-focused on drug-adverse event relationships

Reasonable identification of drug and event (some naming and coding problems)

Timely, centralized collection and processing

Reporting by healthcare professionals is voluntary, with significant underreporting

Biases in reporting rate (varies by seriousness of event, length of time drug has been on market, manufacturer, publicity)

Lack of corresponding drug exposure data

Uneven quality of reports from multiple sources

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3 types of data: Clinical Trials

Strengths Weaknesses

High-quality treatment and control data consistently collected according to a defined protocolPatient randomization effectively eliminates biasGood identification of drugs and major events

Laboratory values (elevated liver enzymes) and other adverse event precursor data available

Relatively small total numbers of patients – cannot reliably detect low-frequency events

Included patient populations often under-represent important subgroups (pediatric, elderly, women)

Little data on long-term exposure; little long-term follow-up

High costs and long time periods before results are available

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3 types of data: E-Healthcare dataStrengths Weaknesses

Large patient populationsBalanced representation of patient subgroupsDrug dosing information available, can measure adverse event incidence and extent of risk, can potentially detect and quantify contraindicated and off-label usesReflects actual medical practice – realistic patients with varied demographics, existing medical conditions, concomitant therapies

Original data typically quite “raw” – time-stamped interventions, procedures, diagnoses; medical narratives; billing codes

Adverse events not typically identified per se

Substantial coding issues – diagnosis codes not oriented to capture adverse events, differences in individual and institutional coding practices (including “upcoding”)

Potential for gaps in data coverage (over-the-counter as well as prescription medications, inpatient as well as outpatient data, periods when patient is unenrolled or enrolled in some other healthcare plan)

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Challenge: recognizing AEs

Unlike clinical trials or spontaneous reports, AE’s are typically not explicitly noted in the data

Need to infer the occurrence of AE’s:• Diagnosis and procedure codes• Laboratory test results (if available)• Drugs used to treat AE’s

Significance of temporal patterns:• Treatment-emergent diagnoses• Grouping temporally-related diagnoses• Trajectories over time (worsening conditions or lab values)

Many AE’s masquerade as common illnesses

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Challenge: working with available coding

Excessive size and granularity of reimbursement-oriented healthcare coding space

Biases in diagnosis and procedure coding• Potential for billing-related “upcoding”• Opposite potential for insufficient care in coding

No easy correspondence between typical healthcare coding (ICD9) and pharmaceutical industry safety vocabulary (MedDRA)

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Challenge: the confounding problem in observational data

Data not coming from a designed experiment• No direct analog of randomization used for clinical trials• No protocol enforcing what observations are made and when

Many visible (and invisible) factors correlated with a patient’s treatment selection• Age, gender, race• Disease severity, medical and medication history, economics• Attitude and behavior (e.g., health-oriented individuals seeking the

latest and greatest medications)

These factors may be correlated with safety outcome

May be difficult to tease out true causes of differential safetyoutcomes

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WebVDME Healthcare – Overall Flow1. Longitudinal Medical Record Data

Filter

Events of interest

2. Derived Adverse Event Report Database

3. Drug-Event Signal Score Data Mart

4. Filtered Scores for Comparative Analysis

5. Drilldown to Subgroups and Patient Profiles

MGPS Data Mining Algorithm

Temporal Abstraction (generation of “reports”)

6. Risk Management Decisions

Case Series

Direct Access by Decision-Makers

Drug-event combinations not of interest; e.g.METFORMIN and DiabetesZOLOFT and Depression…etc.

Filter

Drugs of interestFilter

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Data transformation engineAutomatically generate “reports” from the healthcare data which capture the association in time between drugs and events

User-supplied parameters/choices determine precisely how the reports database is constructed from the original longitudinal database

Automatically create WebVDME configuration for transformed reports database

• New database available and ready for data mining analysis

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Data transformation engineMethod 1: Reports based on fixed time windows

• Divide each patient’s continuous data into (for example) monthly periods• Group all of the drugs and events occurring in each month into a “report” (i.e.

generating 12 reports per patient per year)• May also include drugs taken in preceding or subsequent months (with

distinct naming, depending on whether the drug is being take before, during, or after the monthly interval in question)

