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Nonparametric maximum likelihood estimation (MLE) for bivariate censored data Marloes H. Maathuis advisors: Piet Groeneboom and Jon A. Wellner

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Motivation Estimate the distribution function of the incubation period of HIV/AIDS: –Nonparametrically –Based on censored data: Time of HIV infection is interval censored Time of onset of AIDS is interval censored or right censored

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Approach Use MLE to estimate the bivariate distribution Integrate over diagonal strips: P(Y-X ≤ z) X (HIV) Y (AIDS) z

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Main focus of the project MLE for bivariate censored data: –Computational aspects –(In)consistency and methods to repair the inconsistency

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Main focus of the project MLE for bivariate censored data: –Computational aspects –(In)consistency and methods to repair the inconsistency

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1980 1992 1996 198019831986 X (HIV) Y (AIDS) Interval of onset of AIDS Interval of HIV infection

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1980 1992 1996 198019831986 X (HIV) Y (AIDS) Interval of onset of AIDS Interval of HIV infection Observation rectangle R i

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X (HIV) Y (AIDS) Observation rectangle R i

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X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i

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X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i

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X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i

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X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i

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X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i

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α1α1 α2α2 α3α3 α4α4 X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i s.t.and

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3/5 0 0 2 5 X (HIV) Y (AIDS) Maximal intersections Observation rectangle R i The α i ’s are not always uniquely determined: mixture non uniqueness s.t.and

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Computation of the MLE Reduction step: determine the maximal intersections Optimization step: determine the amounts of mass assigned to the maximal intersections

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Computation of the MLE Reduction step: determine the maximal intersections Optimization step: determine the amounts of mass assigned to the maximal intersections

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Existing reduction algorithms Betensky and Finkelstein (1999, Stat. in Medicine) Gentleman and Vandal (2001, JCGS) Song (2001, Ph.D. thesis) Bogaerts and Lesaffre (2003, Tech. report) The first three algorithms are very slow, the last algorithm is of complexity O(n 3 ).

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New algorithms Tree algorithm Height map algorithm: –based on the idea of a height map of the observation rectangles –very simple –very fast: O(n 2 )

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1 1 1 1 0 0 0 0 1 1 3 3 2 1 1 0 0 2 1 3 3 2 1 2 1 0 2 1 2 2 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 0 2 1 1 1 2 2 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 Height map algorithm: O(n 2 ) 1 2 2 2 1 1 0 0 2

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Main focus of the project MLE of bivariate censored data: –Computational aspects –(In)consistency and methods to repair the inconsistency

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HIV AIDS u1u1 u2u2 Time of HIV infection is interval censored case 2

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HIV AIDS u1u1 u2u2 Time of HIV infection is interval censored case 2

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HIV AIDS u1u1 u2u2 Time of HIV infection is interval censored case 2

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HIV AIDS t = min(c,y) u1u1 u2u2 Time of onset of AIDS is right censored

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HIV AIDS t = min(c,y) u1u1 u2u2 Time of onset of AIDS is right censored

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HIV AIDS t = min(c,y) u1u1 u2u2 Time of onset of AIDS is right censored

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t = min(c,y) HIV AIDS u1u1 u2u2

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HIV AIDS u1u1 u2u2 t = min(c,y)

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HIV AIDS u1u1 u2u2 t = min(c,y)

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HIV AIDS u1u1 u2u2 t = min(c,y)

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Inconsistency of the naive MLE

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Methods to repair inconsistency Transform the lines into strips MLE on a sieve of piecewise constant densities Kullback-Leibler approach

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cannot be estimated consistently X = time of HIV infection Y = time of onset of AIDS Z = Y-X = incubation period

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X = time of HIV infection Y = time of onset of AIDS Z = Y-X = incubation period An example of a parameter we can estimate consis- tently is:

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Conclusions (1) Our algorithms for the parameter reduction step are significantly faster than other existing algorithms. We proved that in general the naive MLE is an inconsistent estimator for our AIDS model.

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Conclusions (2) We explored several methods to repair the inconsistency of the naive MLE. cannot be estimated consistently without additional assumptions. An alternative parameter that we can estimate consistently is:.

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Acknowledgements Piet Groeneboom Jon Wellner

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