Utilizing Residual Plots in Accelerated Life Testing Data Analysis
When analyzing accelerated life testing data, it is important to assess model assumptions, discover inadequacies in the model, note extreme observations and assess the possibility that the test did not account for important factors. One way to perform such verifications is through the use of residual analysis for reliability, which consists of analyzing the results of a regression analysis by assigning residual values to each point in the data set and plotting these residuals. Three types of residual plots are available in ALTA 6: the Standardized Residuals plot, the Cox-Snell Residuals plot and the Standardized vs. Fitted Value plot. This article presents descriptions of the plot types and their benefits.
Standardized Residuals (SR) Plot
The standardized residuals plot is useful for determining the adequacy of the distribution for the data. The plot line has a mean of zero and negative values are possible. The appropriate probability transformation is plotted on the y-axis and the value of the residual is plotted on the x-axis. The plotted points (residuals) are based on the data, which are transformed using an appropriate transformation based on the selected life-stress relationship and distribution. If the model adequately fits the data, the points should track the plot line.
SR for the Weibull Distribution
Under the assumed model, these residuals should look like a sample from an extreme value distribution with a mean of zero. For the Weibull distribution, the standardized residuals are plotted on a smallest extreme value probability paper. If the Weibull distribution adequately describes the data, then the standardized residuals should appear to follow a straight line on such a probability plot. Note that when an observation is censored (suspended), the corresponding residual is also censored.
Figure 1: Probability Plot of Standardized Residuals for the Weibull Distribution
SR for the Lognormal Distribution
Under the assumed model, the standardized residuals should be normally distributed with a mean of zero and a standard deviation of one (~N(0,1)). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.
Figure 2: Probability Plot of Standardized Residuals for the Lognormal Distribution
Cox-Snell Residuals Plot
The Cox-Snell residuals are given by:
where R(Ti) is the calculated reliability value at failure time Ti.
Figure 3: Probability Plot of the Cox-Snell Residuals
Standardized vs. Fitted Value Plot
Figure 4: Standardized vs. Fitted Value Residuals Plot
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