Method 2: Reports based on first occurrences of diagnoses• Consider every day in which a patient is diagnosed with one or more ICD-9

codes that are new to that patient as generating a report• Capture in the report the newly diagnosed ICD-9 codes as well as the drugs

taken by the patient in the previous m days and the subsequent n days• Use separate naming conventions for drugs taken in the previous m days

(“:p” drugs) and the subsequent n days (“:n” drugs)

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Patient Profile: Example of a Patient Taking Isoniazid (for Tuberculosis) and Experiencing a Hepatic Adverse Event

1

2

3

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Analysis Issues: Three Aspects of Sparseness Affecting Regression Modeling

Response sparseness• Each event is in a very small proportion of reports• The MedDRA dictionary has over 10,000 preferred terms (PTs)• Event frequencies of one per thousand or less are common

Predictor sparseness• Any particular drug is in a small proportion of reports• About 3,000 drugs (single ingredients) in many databases

Effect sparseness• Most drugs are unrelated to most events• Commonly use 200 or more predictors (demographic vars and drugs)• Bayesian shrinkage (regularization) useful

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Drug Effects: Additive or Multiplicative?

Logistic function is nearly linear in the region (0.2 < P < 0.8)• Effects of two drugs are nearly additive in this region

For small probabilities (P < 0.05) logistic is near exponential• Effects of two drugs are nearly multiplicative for small P

How to fit an additive effects model when P is small?• Would like to fit either type of effect combination and let data decide

-3 -2 -1 0 1 2 3

0.2

0.4

0.6

0.8

Z = Xb

Pro

babi

lity

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0

0.01

0.02

0.03

0.04

Z = Xb

Pro

babi

lity

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“Extended” Logistic Spline Link Function

“Stretch” standard logistic function• Move point of inflection from α = 0.5 to α = 0.1 (or any 0 < α < 1)• Pα(z) = 2α /[1 + exp(−2(1 − α)z)] [z < 0]• Pα(z) = 2α − 1 + 2(1− α)/[1 + exp(−2αz)] [z > 0]

Spline link family has two continuous derivatives wrt z, α

0 5 10 15

0.0

0.2

0.4

0.6

0.8

Logistic Spline with Knot at Probability = 0.1

Z = Xb

Pro

babi

lity

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Point of Inflection P = α DeterminesRegion of Additivity

Since Pα(z) is linear near P = α, if α = p = (#events)/(#reports)then predictors will have a nearly additive effect on P

It is natural to compare α = 0.5 with α = p

Goodness of fit of multiplicative vs additive models• Compare maximized likelihood functions• Compare need for interaction terms

Examples later

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Defining Odds Ratios If α ≠ ½

For the standard logistic link, the coefficients have the interpretation of being the logarithms of odds ratios

• exp(β) = OR(drug present, event present) [other predictors constant]• Assuming predictors are 0 or 1 and β is coefficient of the drug

For α ≠ ½, odds ratios depend on values of other predictors

Define “typical” OR based on setting other Xβ = z0• Define z0 = z0(α) by Pα(z0) = p

OR(β; α) = [Pα(z0+β)/(1−Pα(z0+β))] / [p/(1-p)] • Reduces to exp(β) if α = ½• Allows comparison of estimated effects for different values of α• Confidence intervals for OR based on confidence intervals for β

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Screening for InteractionsWhen there are hundreds of predictors, there are tens of thousands of potential interactions

• Not feasible to get MLE of all interaction coefficients either sequentially or simultaneously

One-pass algorithm to compare observed and expected counts for all pairs of dichotomous predictors

• Njk = number of response events when predictors j and k are present• Ejk = expected number based on fitted probabilities from model

Empirical Bayes gamma-Poisson model shrinks ratios Njk/Ejk

• True ratios λjk ~ Γ(γ, δ) with γ and δ estimated from all pairs (j, k)

• Posterior mean of λjk = (Njk + γ)/(Ejk + δ) estimates multiplicative effect of interaction due to lack of fit of model

• Estimates are robust to multiple comparisons fallacy

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Decomposing Drug Interaction Estimates

Physician’s definition of drug interaction• Is it more risky to take both drugs together than either separately?• Even if model fits, interest in ratio of joint risk to maximum of two risks• Due-to-regression Interaction = Pα(z0+βj+βk) / Pα(z0+max(βj,βk))• Note that this ratio depends importantly on the shape parameter α

