Distribution, Quantile, Hazard, Density, and Acceleration Factor profilers, along with criteria values under Comparison Criterion can be viewed and compared.
Parametric estimates, covariance matrices, nested model tests, and diagnostics can be examined and compared for each of the selected distributions.
The Summary of Data section gives the total number of observations, the number of uncensored values, and the number of censored values (right, left, and interval). Summary of Data Example shows the summary data for the Devalt.jmp sample data table.
The Scatterplot of the lifetime event versus the explanatory variable is shown at the top of the report window. For the Devalt.jmp sample data, the Scatterplot shows Hours versus Temp. Scatterplot Representation for Failure and Censored Observations indicates how each type of failure is represented on the Scatterplot in the report window. To increase the size of the markers on the graph, right-click the graph, select Marker Size and then select one of the marker sizes listed.
Specify the density curve that you want, one at a time, by entering any value within the range of the accelerating factor. You can then select different distributions by selecting the appropriate check box(es) that appear after you add a curve.
Displays previously entered density curve values. Remove curves by selecting the appropriate check box.
Select to show the density curves. If the Location or the Location and Scale model is fit, or if Nested Model Tests is selected in the launch window, then the density curves for all of the given explanatory variable levels are shown. After the curves have been created, the Show Density Curves option toggles the curves on and off the plot.
Specify the quantile lines that you want, three at a time. You can add more quantiles by continually selecting Add Quantile Lines. Default quantile values are 0.1, 0.5, and 0.9. Invalid quantile values, such as missing values, are ignored. If desired, you can enter just one quantile value, leaving the other entries blank.
Swaps the X and Y axes.
The default view of the scatterplot incorporates the transformation scale. Select this option to switch between the linear and nonlinear scales for the x axis.
Scatterplot of Hours versus Temp shows the initial scatterplot; Scatterplot with Density Curve and Quantile Line Options shows the resulting scatterplot with the Show Density Curves and Add Quantile Lines options selected displaying the curves and the lines for the various Temp levels for the Weibull distribution. You can also view density curves across all the levels of Temp for the other distributions. These distributions can be selected one at a time or can be viewed simultaneously by checking the boxes to the left of the desired distribution name(s).
The Nonparametric Overlay plot is displayed after the scatterplot. Differences among groups can readily be detected by examining this plot. For the Devalt.jmp sample data, you can view these differences for Hours on different scales. You can also change the interval type on a Nonparametric fit probability plot between Simultaneous and Pointwise (results displayed when Show Nonparametric CI is selected), and select whether to Show Parametric CI or Show Nonparametric CI confidence intervals.
Pointwise estimates show the pointwise 95% confidence bands on the plot while simultaneous confidence intervals show the simultaneous confidence bands for all groups on the plot. Meeker and Escobar (1998, chap. 3) discuss pointwise and simultaneous confidence intervals and the motivation for simultaneous confidence intervals in a lifetime analysis.
For this example, the Wilcoxon Group Homogeneity Test, shown in Nonparametric Overlay Plot and Wilcoxon Test for Devalt.jmp, indicates that there is a difference among groups. The high chi-square value and low p-value are consistent with the differences seen among the Temp groups in the Nonparametric Overlay plot.
The Comparisons report section, shown in Distribution Profiler, shows profilers for the selected distributions in the Nonparametric Overlay section, and includes the following tabs:
To show a specific profiler, select the appropriate distribution option in the Nonparametric Overlay section.
The first five tabs show profilers for the selected distributions. Curves shown in the first four profilers correspond to both the time-to-event and explanatory variables. The Acceleration Factor profiler tab only corresponds to the acceleration factor (explanatory variable). Distribution Profiler shows the Distribution Profiler for the Weibull, Lognormal, Fréchet, and Loglogistic distributions.
Comparable results appear on the Quantile, Hazard, and Density tabs. The Distribution, Quantile, Hazard, Density, and Acceleration Factor Profilers behave similarly to the Prediction Profiler in other platforms. For example, the vertical lines of Temp and Hours can be dragged to see how each of the distribution values change with temperature and time. For a detailed explanation of the Prediction Profiler, see the Profilers book.
