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Nonparametric maximum likelihood estimation (MLE) for bivariate censored data

Nonparametric maximum likelihood estimation (MLE) for bivariate censored data. Marloes H. Maathuis advisors: Piet Groeneboom and Jon A. Wellner. Motivation. Estimate the distribution function of the incubation period of HIV/AIDS: Nonparametrically Based on censored data:

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Nonparametric maximum likelihood estimation (MLE) for bivariate censored data

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

  2. 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

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

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

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

  6. Y (AIDS) 1996 Interval of onset of AIDS 1992 1980 1980 1983 1986 X (HIV) Interval of HIV infection

  7. Observation rectangle Ri Y (AIDS) 1996 Interval of onset of AIDS 1992 1980 1980 1983 1986 X (HIV) Interval of HIV infection

  8. Observation rectangle Ri Y (AIDS) X (HIV)

  9. Observation rectangle Ri Maximal intersections Y (AIDS) X (HIV)

  10. Observation rectangle Ri Maximal intersections Y (AIDS) X (HIV)

  11. Observation rectangle Ri Maximal intersections Y (AIDS) X (HIV)

  12. Observation rectangle Ri Maximal intersections Y (AIDS) X (HIV)

  13. Observation rectangle Ri Maximal intersections Y (AIDS) X (HIV)

  14. Observation rectangle Ri Maximal intersections Y (AIDS) α1 α2 α3 α4 s.t. and X (HIV)

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

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

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

  18. 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(n3).

  19. New algorithms • Tree algorithm • Height map algorithm: • based on the idea of a height map of the observation rectangles • very simple • very fast: O(n2)

  20. Height map algorithm: O(n2) 1 1 1 1 1 0 0 0 0 1 2 2 2 1 0 0 0 0 1 2 3 3 2 1 1 1 0 1 2 3 2 3 1 2 1 0 1 2 2 2 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 2 1 1 2 1 1 0 0 0 1 1 1 2 1 1 0 0 0 0 0 0 1 0 0

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

  22. u1 u2 Time of HIV infection is interval censored case 2 AIDS HIV

  23. u1 u2 Time of HIV infection is interval censored case 2 AIDS HIV

  24. u1 u2 Time of HIV infection is interval censored case 2 AIDS HIV

  25. t = min(c,y) u1 u2 Time of onset of AIDS is right censored AIDS HIV

  26. u1 u2 Time of onset of AIDS is right censored AIDS t = min(c,y) HIV

  27. u1 u2 Time of onset of AIDS is right censored AIDS t = min(c,y) HIV

  28. t = min(c,y) AIDS u1 u2 HIV

  29. t = min(c,y) AIDS u1 u2 HIV

  30. t = min(c,y) AIDS u1 u2 HIV

  31. t = min(c,y) AIDS u1 u2 HIV

  32. Inconsistency of the naive MLE

  33. Inconsistency of the naive MLE

  34. Inconsistency of the naive MLE

  35. Inconsistency of the naive MLE

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

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

  38. 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:

  39. 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.

  40. 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: .

  41. Acknowledgements • Piet Groeneboom • Jon Wellner

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