2.5.3.2 Algorithm: Regression with Life Data

Uncensor/Right Censor data: The data are represented as time and censoring indicator pairs \((t_i, c_i)\):

• \(t_i\): The observed time for each unit. This could be Exact failure time (if the unit failed) or Time to censoring (if the unit did not fail within the observation period)
• \(c_i\): Indicates whether the unitis a failure or censored. e.g. 1 = failure (uncensored), 0 = censored

Uncensor/Arbitrary Censor data: The data are represented as time intervals \((tl_i, tr_i)\):

• \(tl_i\): lower bound (time of last inspection or last known survival)
• \(tr_i\): upper bound (time when failure was first detected)
• If \(tr_i = \infty\), it represents right-censoring.
• If \(tl_i = tr_i\), it represents exact failure (uncensored).

Regression Table

The lifetime regression function can be derived from the inverse cumulative distribution function:

\[Y_p = b_0 + b_1x_1 + b_2x_2 + ... + b_kx_k + \sigma\Phi^{-1}(p)\]

• \(Y_p\): failure time (Normal, Logistic, Smallest Extreme Value) or log failure time (Lognormal, Loglogistic, Exponential, Weibull)
• \(p\): cumulative probability \(P(T\le t)\)
• \(\Phi^{-1}(p)\): standardized inverse cumulative distribution
• \(x_i\): predictor values, where \(i = 1,2,...,k\)
• \(b_i\): coefficient of \(x_i\), where \(i = 1,2,...,k\)
• \(b_0\): intercept
• \(\sigma\): \(1/shape\) for weibull distribution or \(scale\) for other distributioln.

MLE (maximum likelihood estimation) method is then used to estimate parameters \(b_0\), \(b_i\) and \(\sigma\). For MLE, the standard error of the fitting parameters can be calculated by Fisher information matrix (FIM).

Anderson-Darling Test

Refer to Anderson-Darling Test Page for calculation.

Probability Plots

Probability Plot for Standardized Residuals

Standardized Residuals: \(\frac{y_i-x_i\hat{b}}{\hat{\sigma}}\)

• \(y_i\): the \(i^{th}\) response value
• \(x_i\): the \(i^{th}\) predictor values
• \(\hat{b}\): estimated regression coefficients
• \(\sigma\): estimated scale parameter

The P-P plot is to check if standardized residuals follow Smallest Extreme Value distribution.

Probability Plot for Cox-Snell Residuals

Cox-Snell residuals: \(-ln(\hat{R}(y_i))\)

• \(\hat{R}(y_i)\): estimated survival probability for the response \(y_i\).

The P-P plot is to check if Cox-Snell residuals follow Exponential distribution.