You can use the Quantile profiler for extrapolation. Suppose that the data are represented by a Weibull distribution. From viewing the Weibull Acceleration Factor Profiler in Acceleration Factor Profiler for Devalt.jmp, you see that the acceleration factor at 45 degrees Celsius is 17.18683 for a baseline temperature of 10 degrees Celsius. Select the Quantile tab to see the Quantile Profiler for the Weibull distribution. Select and drag the vertical line in the probability plot so that the probability reads 0.5. From viewing Weibull Quantile Profiler for Devalt.jmp, where the Probability is set to 0.5, you find that the quantile for the failure probability of 0.5 at 45 degrees Celsius is 13849.01 hours. So, at 10 degrees Celsius, you can expect that 50% of the units fail by 13849.01 * 17.18683 = 238021 hours.
Weibull Quantile Profiler for Devalt.jmp
Selecting the Acceleration Factor tab shows the Acceleration Factor Profiler for the time-to-event variable for each specified distribution. To produce Acceleration Factor Profiler for Devalt.jmp, select Fit All Distributions from the Fit Life by X red triangle menu. Modify the baseline value for the explanatory variable by selecting Set Time Acceleration Baseline from the Fit Life by X red triangle menu and entering the desired value. Note that the explanatory variable and the baseline value appear beside the profiler title.
Acceleration Factor Profiler for Devalt.jmp
The Acceleration Factor Profiler lets you estimate time-to-failure for accelerated test conditions when compared with the baseline condition and a parametric distribution assumption. The interpretation of a time-acceleration plot is generally the ratio of the pth quantile of the baseline condition to the pth quantile of the accelerated test condition. This relation applies only when the distribution is Lognormal, Weibull, Loglogistic, or Fréchet, and the scale parameter is constant for all levels. This relation does not apply for a Normal, SEV, Logistic, or LEV distribution.
Note: The Acceleration Factor Profiler does not appear in the following instances: when the explanatory variable is discrete; the explanatory variable is treated as discrete; a customized formula does not use a unity scale factor; or the distribution is Normal, SEV, Logistic, or LEV.
The Comparison Criterion tab shows the -2Loglikelihood, AICc, and BIC criteria for the distributions of interest. Comparison Criterion Report Tab shows these values for the Weibull, Lognormal, Loglogistic, and Fréchet distributions. Distributions providing better fits to the data are shown at the top of the Comparisons report, sorted by AICc.
This report suggests that the Lognormal and Loglogistic distributions provide the best fits for the data, because the lowest criteria values are seen for these distributions. For a detailed explanation of the criteria, see in the Life Distribution section.
The Results section of the report window shows more detailed statistics and prediction profilers than those shown in the Comparisons report. Separate result sections are shown for each selected distribution. Weibull Distribution Nested Model Tests for Devalt.jmp Data shows a portion of the Weibull results, Nested Model Tests, and Diagnostics plots for Devalt.jmp.
Statistical results, diagnostic plots, and Distribution, Quantile, Hazard, Density, and Acceleration Factor Profilers are included for each of your specified distributions. The Custom Estimation tab lets you estimate specific failure probabilities and quantiles, using both Wald and Profile interval methods. When the Box-Cox Relationship is selected on the platform launch window, the Sensitivity tab appears. This tab shows how the Loglikelihood and B10 Life change as a function of Box-Cox lambda.
Weibull Distribution Nested Model Tests for Devalt.jmp Data
For each parametric distribution, there is a Statistics section that shows parameter estimates, a covariance matrix, confidence intervals, summary statistics, and diagnostic plots. You can save probability, quantile, and hazard estimates by selecting any or all of these options from the Statistics red triangle menu for each parametric distribution. The estimates and the corresponding lower and upper confidence limits are saved as columns in your data table. Save Options for Parametric Distribution shows the save options available for any parametric distribution.
Nested Model Tests are included, if you selected the option on the platform launch window. The Nested Model Tests include statistics and diagnostic plots for the Separate Location and Scale, Separate Location, Regression, and No Effect models. Separate Location and Scale, Separate Location, and Regression analyses results are shown by default. Regression parameter estimates and the location parameter formula are shown under the Estimates section, by default. The Diagnostics plots for the No Effect model can be displayed by selecting the check box to the left of No Effect under the Nested Model Tests title.
