Life Testing

Parametric regression models for censored data are used in a variety of contexts ranging from manufacturing to studies of environmental contaminants. Because of their frequent use for modeling failure time or survival data, they are often referred to as parametric survival models. In this context, they are used throughout engineering to discover reasons why engineered products fail. They are called accelerated failure time models or accelerated testing models when the product is tested under more extreme conditions than normal to accelerate its failure time.

The Parametric Survival and Life Testing dialogs fit the same type of model. The difference between the two dialogs is in the options available. The Life Testing dialog is motivated more by manufacturing applications while the Parametric Survival dialog is motivated more by biomedical applications.

To perform life testing

Choose Statistics __image\ebd_ebd66.gif Survival __image\arrow5.gif Life Testing. The dialog shown below appears.

Model page

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In the Life Testing dialog, the Model page has the following options:

Data

Data Set

Select a data set from the dropdown list or type the name of a data set. You can also type into the Data Set edit field any expression that evaluates to a data set.

Weights

Enter the column that specifies weights to be applied to all observations used in the analysis. To weight all rows equally, leave this blank.

Subset Rows

Enter an S-PLUS expression that identifies the rows to use in the analysis. To use all the rows in the data set, leave this field blank.

Omit Rows with Missing Values

Select this box to omit from the analysis any rows in the data set that contain missing values for any of the variables in the model.

Threshold Parameter

Threshold Parameter Methods

Choose the method to use for specifying the threshold. Select Value to specify a value. Select Linearized-qq to obtain a value using optimization to minimize the curvature of a quantile-quantile plot of the quantiles of the response versus the quantiles obtained from a Kaplan-Meier estimate of survival.

Formula

Formula

Enter a formula specifying the desired model.

Example:

censor(days,event)~voltage

Create Formula

Click this to open a formula builder dialog used to construct a formula specifying the desired model. See the Cox Proportional Hazards - Formula for more information on the Formula dialog for survival functions.

Model

Model Distribution

Select the assumed distribution for the transformed response variable.

Truncation

Truncation Formula

Formula Enter a formula specifying the truncation model.

Example:

censor(days,event)

Create Formula Click this to open a formula builder dialog used to construct a formula specifying the desired model. See the Cox Proportional Hazards - Formula for more information. The Truncation Formula builder is a modified version of the formula builder shown in the example.

Save Model Object

Save Model Object

In the Save As field, enter the name for the object in which to save the results of the analysis. If an object with this name already exists, its contents are overwritten. The model object can be used in later functions such as plotting.

Options page

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In the Life Testing dialog, the Options page has the following options:

Optimization Parameters

Convergence Tolerance

Enter a number specifying the convergence tolerance. Iteration continues until the relative change in the log-likelihood is less than this number.

Maximum Iteration

Enter a numeric value specifying the maximum number of iterations to perform for the maximum likelihood estimation procedure. If convergence has not been reached after this number of iterations, the procedure stops. The default value appears in the field.

Results page

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In the Life Testing dialog, the Results page has the following options:

Short Output for Life Testing

Select this to print a summary of the model results in the designated output window. This includes estimates of the coefficients, dispersion (scale), degrees of freedom, and -2*log-likelihood.

Long Output for Life Testing

Select this to print a long summary of the model results in the designated output window. This includes summary statistics for the deviance residuals, standard errors and z-values for the coefficients, number of iterations, and correlation of coefficients.

Saved Results

Save In

Enter the name of a data set in which a part of the analysis, such as fitted values and residuals, predictions, confidence intervals, or standard errors, is saved.

Fitted Values

Save the fitted values from the model in the object specified in Save In.

Standardized Residuals

The standardized residuals are the standardized Cox-Snell censored residuals described in Meeker and Escobar (1998). They are identical to the Pearson residuals for non-logged distributions except for censored values.

Deviance Residuals

Select to save the deviance residuals. These residuals are reasonable for use in detecting observations with unduly large influence in the fitting process, since they reflect the same criterion as used in the fitting.

