The LIFEREG Procedure

Overview

The LIFEREG procedure fits parametric models to failure time data that can be right, left, or interval censored. The models for the response variable consist of a linear effect composed of the covariates and a random disturbance term. The distribution of the random disturbance can be taken from a class of distributions that includes the extreme value, normal, logistic, and, by using a log transformation, the exponential, Weibull, lognormal, loglogistic, and gamma distributions. The model assumed for the response y is

$y = {X \beta} + \sigma {\epsilon}$

where y is a vector of response values, often the log of the failure times, X is a matrix of covariates or independent variables (usually including an intercept term), ${\beta}$ is a vector of unknown regression parameters, $\sigma$ is an unknown scale parameter, and ${\epsilon}$ is a vector of errors assumed to come from a known distribution (such as the standard normal distribution). The distribution may depend on additional shape parameters. These models are equivalent to accelerated failure time models when the log of the response is the quantity being modeled. The effect of the covariates in an accelerated failure time model is to change the scale, and not the location, of a baseline distribution of failure times. The LIFEREG procedure estimates the parameters by maximum likelihood using a Newton-Raphson algorithm. PROC LIFEREG estimates the standard errors of the parameter estimates from the inverse of the observed information matrix. The accelerated failure time model assumes that the effect of independent variables on an event time distribution is multiplicative on the event time. Usually, the scale function is $\exp(x^'{\beta})$ , where x is the vector of covariate values and ${\beta}$ is a vector of unknown parameters. Thus, if T₀ is an event time sampled from the baseline distribution corresponding to values of zero for the covariates, then the accelerated failure time model specifies that, if the vector of covariates is x, the event time is $T = \exp(x^'{\beta}) T_0$ . If y = log(T) and y₀ = log(T₀), then

$y = x^'{\beta} + y_0$

This is a linear model with y₀ as the error term.

In terms of survival or exceedance probabilities, this model is

$\Pr(T \gt t | x) = \Pr(T_0 \gt \exp(-x^'{\beta})t)$

The probability on the left-hand side of the equal sign is evaluated given the value x for the covariates, and the right-hand side is computed using the baseline probability distribution but at a scaled value of the argument. The right-hand side of the equation represents the value of the baseline Survival Distribution Function evaluated at $\exp(-x^' \beta)t$ .

Usually, an intercept parameter and a scale parameter are allowed in the model. In terms of the original untransformed event times, the effects of the intercept term and the scale term are to scale the event time and power the event time, respectively. That is, if

$\log(T) = \mu + \sigma \log(T_0)$

then

$T = \exp(\mu) T_0^{\sigma}$

Although it is possible to fit these models to the original response variable using the NOLOG option, it is more common to model the log of the response variable. Because of this log transformation, zero values for the observed failure times are not allowed unless the NOLOG option is specified. Similarly, small values for the observed failure times lead to large negative values for the transformed response. The NOLOG option should only be used if you want to fit a distribution appropriate for the untransformed response, the extreme value instead of the Weibull, for example.

The parameter estimates for the normal distribution are sensitive to large negative values, and care must be taken that the fitted model is not unduly influenced by them. Likewise, values that are extremely large even after the log transformation have a strong influence in fitting the extreme value (Weibull) and normal distributions. You should examine the residuals and check the effects of removing observations with large residuals or extreme values of covariates on the model parameters. The logistic distribution gives robust parameter estimates in the sense that the estimates have a bounded influence function.

The standard errors of the parameter estimates are computed from large sample normal approximations using the observed information matrix. In small samples, these approximations may be poor. Refer to Lawless (1982) for additional discussion and references. You can sometimes construct better confidence intervals by transforming the parameters. For example, large sample theory is often more accurate for $\log(\sigma)$ than $\sigma$ . Therefore, it may be more accurate to construct confidence intervals for $\log(\sigma)$ and transform these into confidence intervals for $\sigma$ . The parameter estimates and their estimated covariance matrix are available in an output SAS data set and can be used to construct additional tests or confidence intervals for the parameters. Alternatively, tests of parameters can be based on log-likelihood ratios. Refer to Cox and Oakes (1984) for a discussion of the merits of some possible test methods including score, Wald, and likelihood ratio tests. It is believed that likelihood ratio tests are generally more reliable in small samples than tests based on the information matrix. The log-likelihood function is computed using the log of the failure time as a response. This log likelihood differs from the log likelihood obtained using the failure time as the response by an additive term of $\sum \log(t_i)$ , where the sum is over the noncensored failure times. This term does not depend on the unknown parameters and does not affect parameter or standard error estimates. However, many published values of log likelihoods use the failure time as the basic response variable and, hence, differ by the additive term from the value computed by the LIFEREG procedure.

The classic Tobit model (Tobin 1958) also fits into this class of models but with data usually censored on the left. The data considered by Tobin in his original paper came from a survey of consumers where the response variable is the ratio of expenditures on durable goods to the total disposable income. The two explanatory variables are the age of the head of household and the ratio of liquid assets to total disposable income. Because many observations in this data set have a value of zero for the response variable, the model fit by Tobin is

$y = \max(x^' \beta + \epsilon, 0)$

which is a regression model with left censoring.

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