Issue 51, May 2005

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
Once the parameters have been estimated, the standardized residuals for the Weibull distribution can be calculated by:

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
Once the parameters have been estimated, the fitted or calculated responses can be calculated by:

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 purpose of this plot is to determine the adequacy of the distribution for the data. The line is plotted on an exponential probability plotting paper and is on the positive domain. The plotted points (residuals) are based on the data, which are transformed using an appropriate transformation based on the selected life-stress relationship. If the model adequately fits the data, the plotted points should track the plot line.
 

The Cox-Snell residuals are given by:

where R(T_i) is the calculated reliability value at failure time T_i.

Figure 3: Probability Plot of the Cox-Snell Residuals

Standardized vs. Fitted Value Plot
This plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Residual vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, in the case of the Weibull distribution, the standardized residuals are plotted versus h(V), for the lognormal versus m′(V), and for the exponential versus m(V).

Figure 4: Standardized vs. Fitted Value Residuals Plot

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