Jakobs Duke 2015 Analysis of recurrent adverse events of ... · Aalen OO, Borgan Ø, Gjessing HK....

CONFIDENTIAL © 2015 PAREXEL INTERNATIONAL CORP.

ANALYSIS OF RECURRENT

ADVERSE EVENTS OF

SPECIAL INTEREST:

AN APPLICATION FOR

HAZARD-BASED MODELS

23 Oct 2015 – Duke Industry Symposium

Peter Jakobs

Sen Dir Biostatistics, PAREXEL

© 2014 PAREXEL INTERNATIONAL CORP. / CONFIDENTIAL2

AGENDA

TOPICS FOR

TODAY

� Motivation (limitations of standard practice)

� Basic multistate models for safety event history

� Brief introduction to multistate models

� Example and estimations

� Summary and conclusions

� Some references


MOTIVATION (1)

Typical table for adverse events:

N’s based on “received at least one dose of study treatment”

Safety analysis set


MOTIVATION (2)

Such tables are lacking information about

� relationship to exposure / duration of follow-up

� duration of events

� potential resolution (or any other outcomes) of events

� event recurrence: 2nd, 3rd, .. AE?

� timing of event occurrence


MOTIVATION (3)

Can not infer from the red curve: higher risk at start of Drug A with subsequent development of tolerance

Time to first safety event might be misleading:

Source: Novartis 2008


BASIC MULTISTATE MODELS FOR SAFETY EVENT HISTORY (1)

standard AE incidence approach (like the table on slide 3):

� πAEX is the probability to experience at least one AE X of special interest (at any point in time) after start of study treatment

� irrespective of duration of exposure or length of observation !

� may work for AEs that usually occur very early

� poor approach for AEs that occur rather late and in studies where length of observation differs between subjects and treatment groups

� provides no information about event duration, event resolution, event recurrence

No AE X AE XπAEX



More realistically to assume the “risk“ to experience a specific AE (AE X) changes over time t (t=0 denotes start of study treatment)

� can be modeled by hazard function hAEX(t) for event occurrence

No AE X AE XhAEX(t)

For consideration:

� May the hazard function change for the re-occurrance of same AE of special interest? Hazards h1st AEX(t) and hrecur AEX (t*)

� What about the time axis?

� t*=0 at start of study treatment (i.e., continuation of time) or

� t*=0 at resolution day of previous AE X (i.e., “reset the clock“)?



Resolution and duration of an AE matter!

� can be modeled by additional hazard function hAEX,no AEX(t) for resolution of AE X

No AE X AE X

hAEX(t)

For consideration (similar as one previous slide):

� Hazards for resolution same or different for re-occurrance of same AE of special interest? Hazards hfirst AEX, no AEX(t) and hrecur AEX, no AEX(t*)?

� What about the time axis?

� t*=0 at start of study treatment (i.e., continuation of time) or

� t*=0 at respective onset day of AE X (i.e., “reset the clock“)?

hAEX,no AEX(t)



Better understanding of a process:

� AE X itself is not a significant AE but suspected/known to increase the risk for life threatening AE Y (with or without AE X)

Neither AE X nor AE Y

AE Y (with / without AE X)

hAEX,AEY(t)

hAEY,AEX(t)

AE X

hAEX(t)

hAEX,resol(t)

hAEY(t)

hAEY,resol(t)


MULTISTATE STOCHASTIC PROCESS

� Stochastic process (Xj,t)t≥0 denotes the state (one of a small number of clinically defined conditions) a subject j is in at time (day) t

� Of interest are transition probabilities (s < t, m ≠ k):

� �� = � �� = � , �� (“past” represents process history up to time s)

� Process (Xt)t≥0 fulfills the Markov property, if:

(transition probabilities depend only on the current state, not on the past)

� �� = � �� = � , �� = � �� = � �� = �


TRANSITION HAZARDS FOR A MULTISTATE STOCHASTIC PROCESS

� Process (Xt)t≥0 fulfills the Markov property, if:

� �� = � �� = � , �� = � �� = � �� = �(transition probabilities depend only on the current state, not on the past)

A multistate Markov process (Xt)t≥0 can be specified by:

� an initial distribution for X0 (can be degenerated)

� and transition hazard functions from state [k] to [m] at time t:

or cumulative transition hazards from state [k] to [m] up to time t:

� � [�] � = ��→�� = � ��= [�]

��

� � � � = � � � [�] � ��

�


TRANSITION PROBABILITIES AND CUMULATIVE HAZARDS

d ×××× d transition probability matrices P(s,t) in a d-state model are related to cumulative transition hazard functions as follows:

where

� s = t0 < t1 < t2 <… tN-1 < tN = t is a partition of the time interval [s,t ]

� I is the d ×××× d identity matrix

� ∆H(tn) is a d ×××× d matrix with (k,m)-th element defined as

o H[k][m](tn) − H[k][m](tn-1) and

o � � [�] = −∑ � � [�] �� [�]

� , ! ≈ # $ + ∆' !()

(*+


ESTIMATION IN MULTISTATE MARKOV MODELS

� Nelson-Aalen estimate for cumulative transition hazards

� can be obtained with R-package mvna

� treatment effects can be expressed as hazard ratios for the model-specific

transitions, i.e., a semi-parametric model.

