Transcript Slide 1 - scharp
Slide 1
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 2
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 3
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 4
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 5
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 6
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 7
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 8
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 9
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 10
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 11
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 12
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 13
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 14
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 15
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 16
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 17
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 18
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 19
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 20
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 2
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 3
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 4
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 5
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 6
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 7
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 8
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 9
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 10
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 11
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 12
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 13
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 14
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 15
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 16
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 17
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 18
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 19
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling
Slide 20
Biostat/Stat 576
Lecture 16
Selected Topics on Recurrent
Event Data Analysis
Regression Analysis
• Regression analysis of gap times
– Challenges: last censored observations will
complicate any analysis due to biased
sampling embedded
– One approach is to simply focus on complete
gap times
• Complete gap times are all right-truncated
• Regression methods for right-truncated time-toevent can be adapted
• Comparability of complete recurrence times
• Comparable complete gap times of
– For
, the observation interval is
–
should satisfy
– Similarly,
• References
– Bhattacharya, et al (1983, Ann. Stat.)
– Efron and Petrosian (1999, JASA)
• Comparability of
– Total censoring time
– For
,
• observation time should satisfy
– Similarly for
,
• Conditional on
– Total censoring time
– Observation time area
• Joint density of
is proportional to
• Due to this comparability, we would have
• Can we take advantage of this probability
structure of comparable complete gap
times?
• Assume accelerated failure time model
–
–
: is subject-specific intercept
: is mean-zero random error of same
distributions
• Random errors
– For jth and kth recurrences
– Observation interval for random errors
Recurrence time
Observation interval
• Comparability conditions:
– Symmetrically
– Therefore
• This motivates estimating equations
– Satisfying
• Reference:
– Wang and Chen (2000, Bmcs)
• Complete gap times
– Right-truncated
– Regression analysis for right-truncated failure
times
• proportional reverse-time hazards models
• References:
– Lagakos, et al. (1988, Bmka)
– Kalbfleisch and Lawless (1991, JASA)
• Reverse-time hazard function
• Proportional reverse-time hazards model
• Challenges
– How do we construct proper risk sets for righttruncated outcomes?
– How do we construct proper risk sets for complete
gap times?
• For usual right-truncated
outcomes
– Risk set for
– Interpretation
• Those in the risk set should
have fair chance to fail at
– Probability of
the risk set
given
• For right-truncated complete gap times,
– Usual risk sets may not be proper
– Observation intervals for those in the risk sets
may not be comparable
– Right-truncated gap times may become
seemingly smaller and smaller just because
they occur later in the process
• Risk sets need to be adjusted so that those in the
risk set should have fair chance to fail not just
because they occur later
• How do we adjust risk sets?
– A comprehensive approach
• At
• This approach is expected to be cumbersome
– A reduced risk set approach
• To include only two gap times in the risk set at a
time
• Two scenarios:
– Include a later gap time in risk set
» Later gap time tends to be smaller. So as long as the
observation interval is large enough for this gap time
to fail at the risk set time, it can be included
– Include an earlier gap time in risk set
» Earlier gap time tends to be longer. So their
advantage in observation time should be offset by
early advantage
To include a later gap time
To include an earlier gap time
• Conditional on the reduced risk set, the
conditional likelihood contribution under the
proportional reverse-time model becomes
• This is similar to the conditional likelihood in twin
studies
• Reference
– Chen, et al. (2004, Biostat.)
• Other approaches
– Conditional approaches
– Marginal approaches
– Joint modeling