To see results for each of the models (independently of the other models), click the underlined model of interest (listed under Nested Model Tests) and then uncheck the check boxes for the other models. Nested models are described in Nested Model Tests. Separate Location and Scale, Separate Location, Regression, and No Effect models, using a Weibull distribution for Devalt.jmp, are shown in Separate Location and Scale Model with the Weibull Distribution for Devalt.jmp Data, Separate Location Model with the Weibull Distribution for Devalt.jmp Data, Regression Model with the Weibull Distribution for Devalt.jmp Data, and No Effect Model with the Weibull Distribution for Devalt.jmp Data, respectively.
If the Nested Model Tests option was not checked in the launch window, then the Separate Location and Scale, and Separate Location models are not assessed. In this case, estimates are given for the regression model for each distribution that you select, and the Cox-Snell Residual P-P Plot is the only diagnostic plot.
Assumes that the location and scale parameters are different for all levels of the explanatory variable and is equivalent to fitting the distribution by the levels of the explanatory variable. The Separate Location and Scale Model has multiple location parameters and multiple scale parameters.
Assumes that the location parameters are different, but the scale parameters are the same for all levels of the explanatory variable. The Separate Location Model has multiple location parameters and only one scale parameter.
Assumes that the explanatory variable does not affect the response and is equivalent to fitting all of the data values to the selected distribution. The No Effect Model has one location parameter and one scale parameter.
The Multiple Probability Plots shown in Weibull Distribution Nested Model Tests for Devalt.jmp Data are used to validate the distributional assumption for the different levels of the accelerating variable. If the line for each level does not run through the data points for that level, the distributional assumption might not hold. Side-by-side comparisons of the diagnostic plots provide a visual comparison for the validity of the different models. See Meeker and Escobar (1998, sec. 19.2.2) for a discussion of multiple probability plots.
The Cox-Snell Residual P-P Plots are used to validate the distributional assumption for the data. If the data points deviate far from the diagonal, then the distributional assumption might be violated. The Cox-Snell Residual P-P Plot red triangle menu has an option called Save Residuals that enables you to save the residual data to the data table. See Meeker and Escobar (1998, sec. 17.6.1) for a discussion of Cox-Snell residuals.
Separate Location Model with the Weibull Distribution for Devalt.jmp Data
Regression Model with the Weibull Distribution for Devalt.jmp Data
No Effect Model with the Weibull Distribution for Devalt.jmp Data
In addition to a statistical summary and diagnostic plots, the Fit Life by X report window also includes profilers and surface plots for each of your specified distributions. To view the Weibull time-accelerating factor and explanatory variable profilers, click the Distribution tab under Weibull Results. To see the surface plot, click the disclosure icon to the left of the Weibull title (under the profilers). The profilers and surface plot behave similarly to other platforms. See the Profilers book.
The report window also includes a tab labeled Acceleration Factor. Clicking the Acceleration Factor tab shows the Acceleration Factor Profiler. This profiler is an enlargement of the Weibull plot shown under the Acceleration Factor tab in the Comparisons section of the report window. Weibull Acceleration Factor Profiler for Devalt.jmp shows the Acceleration Factor Profiler for the Weibull distribution of Devalt.jmp. The baseline level for the explanatory variable can be modified by selecting the Set Time Acceleration Baseline option in the Fit Life by X red triangle menu.
Weibull Acceleration Factor Profiler for Devalt.jmp
If you want to use a custom transformation to model the relationship between the lifetime event and the accelerating factor, use the Custom option. This option is found in the list under Relationship in the launch window. Enter comma delimited values into the entry fields for the location (μ) and scale (σ) parameters. For the Devalt.jmp sample data, an example entry for μ could be “1, log(:Temp), log(:Temp)^2,” and an entry for σ could be “1, log(:Temp),” where 1 indicates that an intercept is included in the model. Select the Use Exponential Link check box to ensure that the sigma parameter is positive.
Custom Relationship Specification in Fit Life by X Launch Window
After selecting OK, location and scale transformations are created and included at the bottom of the Estimates report section.
For an example of how to use a custom transformation, see Custom Relationship Example. Analysis proceeds similarly to the Example of the Fit Life by X Platform, where the Arrhenius Celsius Relationship was specified.