Semiparametric Methods of Time Scale Selection

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Semiparametric Methods of Time Scale Selection

• by Thierry Duchesne,
• University of Toronto
• MMR2000
• Bordeaux, July 6, 2000

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Outline

• 1. Introduction
• 2. Models of failure
• 3. Semiparametric inference
• 4. Model assessment
• 5. Further research and concluding remarks
• 6. References

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Introduction

• How should we measure age at failure?
• Examples
• Automobiles (Lawless et al., LIDA 1995)
• days since initial purchase
• cumulative mileage
• some hybrid of (i) and (ii)
• Asbestos miners (Oakes, LIDA 1995)
• biological age
• cumulative amount of exposure to asbestos
• Some hybrid of (i) and (ii)

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1. Introduction

• Possible approach always use chronological time
  and treat other measures as time-varying
  covariates.
• May be advantages in adopting other scale, say t
  (e.g., Farewell and Cox, App. Stat. 1979,
  Kordonsky and Gertsbakh, LIDA 1997, Duchesne and
  Lawless, LIDA 2000)
• failure mechanism may depend primarily on t
• chronological time may be irrelevant in context
• effect of covariates may be most simply
  expressed on some scale other than real time

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2. Models of failure

• Notation
• x denotes chronological time, X is the r.v. of
  chronological time at failure
• z(x) is value of usage measure at time x (usage
  measure z(0)0, z() left-continuous and
  non-decreasing, z() external covariate in that
  its path is determined independently of the
  failure time X)
• Zz(x) x?0 is usage path or usage history
• Z space of all paths
• time scale
• to emphasize that ? is age/time in other scale,
  we often use the notation

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2. Models of failureIdeal time scale (ITS)
Kordonsky and Gertsbakh (LIDA 1997) load
invariant scale Bagdonavicius and Nikulin
(Biomtka 1997) transfer functional. Let
, a r.v. of time to failure in scale ?.
Then ? is an ITS if such that
for some , (1) where G()
is a strictly decreasing survivor function. A
useful consequence of (1) for model selection
if ? is an ITS, (2) (Duchesne and Lawless, LIDA
2000)
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2. Models of failureCollapsible models (Oakes,
LIDA 1995)
Class of models with simple form for ?. The
model PXgtxZ is collapsible in x and z(x)
if (3) Where is
non-increasing in x and z(x). (3) ? ITS of the
form ?x, z(x), and probability of surviving
past x only depends on amount of usage at x, not
on past history. Examples
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3. Semiparametric inference
Model (4) When G(?) and ?? specified
parametrically ? maximum likelihood (Oakes,
1995) data ,
where if individual i failed at ,
if i is right-censored at
. where
is the hazard function corresponding to
survivor function G(?).
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3. Semiparametric inference

• Estimating ? without specifying G() in (4) is
  sometimes desirable
• misspecification in ? and G() may offset each
  other
• to find parametric form of G(), need to look at
  probability plots of failure times in scale ? .
• Semiparametric inference about ?
• Minimum CV method
• more difficult when censoring is present
• intervals and tests about ? not easy

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3. Semiparametric inferenceRank-based method
Inspired from Robins and Tsiatis (Biomtka 1992)
and Lin and Ying (Stat. Plan. and Infer.
1995). Idea (a) use a rank test statistic as an
estimating function (b) replace G() in
log-likelihood by an empirical estimate of it for
each fixed ?. For our model, we obtain the
score (5) where
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3. Semiparametric inference

• Variance of score (5)
• (1-?)100 conf. region for ?
• Conditions for precise inference
• large variability in observed usage paths
• distribution of XZ with small variation

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3. Semiparametric inference
Example where (i) is label of ith
individual to fail in scale is the
statistic we would have obtained, had we wanted
to test no association between failure
times and covariates
, using rank regression and
exponential scores.
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3. Semiparametric inferencePros and cons of
this method

• Pros
• estimator has nice asymptotic properties
• easily handles right-censoring
• scores consistent with definition of ITS
• quite efficient in many settings
• easy to get tests and confidence intervals
• Cons
• computationally intensive
• neither smooth nor monotone in
• covariates must be observed continually in order
  to compute values of

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4. Model assessment
Collapsibility is a strong assumption. How do we
check if a simple scale is ideal? Generalized
residuals From definition of ITS, s
are independent of Z. Thus, plots of
against features of should show no
trend. If there is a trend appearing, time scale
is not ideal.
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4. Model assessment
Example Data from Kordonsky and Gertsbakh
(R.E.S.S. 1995) on fatigue life of steel
specimens. Two scales Scale 1 Scale
2 ? Plots of failure times in scale 1 vs
shows no trend, whereas clear quadratic
trend appears when plotting failure times in
scale 2.
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5. Further research and concluding remarks

• Nonparametric age curves
• For collapsible models, we can draw age curves
  non-parametrically.
• useful for time scale selection
• useful for model assessment
• idea similar to nonparametric regression

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5. Further research and concluding remarks

• Ideas for further research
• properties of generalized residuals
• generalized residuals under censoring
• formal test of the collapsibility assumption
• more efficient method to draw age curves
• more efficient semiparametric estimation
• use of collapsible models in prediction

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6. References
Bagdonavicius, V.B. and Nikulin, M.S. (1997).
Transfer functionals and semiparametric
regression models, Biometrika, 84,
365-378. Duchesne, T. and Lawless J. (2000).
Alternative time scales and failure time
models, Lifetime Data Analysis, 6,
157-179. Farewell, V.T. and Cox, D.R. (1979). A
note on multiple time scales in life testing,
Applied Statistics, 28, 73-75. Kordonsky, K.B.
and Gertsbakh, I. (1995). System state
monitoring and lifetime scales-I, Reliability
Engineering and System Safety, 47,
1-14. Kordonsky, K.B. and Gertsbakh, I. (1997).
Multiple time scales and the lifetime
coefficient of variation Engineering
applications, Lifetime Data Analysis, 2,
139-156. Lawless, J.F., Hu, J. and Cao, J.
(1995). Methods for the estimation of failure
distributions and rates from automobile warranty
data, Lifetime Data Analysis, 1, 227-240. Lin,
D.Y. and Ying, Z. (1995). Semiparametric
inference for the accelerated life model with
time-dependent covariates, Statistical Planning
and Inference, 44, 47-63. Oakes, D. (1995).
Multiple time scales in survival analysis,
Lifetime Data Analysis, 1, 139-156. Robins, J.
and Tsiatis, A.A. (1992). Semiparametric
estimation of an accelerated failure time model
with time-dependent covariates, Biometrika, 79,
311-319.