Reliability Data Analysis

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 37

Introduction¶

It has always been assumed that software would be added to ASL through the efforts of individuals or groups. However it would not be wise if no opportunity were given for others beyond the authors to comment at an early stage on the material and the functions developed.

In this article a brief description of the functions which are contained within the Reliability Data Analysis volume is given. A beta-test version of the software is to be produced in the near future. Before this though it would be helpful to have the views of others on the functions included and those they feel should be added.

The software discussed is part of a larger project which is the development of an Expert System for Reliability Data Analysis. The Expert System is being developed in APL. A subsequent article may describe the details of the Expert System.

APL is a natural language for this volume since most of the algebra is linear in form, either based on raw data or on the ordered statistics. Hence for the most part the routines follow the mathematics without the need for great sophistication. There is though the need for iterative procedures, especially Newton-Raphson-based approaches for estimation.

The data analysis volume is assumed to be part of a larger shelf where there would be routines for Reliability Analysis and Modelling. Fault tree programmes might be written based on APL routines for logic manipulations for analysing systems. Also a Stochastic Model could be added following work done at Swansea under the supervision of Alan Hawkes.

In the future it may be that the Data Analysis volume might share code with related shelves, such as a Survival Analysis Shelf or a more general Medical Statistics Shelf. This emphasises that the software described is only a first attempt, and not the definitive model for the volume. Contributions are welcomed and as authors we would be delighted to see the shelf grow. The names used for functions might equally transform with suggestions from others.

At present we see the possibility of six sections to the volume. These are:

• Elementary Analysis
• Graphical Analysis
• Distributions
• Covariate Analysis
• Growth Models
• Dependency Analysis

Elementary Analysis¶

As with the bottom shelf this is seen as a collection of routines which have been found to be useful in the analysis of reliability data, especially in the initial stages of such an analysis. It augments the Bottom Shelf. For example the coefficient of variation (COEFVAR) and the index of dispersion (INDIS) are added since they indicate whether the data is under- or over-dispersed compared to the Poisson Process. This aids in model selection. Also included is a function to calculate the cumulative mean time to failure, since again many practitioners find it a useful measure.

For chronological data Ascher has frequently suggested the need to test for trend. Hence the volume contains two of the standard trend test measures, the Laplace Trend (TRENDLP) and the MIL-HNBK-189 (TRENDMIL) test.

Graphical Analysis¶

Graphical analyses are often used in Reliability. Engineers find them comprehensible and informative. They can give rise to parameter estimates, but more appropriately can indicate the validity of the model. It is assumed that the volume will provide just the values to be plotted and not the plotting routines, which should exist within ASLGRAPH. Hence the volume will produce X-Y coordinates for plotting.

The routines cover the standard plots to assess lifetime distributions such as Weibull (WEIBPLOT), Hazard (HAZPLOT) and TTT (TTTPLOT) plots. These plots often shed light on the appropriateness of a distribution but may also indicate possible problems with the data. A function also exists to produce a Duane Growth Plot (DUANEPLOT). This plot is often used to assess whether time between failures is increasing (good) or decreasing (bad).

Beyond these there will be functions to produce plots for more sophisticated analysis, such as residual plots for covariate analysis (see Covariate section) and for examining dependency such as Walls’ method (WALLSPLOT).

Lifetime Distributions¶

Obviously looking into the future there should be a separate shelf for distributions, covering the common distributions. Already, though, certain aspects of some distributions have been covered by the Bottom Shelf. In the interim it seems sensible to include some routines associated with lifetime distributions in this volume. (A lifetime distribution is generally taken to be a distribution which takes only positive values.) Hence included in this volume are routines for parameter estimation for a number of lifetime distributions.

Given the reliability context these functions allow for the possibility of censoring, currently right-hand censoring. The distributions considered are the Exponential, both one- and two-parameter forms (EXPEST, EXP2EST), the Weibull, two- and three-parameter forms (WEIBEST, WEIB3EST), the Gamma (GAMMAEST) and Log Gamma (LGAMMAEST).

Since it is also possible to obtain empirical estimates for distributions we have included Kaplan-Meier for reliability function (KMEST) and Nelson-Altschuler for cumulative hazard function (NAEST).

Covariate Analysis¶

Cox’s Regression Model, referred to as the Proportional Hazard Model (PHM), has led to a resurgence of interest in modelling lifetimes through the use of covariates. There is a wide range of possible models. The current volume contains functions for two models: PHM and Weibull Regression. These functions include estimation of parameters with approximate standard deviations, (PHMBETA, WEIBREGBETA), likelihood values, (PHMLIKE, WEIBREGLIKE), and residuals, (PHMRESID[I], WEIBREGRESID). (The [I] indicates more than one residual available.)

Specific formulations of other models are covered by ASLREG either directly or by transformation, for example Gamma, Log Gamma and Log Normal Regression. Hence it seems inappropriate to repeat them in this shelf.

Growth Models¶

It had been assumed that as technology develops, systems and components will improve especially in the context of hardware. The more pragmatic will realize this is not the case. In Reliability, however, there are a number of models for describing growth which assume positive growth for hardware: the Duane Growth Model, ARINC and others. Most of these models should contain a governmental health warning. For some of the models there is more than one estimator for the parameters of the models. For the Duane Growth model it is possible to use either graphical methods or Crow’s estimate for Non-Homogeneous Poisson Process, (CROWEST).

Beyond hardware models there are a stream of software models such as Jellinski-Moranda (JMEST) and Littlewood-Verrall (LVEST). These models equally have drawbacks, but the Littlewood-Verrall model has the advantage that improvement is only expected at every failure, not required.

Dependency Analysis¶

There is a wide range of possible approaches, for example Engineering yields the beta-Factor model and its extensions, whilst Statistics provides a rich medium for analysis of dependency. Currently there is not a single approach regularly used and hence at this stage we have decided not to include any specific methods except the graphical approach to analysis suggested by Walls (WALLSPLOT).

Conclusions and Comments¶

This article has briefly covered what is contained within the volume. It is hoped to release the volume for beta-test later in the year; it will be supplied with limited documentation. If you have comments on the material so far included, or wish to suggest further functions which might be added, please write to me at the address given in the editorial on page 30.