Methods and formulas for equations in Regression with Life Data

Lifetime regression

The regression model estimates the percentiles of the failure time distribution:

Y_p = β₀ +β₁x₁ + β₂x₂ + ... +β_kx_k + σ Φ^-1(p)

Term	Description
Y_p	either failure time or log(failure time)
β₀	y-intercept (constant)
β₁...β_k	regression coefficients
x₁...x_k	predictor values
σ	1/shape (Weibull distribution) or scale (other distributions)
Φ^-1(p)	p^th quantile of the standardized life distribution

Depending on the distribution, Y_p = failure time or log (failure time):

For the Weibull, exponential, lognormal, and loglogistic distributions, Y_p = log (failure time)
For the normal, extreme value, and logistic distributions, Y_p = failure time

When Y_p = log (failure time), Minitab takes the antilog to display the percentiles on the original scale.

The value of the error distribution also depends on the distribution chosen.

For the normal distribution, the error distribution is the standard normal distribution – normal (0,1). For the lognormal distribution, Minitab takes the natural log of the data and then also uses a normal distribution.
For the logistic distribution, the error distribution is the standard logistic distribution – logistic (0, 1). For the loglogistic distribution, Minitab takes the natural log of the data and then also uses a logistic distribution.
For the extreme value distribution, the error distribution is the standard extreme value distribution – extreme value (0, 1). For the Weibull distribution and the exponential distribution (a type of Weibull distribution), Minitab takes the natural log of the data and then also uses the extreme value distribution.