Template:Monitoring warranty returns using statistical process control

Monitoring Warranty Returns Using Statistical Process Control (SPC)
By monitoring and analyzing warranty return data, the user now has the ability to detect specific return periods and/or batches of sales or shipments that may deviate (differ) from the assumed model. This provides the analyst (and the organization) the advantage of early notification of possible deviations in manufacturing, use conditions and/or any other factor that may adversely affect the reliability of the fielded product.

Obviously, the motivation for performing such analysis is to allow for faster intervention to avoid increased costs due to increased warranty returns or more serious repercussions. Additionally, this analysis can also be used to uncover different sub-populations that may exist within this population.

Analysis Method

For each sales period  $$i$$  and return period  $$j$$, the prediction error can be calculated as follows:


 * $${{e}_{i,j}}={{\hat{F}}_{i,j}}-{{F}_{i,j}}$$

where $${{\hat{F}}_{i,j}}$$  is the estimated number of failures based on the estimated distribution parameters for the sales period  $$i$$  and the return period  $$j$$, which is calculated using the equation for the conditional probability, and  $${{F}_{i,j}}$$  is the actual number of failure for the sales period  $$i$$  and the return period  $$j$$.

Since we are assuming that the model is accurate, $${{e}_{i,j}}$$  should follow a normal distribution with mean value of zero and a standard deviation  $$s$$, where:


 * $${{\bar{e}}_{i,j}}=\frac{\underset{i}{\mathop{\sum }}\,\underset{j}{\mathop{\sum }}\,{{e}_{i,j}}}{n}=0$$

and $$n$$  is the total number of return data (total number of residuals).

The estimated standard deviation of the prediction errors can then be calculated by:


 * $$s=\sqrt{\frac{1}{n-1}\underset{i}{\mathop \sum }\,\underset{j}{\mathop \sum }\,e_{i,j}^{2}}$$

and $${{e}_{i,j}}$$  can be normalized as follows:


 * $${{z}_{i,j}}=\frac{s}$$

where $${{z}_{i,j}}$$  is the standardized error. $${{z}_{i,j}}$$ follows a normal distribution with  $$\mu =0$$  and  $$\sigma =1$$.

It is known that the square of a random variable with standard normal distribution follows the $${{\chi }^{2}}$$  (Chi Square) distribution with 1 degree of freedom and that the sum of the squares of  $$m$$  random variables with standard normal distribution follows the  $${{\chi }^{2}}$$  distribution with  $$m$$  degrees of freedom $$.$$  This then can be used to help detect the abnormal returns for a given sales period, return period or just a specific cell (combination of a return and a sales period).


 * For a cell, abnormality is detected if $$z_{i,j}^{2}=\chi _{1}^{2}\ge \chi _{1,\alpha }^{2}.$$
 * For an entire sales period $$i$$, abnormality is detected if  $$\underset{j}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{J}^{2}\ge \chi _{\alpha ,J}^{2},$$  where  $$J$$  is the total number of return period for a sales period  $$i$$.
 * For an entire return period $$j$$, abnormality is detected if  $$\underset{i}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{I}^{2}\ge \chi _{\alpha ,I}^{2},$$  where  $$I$$  is the total number of sales period for a return period  $$j$$.

Here $$\alpha $$  is the criticality value of the  $${{\chi }^{2}}$$  distribution, which can be set at critical value or caution value. It describes the level of sensitivity to outliers (returns that deviate significantly from the predictions based on the fitted model). Increasing the value of  $$\alpha $$  increases the power of detection, but this could lead to more false alarms.

Example 5:

Example 6: