Chapter 6: Left Truncated Data (LTD) Module
The Left Truncated Data is treated as a special type of Life Data Analysis, and it is not the same as censored data, as there is no information on how many items of this sample have failed in the past. But this operation time cannot be discarded, as the items present burnin.
6.1 LTD Analysis List
The dashboard follows the same structure as other modules, as shown below:
On the right side of the screen, there are 4 symbols; they are also present on the RGA chart dashboard and present the exact same functions:
This is an explanation of their functions, following the order from left to right:

Configure the dashboard to show relevant information in the list, such as name, quantity of items or creation date.

Copy the dashboard list’s information so that the user can then paste it in another document of his choosing.

Download a spreadsheet containing the dashboard information.

Generate a .pdf archive of the dashboard at hand.

This chart will also sort the analysis as the preference of the user: by name, quantity of items, creation date, or by last update date, and this can be done by clicking on each feature, on the top side of the dashboard.
6.2 LTD – Data tab
The LTD dashboard has the same inputs as the LDA dashboard, except for a new fixed column to the left (Left Truncated Age), which refers to the operating time with no information.
Data type can be exact time or interval time, with the options to have suspended items or grouped items.
6.3 LTD – Plot tab
On the “Plot” tab, the generated graphs will be available after the parameters are shown in the “Analysis Results” box (upon hitting the “Calculate” button on the “Data” tab).
For the LTD analysis, there are available six graphs:

Prob. of Failure vs Time (distribution scale)

Reliability vs Time

Prob. of Failure vs Time

Pdf vs Time

Failure Rate vs Time

Optimum Replacement Interval Time
Besides the plots, the settings, such as scale and confidence bound methods and interval, can be changed, and “Analysis Results” are available as follows:
The “Confidence Bounds” setting button will open a new window with the options for the bound’s methods, axis reference and interval.
6.4 LTD – Overlay Plot tab
On this tab, it is possible to plot on the same graph, different items (if they are the same type) so that the user can compare the reliability results of different assets.
At first, there are no plots being displayed on the screen. Note that’s because the user must select which items he wants to plot on the screen, by moving the desired item from the box on the left to the other box on the right, as shown as follows:
6.5 LTD – Report tab
This feature displays all the current item’s graphs, and the information related to analysis parameters, plotting methods, parameter estimation and fitting model, and of course, the parameters’ values, number of suspensions and failures in the data set.
6.6 LTD – Calculator
The Forecast tab presents the warranty returns for future periods based on the items that are still working for each production period.
This WANRI module applies the conditional probability of failure to determine the number of items that may fail for each production period. The system multiplies the conditional probability of failure by the number of units that are still working in a determined period and then the number of warranty returns will be calculated.
In the result box, for this specific case, where the function selected is the Reliability function R(t), the LTD Calculator returns what the reliability of the item after working for a given time is. It also shows the confidence bounds value for this time and, in this case, the user is also able to reconfigure how the bounds are calculated and which bound he wants to see. The precision refers to how many decimal places will be considered for the results.
While exporting the calculator results history, the calculator stays locked and can only operate again after the opened document is closed.
As all the reliability functions have a range input, the LTD Calculator will inform the user if the entered value is valid for the given operation, as in the example below:
The function for the BX% Life needs a percentage parameter, so it only accepts values between 0 and 100.
The functions are placed side by side with their inverted functions, and some may need a new input due to their parameters, like the Conditional Reliability Function R(t/T), which also asks for the working time performed by the item without presenting any failures.
Note that F(T) (cumulative probability of failure) and F(t/T) (conditional cumulative probability of failure) are placed beside R(T) and R(t/T), which are their complementary functions.
T(R) is a function which returns the time for a desired reliability value.
The Mean Function is the only one which requires no parameters, because it is the average value of the modeling distribution.
The function λ(t) returns the failure rate for a specific time value.
The Parameter Bounds will show the values estimated for the current model parameters with their upper and lower bounds, based on the bounds configuration which can be altered at any time inside the calculator on the Confidence Bounds button.
Note that F(T) (cumulative probability of failure) and F(t/T) (conditional cumulative probability of failure) are placed beside R(T) and R(t/T), which are their complementary functions.
T(R) is a function which returns the time for a desired reliability value.
Optimum Replacement Interval is applied to plan preventive maintenances, as it returns the interval between preventive actions. It can be applied for items presenting increasing failure rates over time and with an unplanned replacement cost higher than the scheduled replacement cost. It can be used to minimize the shortterm cost per unit time or to minimize the longterm cost per time.