Pearson Residuals

Select to save the Pearson residuals. They are a rescaled version of the working residuals. Their sums-of-squares is the chi-squared statistic.

Response Residuals

Select to save the response residuals. These are the ordinary residuals (the response minus the fitted value).

Plot page

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In the Life Testing dialog, the Plot page has the following options:

Plots

Probability Plot of Failure Time

Select this to plot failure probability versus failure time in models with one or fewer covariates or stratification variables.

Stress vs Failure Time

Select this to plot stress versus failure time.

Six Distributions Plot

Select this for comparative probability plots of the fitted values. The first three of these plots can be adorned with smooth lines, a rugplot, or labels for a specified number of the largest points (in absolute value), depending on the options chosen. Censored observations can be added to the first four plots. The last three plots are available only for models with a single covariate.

Response vs Fit

Display a plot of the response variable versus the fitted values. The line y = x is also drawn on the graph.

Residuals vs Fit

Select this to plot the deviance residuals versus the fitted values.

Sqrt Abs Residuals vs Fit

Display a plot of the square root of the absolute values of the residuals versus the fitted values. This plot is useful for checking for the constant variance assumption of the model.

Probability Plot of Residuals

Select this to create a probability plot of the standardized residuals.

Probability Plot Options

Add Maximum likelihood estimate Select this to plot the maximum likelihood estimate of the failure probability on the fit plot.

Add Legend Select to add a legend to the plot. The legend gives the quantiles associated with each line and color type.

Add Grid Lines Select to add grid lines to the plot.

Plot Methods Specify the method for computing the quantiles of the plotted points on the graph. In addition to probabilities for the model in object, probabilities for one other model can be plotted as points. Let the covariate be x, and let the response be censor(y). Then the possible methods for these alternative models are:

KM censor(y) ~ strata(x) Turnbulls generalization of Kaplan-Meier estimates

Separate censor(y) ~ strata(x) censorReg estimates with separate location and scale for each covariate value.

Factor censor(y) ~ factor(x) censorReg estimates with separate location and the same scale for each covariate value

Regression censor(y) ~ x censorReg model with location predicted by regression on the covariate

One censor(y) ~ 1 censorReg model with no covariates

None The comparison points are not plotted.

The default is method="KM".

Confidence Bands Specify the logical value (T or F) indicating whether to plot confidence bands. If confidence bands are desired for only some of the lines, a vector of logical values (for example T, F, T) may be specified indicating which lines receive confidence bands, where each value corresponds to a unique level of the covariate.

Plot At Specify the values of the covariate for which a probability distribution is to be plotted. Ignored if there is no covariate in the model in object.

Coverage Type the confidence level for the maximum likelihood estimation.

Stress Plot Options

Add Legend Select to add a legend to the plot. The legend gives the quantiles associated with each line and color type.

Jitter Data Select to jitter the points slightly so that duplicate points are not overlaid.

Failure Probabilities Enter a vector of probabilities for which quantiles are to be computed. The quantiles at each covariate value are plotted and quantiles with the same probability are connected with a line.

Predict page

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In the Life Testing dialog, the Predict page has the following options:

New Data

Enter the name of a matrix or data set to use for computing predictions. It must contain the same names as the terms in the right side of the formula for the model. If omitted, the original data are used for computing predictions.

Save In

Enter the name of a data set in which a part of the analysis, such as fitted values and residuals, predictions, confidence intervals, or standard errors, is saved.

Probability and Response Predictions

Probabilities Select this to predict probability of failure.

Response Select this to predict the response based upon the fitted probability distribution.

Response Options

Probability Values A vector of probabilities used to predict the response. This field is enabled only while Predict Response is selected.

Confidence Level Enter the confidence level used for confidence intervals of predicted response values.

Related S-Plus language functions for Life Testing

censorReg, print.censorReg, plot.censorReg, summary.censorReg, residuals.censorReg, censorReg.control, censorReg.fit, censorReg.distributions, anova.censorReg, pftdist, qftdist

Other related S-Plus language functions

formula, lm, solve, censor