Quality and Reliability Methods - SAS

394 Reliability and Survival Analysis II Chapter 20
Nonlinear Parametric Survival Models
Small). The Squamous level is not shown, but it is calculated as the negative sum of the other estimates. Two
example Risk Ratios for Cell Type calculations follow.
Large/Adeno = exp(β Large )/exp(β Adeno ) = exp(-0.2114757)/exp(0.57719588) = 0.4544481
Squamous/Adeno = exp[-(β Adeno + β Large + β Small )]/exp(β Adeno )
= exp[-(0.57719588 + (-0.2114757) + 0.24538322)]/exp(0.57719588) = 0.3047391
Reciprocal shows the value for 1/Risk Ratio.
Nonlinear Parametric Survival Models
This section shows how to use the Nonlinear platform for survival models. You only need to learn the
techniques in this section if:
• The model is nonlinear.
• You need a distribution other than Weibull, lognormal, exponential, Fréchet, or loglogistic.
• You have censoring that is not the usual right, left, or interval censoring.
With the ability to estimate parameters in specified loss functions, the Nonlinear platform becomes a
powerful tool for fitting maximum likelihood models. See the Modeling and Multivariate Methods book for
complete information about the Nonlinear platform.
To fit a nonlinear model when data are censored, you first use the formula editor to create a parametric
equation that represents a loss function adjusted for censored observations. Then use the Nonlinear
command in the Analyze > Modeling menu, which estimates the parameters using maximum likelihood.
As an example, suppose that you have a table with the variable time as the response. First, create a new
column, model, that is a linear model. Use the calculator to build a formula for model as the natural log of
time minus the linear model, that is, ln(time) - B0 + B1*z where z is the regressor.
Then, because the nonlinear platform minimizes the loss and you want to maximize the likelihood, create a
loss function as the negative of the log-likelihood. The log-likelihood formula must be a conditional
formula that depends on the censoring of a given observation (if some of the observations are censored).
Loss Formulas for Survival Distributions
The following formulas are for the negative log-likelihoods to fit common parametric models. Each formula
uses the calculator if conditional function with the uncensored case of the conditional first and the
right-censored case as the Else clause. You can copy these formulas from tables in the Loss Function
Templates folder in Sample Data and paste them into your data table. “Loglogistic Loss Function” on
page 395, shows the loss functions as they appear in the columns created by the formula editor.
Exponential Loss Function
The exponential loss function is shown in “Loglogistic Loss Function” on page 395, where sigma represents
the mean of the exponential distribution and Time is the age at failure.