Distribution, Quantile, Hazard, Density, and Acceleration Factor profilers, along with criteria values under Comparison Criterion can be viewed and compared.
Summary of Data Example
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.
Scatterplot of Hours versus Temp
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.
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).
Scatterplot with Density Curve and Quantile Line Options
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.
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.
Nonparametric Overlay Plot and Wilcoxon Test for Devalt.jmp
The Comparisons report section, shown in Distribution Profiler, shows profilers for the selected distributions in the Nonparametric Overlay section, and includes the following tabs:
Distribution Profiler
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.
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.
Comparison Criterion Report Tab
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
Save Options for 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.
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.
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.
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
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.
Weibull Estimates and Formulas for Custom Relationship
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.