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Parameter inference from hitting times for perturbed Brownian motion.

Tamborrino M, Ditlevsen S, Lansky P - Lifetime Data Anal (2014)

Bottom Line: To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process.Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters.Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, 2100, Copenhagen, Denmark, mt@math.ku.dk.

ABSTRACT
A latent internal process describes the state of some system, e.g. the social tension in a political conflict, the strength of an industrial component or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. the political conflict finishes, the industrial component breaks down or the person dies. Imagine an intervention, e.g., a political decision, maintenance of a component or a medical treatment, is initiated to the process before the event occurs. How can we evaluate whether the intervention had an effect? To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process. Then, the time interval between the start of the process and the final event is divided into two subintervals: the time from the start to the instant of intervention, denoted by S, and the time between the intervention and the threshold crossing, denoted by R. The first question studied here is: What is the joint distribution of (S,R)? The theoretical expressions are provided and serve as a basis to answer the main question: Can we estimate the parameters of the model from observations of S and R and compare them statistically? Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters. Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

No MeSH data available.


Related in: MedlinePlus

Schematic illustration of the single trial. At time 0, an intervention is applied, dividing the observed interval into two subintervals: the time  upto the instant of intervention, and the time  between the intervention and the first crossing after it. The random position of the process at time 0 is denoted by . In the fully observed case, the variables  and  are uncensored (top panel). Under right censoring, a censoring time happens before the event (middle top panel). Under left censoring, the start of the process is not observed (middle lower panel). In presence of truncation, an event happens before the intervention occurs (lower panel), and
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Fig1: Schematic illustration of the single trial. At time 0, an intervention is applied, dividing the observed interval into two subintervals: the time upto the instant of intervention, and the time between the intervention and the first crossing after it. The random position of the process at time 0 is denoted by . In the fully observed case, the variables and are uncensored (top panel). Under right censoring, a censoring time happens before the event (middle top panel). Under left censoring, the start of the process is not observed (middle lower panel). In presence of truncation, an event happens before the intervention occurs (lower panel), and

Mentions: The type of experimental data and the description of the involved quantities are illustrated in Fig. 1. At a time independent of when the process started, an intervention is applied and the time the process has run as well as the time to an event after the intervention are measured. The time of the intervention is set to 0 by convenience. The intervention divides the observed interval into two subintervals: the time from the start of the process to the instant of intervention, denoted by , and the time between the intervention and an event after it, denoted by . Thus, the observed interval has length . The experiment is repeated times. This allows to obtain independent and identically distributed pairs of intervals , for . Note that and are not independent. A common situation for failure time data is the need to accommodate censoring or truncation in data. Left censoring happens when the time of start of the process is not observed, and right censoring when the study ends before an event occurs. In these cases either or are only known to be larger than a given value. Truncation happens if an event occurs before the intervention. In this case is undefined.Fig. 1


Parameter inference from hitting times for perturbed Brownian motion.

Tamborrino M, Ditlevsen S, Lansky P - Lifetime Data Anal (2014)

Schematic illustration of the single trial. At time 0, an intervention is applied, dividing the observed interval into two subintervals: the time  upto the instant of intervention, and the time  between the intervention and the first crossing after it. The random position of the process at time 0 is denoted by . In the fully observed case, the variables  and  are uncensored (top panel). Under right censoring, a censoring time happens before the event (middle top panel). Under left censoring, the start of the process is not observed (middle lower panel). In presence of truncation, an event happens before the intervention occurs (lower panel), and
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4464758&req=5

Fig1: Schematic illustration of the single trial. At time 0, an intervention is applied, dividing the observed interval into two subintervals: the time upto the instant of intervention, and the time between the intervention and the first crossing after it. The random position of the process at time 0 is denoted by . In the fully observed case, the variables and are uncensored (top panel). Under right censoring, a censoring time happens before the event (middle top panel). Under left censoring, the start of the process is not observed (middle lower panel). In presence of truncation, an event happens before the intervention occurs (lower panel), and
Mentions: The type of experimental data and the description of the involved quantities are illustrated in Fig. 1. At a time independent of when the process started, an intervention is applied and the time the process has run as well as the time to an event after the intervention are measured. The time of the intervention is set to 0 by convenience. The intervention divides the observed interval into two subintervals: the time from the start of the process to the instant of intervention, denoted by , and the time between the intervention and an event after it, denoted by . Thus, the observed interval has length . The experiment is repeated times. This allows to obtain independent and identically distributed pairs of intervals , for . Note that and are not independent. A common situation for failure time data is the need to accommodate censoring or truncation in data. Left censoring happens when the time of start of the process is not observed, and right censoring when the study ends before an event occurs. In these cases either or are only known to be larger than a given value. Truncation happens if an event occurs before the intervention. In this case is undefined.Fig. 1

Bottom Line: To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process.Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters.Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, 2100, Copenhagen, Denmark, mt@math.ku.dk.

ABSTRACT
A latent internal process describes the state of some system, e.g. the social tension in a political conflict, the strength of an industrial component or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. the political conflict finishes, the industrial component breaks down or the person dies. Imagine an intervention, e.g., a political decision, maintenance of a component or a medical treatment, is initiated to the process before the event occurs. How can we evaluate whether the intervention had an effect? To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process. Then, the time interval between the start of the process and the final event is divided into two subintervals: the time from the start to the instant of intervention, denoted by S, and the time between the intervention and the threshold crossing, denoted by R. The first question studied here is: What is the joint distribution of (S,R)? The theoretical expressions are provided and serve as a basis to answer the main question: Can we estimate the parameters of the model from observations of S and R and compare them statistically? Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters. Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

No MeSH data available.


Related in: MedlinePlus