Life Tests

• \(n\) : Number of items put on the test.
• The failure times are recorded chronologically.

• \(t_{1:n} < t_{2:n} < \ldots < t_{n:n}\) are called lifetime or survival times.

Lifetime or Survival Time

• Lifetime: The time to the occurrence of some event of interest from a particular starting point.
• Sometimes the events are actual deaths of individuals.
• In other instances ``lifetime’’ is used in a figurative sense.

Example 1

• Manufactured items are often subjected to life tests in order to obtain information on their durability.
• This involves putting items in operation, often in a laboratory setting.
• Observing them until they fail.

Example 1 (Continued)

• 20 sensors are installed.

• The failure times (in years) of the sensors are as follows:

  0.376, 0.510, 0.513, 0.560, 0.563, 0.573, 0.688, 0.702, 0.717, 0.751, 0.987, 1.100, 1.122, 1.162, 1.354, 1.361, 1.477, 1.580, 1.618, 1.675.

Example 2

• Demographers and social scientists are interested in the duration of certain life “states” of humans.
• For example, the marriages formed during the year 2020 in Tripura.
• The lifetime of a marriage would be its duration.
• A marriage may end due to annulment, divorce, or death.

Example 3

• Consider medical studies dealing with potentially fatal diseases.
• Interested in the survival time of individuals with the disease, measured from the date of diagnosis
• To compare treatments for a disease at least partly in terms of the survival time distributions

Example 4

• Laboratory animals are subjected to doses of carcinogenic substance.
• Observed to see if they develop tumors.
• A main variable of interest is the time to appearance of a tumor, measured from when the dose is administered.

Example 5

• Data from fatigue endurance tests for deep-groove ball bearings.
• This data is taken from Lawless (1982).
• 23 ball bearing are put on test.
• The failure times (in millions of revolution) are as follows:

Censoring

• Quite useful technique in reliability life testing.
• Possible termination of experiment before failing all the experimental units.

Type-I Censoring

• \(n\): Number of items put on the test.
• \(\tau\): Pre-fixed time.
• \(\tau^* = \tau\): Experiment termination time.

• Advantage: Pre-fixed experiment termination time.
• Disadvantage: Very few failures, even no failure.

Type-II Censoring

• \(n\): Number of items put on the test.
• \(r (\le n)\): Pre-fixed integer.
• \(\tau^* = t_{r:n}\): Experiment termination time.

• Advantage: Pre-fixed number of failures.
• Disadvantage: Long experimental duration.

Different Censoring Schemes

Basic Censoring Schemes

• Type-I
• Type-II

Other Censoring Schemes

• Hybrid Type-I
• Hybrid Type-II
• Progressive Type-II

Advantages of Censoring

• Lower cost in terms of money and time than complete experiment.
• Survival experimental units can be used for further experiments.

Average Lifetime

• 20 sensors were installed and monitored until failure.
• Observed Failure Times (in years):

• An estimate of average lifetime: 0.969 years.

Statistical Challenge (Type-I Example)

• Experiment terminated at the end of 1st year (\(\tau^* = 1\)).
• Observed failure times:

• The mean of the observed data: 0.630 years

Improved Estimate

• Missing out information: 9 sensors lasts more than 1 year.

• An improve estimate of average lifetime: \[\frac{\text{Sum of observed lifetimes} + (9\times 1)}{20}=0.797 \text{ years}\]

Modelling (Weibull Example)

• Assume lifetimes \(T_1,\, T_2,\, \ldots,\, T_{20} \overset{i.i.d.}{\sim} \text{Weibull}(\alpha, \theta)\).
• The PDF is given by \[f(x) = \frac{\alpha}{\theta} x^{\alpha-1} e^{-\frac{x^\alpha}{\theta}}\quad \text{for } x > 0.\]

Likelihood Estimation

Likelihood function of \((\alpha, \theta)\) based on the censored sample is:

\[L(\alpha, \theta) = \frac{n!}{(n-r)!} \left[ \prod_{i=1}^{r} f(t_{i:n}; \alpha, \theta) \right] [1 - F(\tau^*; \alpha, \theta)]^{n-r}\]

• Maximize likelihood function w.r.t. parameters.
• Maximizers of likelihood function are estimates of the respective parameters.

Revisit Data

• Based on complete sample:
  • \(\hat{\alpha}=2.55,\hat{\theta}=1.27,\hat{\mu}=0.975\).

• Ignoring censoring effect:
  • \(\hat{\alpha}=4.21,\hat{\theta}=0.21,\hat{\mu}=0.627\).

• Considering censoring effect:
  • \(\hat{\alpha}=2.57,\hat{\theta}=1.16,\hat{\mu}=0.941\).

Malignant Melanoma Cancer Data Set

• Melanoma : Skin cancer
• Collected at Odense University Hospital during 1962 to 1977
• Two hundred and five patients
• No of days after the operation
  • until they died
    • until they left the study
    • until the termination of the study in the year 1977

Malignant Melanoma Cancer Data Set

Potential Questions

• What is average lifetime for patients dead due to melanoma?
• How does ulcer effects the survival chance of a patient?
• Does gender has effect on survival chance?

What are Competing Risks?

• Competing Risk is an event whose occurrence precludes the occurrence of the primary event of interest.
• Mutual Exclusivity

Points to Keep in Mind

• Cannot seperate cause of deaths (competing risks).
• Presence of censored observations.
• Information on covariates.

Two Motor Data (Snippet)

What is a Load-Sharing System?

• Components are connected in parallel and share a common workload.
• If one component fails, the remaining components bear the additional load.
• This typically leads to an increased failure rate.

Points to Consider

• Load redistribution rule (Equal Load-Sharing Rule, Spatial/Proximity Effect).
• Dependence between lifetimes of different components.
• Presence of censoring.
• Information on covariates.