Statistician’s definition of interaction• Due-to-lack-of-fit Interaction = λjk = (Njk + γ)/(Ejk + δ)

Total Interaction = (Due-to-regression) × (Due-to-lack-of-fit)• Either component can be larger, depending on the example• Tradeoff: better-fitting choice of α may increase due-to-regression

interaction but decrease due-to-lack-of-fit interaction

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Example: Myocardial Infarction (MI)

Graph of maximized log likelihood versus α• Model: 48 age-sex-year X’s plus the 100 drugs with most MIs• MLE α = 0.37; ∆logLike(0.5) = -94; ∆logLike(p=0.015) = -1279• Multiplicative model fits much better than additive model

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Estimates of Interactions for MIPredictor 1 Odds Ratio 1 Predictor 2 Odds Ratio 2 Total Count Observed Expected Regression Lack of Fit TotalGender_U 0.82 D_Abciximab 3.16 854 217 28.70 0.83 5.39 4.48Gender_U 0.82 D_Heparin 2.11 4578 264 56.46 0.83 3.94 3.26

D_Glyceryl Trinitrate 2.33 D_Sildenafil 3.57 248 70 44.32 2.15 1.45 3.11D_Sildenafil 3.57 D_Abciximab 3.16 1 1 0.14 2.78 1.09 3.04Gender_U 0.82 D_Aspirin 1.35 3332 296 70.12 0.83 3.68 3.04

D_Phenylpropanolamine 4.07 D_Alka-Seltzer Plus Col 9.10 1799 523 526.63 2.75 0.99 2.73D_Phenylpropanolamine 4.07 D_Abciximab 3.16 3 0 0.55 2.74 1.00 2.72

D_Metoprolol 1.54 D_Interferon-Beta-1a 1.98 163 27 9.35 1.51 1.77 2.67D_Clozapine 2.63 D_Abciximab 3.16 1 1 0.17 2.41 1.09 2.63

D_Glyceryl Trinitrate 2.33 D_Interferon-Beta-1a 1.98 72 13 5.92 1.91 1.37 2.62D_Clopidogrel 1.41 D_Interferon-Beta-1a 1.98 120 27 7.92 1.39 1.88 2.62D_Sildenafil 3.57 D_Phenylpropanolamin 4.07 33 5 8.73 3.01 0.86 2.59D_Sildenafil 3.57 D_Alka-Seltzer Plus Col 9.10 1 0 0.37 2.55 1.01 2.56

D_Valdecoxib 3.13 D_Phenylpropanolamin 4.07 13 2 3.46 2.72 0.94 2.56D_Glyceryl Trinitrate 2.33 D_Clozapine 2.63 87 9 6.24 2.19 1.16 2.54

D_Celecoxib 2.57 D_Clozapine 2.63 43 4 3.48 2.39 1.06 2.53D_Valdecoxib 3.13 D_Clozapine 2.63 1 0 0.06 2.41 1.03 2.48D_Sildenafil 3.57 D_Clozapine 2.63 1 0 0.08 2.38 1.03 2.45

Lack of fit estimate: λ = (Observed + 14.7)/(Expected + 14.2)

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Example: Lactic Acidosis (L.A.)

Maximized log likelihood has small “bump” near α = 0.001• Model: 48 age-sex-year X’s plus the 100 drugs with most L.A.s• MLE α = 0.041; ∆logLike(0.5) = -517; ∆logLike(p=0.0014) = -160• Near-additive model fits better than multiplicative model

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Dependence of Odds Ratios on α

Different link functions give similar OR estimates for L.A.

Unadjusted estimate

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SummaryNovel family of extended logistic link functions seems useful

• Fits additive, multiplicative and in-between models for sparse events

Interaction is defined relative to an implicit regression model • Good-fitting model reduces measures of lack-of-fit interaction• Comparing joint risk to single risk may be most important medically

– Can partition this ratio into due-to-regression and due-to-lack-of-fit

Purpose of examples is mainly to demonstrate methods• Spontaneous report associations are often not causal!• Extended LR model could be used with randomized trials

– But very sparse variables are less common with moderate sample sizes

EB model gets shrinkage estimates of lack-of-fit interaction• We are also working on EB models for coefficient estimation

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Thank you

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