� Aalen-Johansen estimates of transition probabilities (“empirical transition matrix“)

� plug-in the Nelson-Aalen estimates into formula on previous slide

� can be obtained with R package etm, e.g., separately for each treatment

� Sojourn times:

� duration of a subject staying in a certain state (e.g., time spent with a

specific AE)


EXAMPLE MODEL

Re-visit the model with 3 states:



hAEX,AEY(t)

hAEY,AEX(t)

AE X

hneither,AEX(t)

hAEX,neither(t)

hneither, AEY(t)

hAEY,neither(t)

1) How to arrange the data for statistical analysis?


EXAMPLE : HOW TO ARRANGE THE DATA

Subject 1 did not experience any of the AEs X or Y, stopped at day 725

subjid start stop censor from to transition time

1 1 725 0 neither AE X neither –> X 725

1 1 725 0 neither AE Y neither –> Y 725



2 231 245 1 AE X AE Y X –> Y 15

2 231 245 0 AE X neither X –> neither 15

2 246 284 1 AE Y neither Y –> neither 28

2 246 284 0 AE Y AE X Y –> X 28




2 490 506 0 AE X AE Y X –> Y 16



Subject 2:

� AE X onset at day 230, then AE Y at day 245, both resolved by day 284

� AE X onset at day 489, resolved at day 506

� Study ended without further AE X or Y at day 730


EXAMPLE MODEL




hAEX,AEY(t)

hAEY,AEX(t)

AE X

hneither,AEX(t)

hAEX,neither(t)

hneither, AEY(t)

hAEY,neither(t)

2) How to estimate the transition hazard functions?


EXAMPLE : CONTRIBUTION TO ESTIMATION OF TRANSITION HAZARDS

subject 1 contributes only to estimation of hAEX(t) and hAEY(t) subject 1 is never „at risk“ for any other transition

subjid start stop censor from to transition time





2 231 245 1 AE X AE Y X –> Y 15


2 246 284 1 AE Y neither Y –> neither 28

2 246 284 0 AE Y AE X Y –> X 28




2 490 506 0 AE X AE Y X –> Y 16



Subject 2 contributes to estimation of all transition hazards, e.g.,:

� for hneither,AEX(t) and hneither,AEY(t) from day 1 to 230, from day 285 to 489 and from day 507 to 730

� for hAEX,AEY(t) and hAEX,neither(t) from day 231 to 245 and from day 490 to 506


EXAMPLE: NELSON-AALEN ESTIMATES FOR TRANSITION HAZARDS

(R PACKAGE MVNA)


EXAMPLE MODEL




hAEX,AEY(t)

hAEY,AEX(t)

AE X

hneither,AEX(t)

hAEX,neither(t)

hneither, AEY(t)

hAEY,neither(t)

2) How to estimate the transition probabilites?


EXAMPLE: ESTIMATED (EMPIRICAL) TRANSITION PROBABILITY MATRIX

(R PACKAGE ETM)

Aalen-Johansen estimate of P(365, 730)

Neither AEX AEY

Neither 0.855 0.107 0.038

AEX 0.821 0.128 0.051

AEY 0.829 0.123 0.048

Interpretation: a patient with AEY at day 365

� has 83% chance to be free of AEs X and Y at day 730

� has 12% chance to have only AE X at day 730

� Has 5% chance to have AEY at day 730


EXAMPLE: ESTIMATED TRANSITION PROBABILITIES OVER TIME

(R PACKAGE ETM)


SUMMARY AND CONCLUSIONS

� Standard incidence tables for AEs may lead to incomplete assessments

� Time to first occurrence of an AE estimated and displayed by Kaplan-Meier plots is somewhat better but still not satisfying

� Transition hazard functions in multi-state models are key in simple to moderate complex situations for AEs of special interest

� Nelson-Aalen estimation of cumulative transition hazards is straighforward

� Functionals of the Nelson-Aalen estimate lead to interpretable estimates for transition probabilities (and to sojourn times, etc)

� Within each multi-state model, a variety of assumptions are to be made:

� Discussion with medical experts recommended

� Complexity needs to be checked against the size / richness of dataset


SOME REFERENCES

� Aalen OO, Borgan Ø, Gjessing HK. Survival and event history analysis – a process point of view. Springer; 2008.

� Allignol A, Beyersmann J, Schumacher M. mvna: An R package for the Nelson-Aalen estimator in multistate models. R News 2008; 8; 48-50.

� Allignol A, Schumacher M, Beyersmann J. Empirical transition matrix of multistate models: the etm package. Journal of Statistical Software 2011; 38; 1-15.

� Andersen PK, Borgan Ø, Gill RD, Keiding N: Statistical models based on counting processes. Springer; 1993.

� Beyersmann J, Allignol A, Schumacher M. Competing Risks and Multistate Models with R. Springer; 2012.

� Kraemer HC. Tutorial in biostatistics – events per person-time (incidence rate): a misleading statistic? Statistics in Medicine 2009; 28; 1028-1039.


THANK YOU


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