An Example of Using Reliability DOE for Life Testing
Design of Experiments (DOE) is one of the important tools in Design for Six Sigma (DFSS) and Design for Reliability (DFR). However, there are difficulties with applying the traditional DOE analysis methods, such as ANOVA or linear regression, for data from life tests. In this article, we will briefly explain why the traditional DOE analysis methods cannot be directly applied to life data, and then provide an example of how to use ReliaSoft's DOE++ to design a life test and analyze the data.
Difficulties of Using Traditional DOE Analysis Methods
Another difficulty arises due to data type. In life tests, the data obtained may be either complete, censored or interval. However, there is no way to calculate the sum of squares for censored and interval data. The sum of squares makes sense only for complete response data. Further, the standard regression techniques applicable to the complete response data in traditional DOE can no longer be used if there are censored and interval data.
Therefore, in order to apply the traditional DOE methods to life data, people usually treat suspensions as failures and use the middle points of the interval data as the failure times. This "adjustment" is not correct and may lead to wrong conclusions. Even after this treatment, the data still may be skewed. So, a transformation is often applied to the "adjusted" data and then ANOVA and linear regression are used for the "transformed" data.
However, even with the adjustment and transformation, problems may still exist . Standard linear regression assumes the following about the data used in the model:
Thus, a single transformation is asked to produce the above two requirements simultaneously, which usually does not occur.
So it is best to reconsider using ANOVA and standard linear regression on life data. The Reliability DOE module in the DOE++ software provides an alternative, and the following sections provide an example of using this technique (Example 1), contrasted with the use of traditional DOE methods for the same data set (Example 2).
1) Design the Experiment
It is reasonable to assume that the 3-way or higher interactions among factors are negligible, so the fractional generator is set to E = ABCD. A 16-run design is generated as shown below.
In Figure 2 above, the code "-1" means a factor is at its low level and "1" means the factor is at its high level. In DOE++, you also can select Actual Values to display the actual settings of these factors.
The experiment is not conducted according to the standard order of the rows; rather, it is conducted by run order. Run order is a randomized order intended to eliminate the effects of some noise factors. Also, the technicians in this case feel that it is not convenient to read the coded values of the factor settings, so the engineer sorts the rows by Run Order and selected Actual Values for the Display Factor, obtaining the following design matrix.
2) Conduct Experiment and Collect Data
The failure times are in units of hours. Notice that there is a suspension (highlighted in Figure 4 above) at the end of the 5 day test.
3) Analyze the Data
In order to understand the results in Figure 5, let's discuss the math a little bit. For a detailed discussion, please refer to [1, 2]. The probability density function of the Weibull distribution is:
In reliability DOE, the scale parameter, , is assumed to be a function of the effects, which is expressed as:
Usually, the high order interactions such as 3-way and 4-way interactions can be ignored in the above equation. Once the effects have been selected for the model, maximum likelihood estimation is used to estimate the parameters of each distribution. From the log-likelihood values highlighted in Figure 5, we can see that the Weibull distribution provides larger likelihood values. Thus, the engineer decides to use the Weibull distribution. To further confirm his choice, he also examines the residual plots for both distributions, which are shown in Figure 6.
The red dashed lines are the thresholds at the given significance level of 0.1 for detecting abnormal residuals. Apparently, the Weibull distribution has a better residual plot, since all of the residuals fall between the high and the low thresholds.
4) Interpret the Results
Factors that are statistically significant are colored red in the results. Although the CD interaction is shown as significant, neither of the main effects, C or D, is significant. So it is questionable whether the 2-way interaction for CD is actually significant. A possible reason why CD is shown as being significant in the results is because it is confounded with other effects that were not in the model. So by checking the generator E=ABCD, the engineer realizes that CD is confounded with ABE, a 3-way interaction. For this interaction, all the main effects, A, B and E, are significant. Therefore, it is reasonable to assume that the interaction of ABE, instead of CD, causes the significance of the confounded effects. So, he decides to only keep A, B, E, AE and ABE in the model and gets the following results.
5) Draw Conclusions
The red dashed lines indicate the best settings for each factor. The blue dashed line is the predicted response [()] under the best settings. Those values are also displayed as X and Y values in the plot.
6) Make Prediction
So . The distribution is:
In DOE++, one can get the predicted and its two-sided confidence bounds by clicking the Prediction icon to open the prediction utility. Figure 9 shows the prediction results using a 90% confidence level.
7) Confirmation Experiments
In Example 1, we illustrated how to use reliability DOE for life tests and data analysis. One additional question is: What are the results if the traditional DOE methods are used for the data in Example 1? Example 2 will give the answer.
From this plot, none of the effects is significant.
The analysis could be stopped here with this conclusion. However, the engineer wants to go further. He decides to drop effects AC, DE and BD which are on the probability line with values close to 0. He believes it to be safe to pool these effects into error.
The results of the reduced model are shown in Figures 12 and 13:
From the results in Figures 12 and 13, the engineer concludes that effects A and AE are significant. At this point, he hears that someone has already performed the analysis using the reliability DOE technique without "adjusting" and "transforming" the original data. He also worries that his adjustment to the original data could result in incorrect conclusions, so he stops the analysis.
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