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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

Nonparametric maximum likelihood estimation (MLE) for bivariate censored data

Marloes H. Maathuis

advisors:

Piet Groeneboom and Jon A. Wellner

motivation
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

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

Y (AIDS)

z

X (HIV)

main focus of the project
Main focus of the project
  • MLE for bivariate censored data:
    • Computational aspects
    • (In)consistency and methods to repair the inconsistency
main focus of the project1
Main focus of the project
  • MLE for bivariate censored data:
    • Computational aspects
    • (In)consistency and methods to repair the inconsistency
slide6

Y (AIDS)

1996

Interval of

onset of AIDS

1992

1980

1980

1983

1986

X (HIV)

Interval of

HIV infection

slide7

Observation rectangle Ri

Y (AIDS)

1996

Interval of

onset of AIDS

1992

1980

1980

1983

1986

X (HIV)

Interval of

HIV infection

slide9

Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)

slide10

Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)

slide11

Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)

slide12

Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)

slide13

Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)

slide14

Observation rectangle Ri

Maximal intersections

Y (AIDS)

α1

α2

α3

α4

s.t.

and

X (HIV)

slide15

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

computation of the mle
Computation of the MLE
  • Reduction step:

determine the maximal intersections

  • Optimization step:

determine the amounts of mass assigned to the maximal intersections

computation of the mle1
Computation of the MLE
  • Reduction step:

determine the maximal intersections

  • Optimization step:

determine the amounts of mass assigned to the maximal intersections

existing reduction algorithms
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).

new algorithms
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)
height map algorithm o n 2
Height map algorithm: O(n2)

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main focus of the project2
Main focus of the project
  • MLE of bivariate censored data:
    • Computational aspects
    • (In)consistency and methods to repair the inconsistency
slide23

u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV

slide24

u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV

slide25

u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV

slide26

t = min(c,y)

u1

u2

Time of onset of AIDS is right censored

AIDS

HIV

slide27

u1

u2

Time of onset of AIDS is right censored

AIDS

t = min(c,y)

HIV

slide28

u1

u2

Time of onset of AIDS is right censored

AIDS

t = min(c,y)

HIV

slide29

t = min(c,y)

AIDS

u1

u2

HIV

slide30

t = min(c,y)

AIDS

u1

u2

HIV

slide31

t = min(c,y)

AIDS

u1

u2

HIV

slide32

t = min(c,y)

AIDS

u1

u2

HIV

methods to repair inconsistency
Methods to repair inconsistency
  • Transform the lines into strips
  • MLE on a sieve of piecewise constant densities
  • Kullback-Leibler approach
slide38

X = time of HIV infection

Y = time of onset of AIDS

Z = Y-X = incubation period

  • cannot be estimated consistently
slide39

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:
conclusions 1
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.
conclusions 2
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: .
acknowledgements
Acknowledgements
  • Piet Groeneboom
  • Jon Wellner
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