Model Specifications

The LIFEREG Procedure

Model Specifications

LIFEREG procedure Suppose there are n observations from the model ${y=X \beta} + \sigma {\epsilon}$ , where X is an n ×k matrix of covariate values (including the intercept), y is a vector of responses, and ${\epsilon}$ is a vector of errors with survival distribution function S, cumulative distribution function F, and probability density function f. That is, $S(t) = \Pr(\epsilon_i \gt t)$ , $F(t) = \Pr(\epsilon_i \leq t)$ , and f(t)=dF(t)/dt, where $\epsilon_i$ is a component of the error vector. Then, if all the responses are observed, the log likelihood, L, can be written as

$L = \sum \log ( \frac{f(w_i)}{\sigma} )$

where $w_i = \frac{1}{\sigma}(y_i-x_i^'{\beta})$ .

If some of the responses are left, right, or interval censored, the log likelihood can be written as

$L = \sum \log ( \frac{f(w_i)}{\sigma} ) + \sum \log ( S(w_i) ) + \sum \log ( F(w_i) ) + \sum \log ( F(w_i) - F(v_i) )$

with the first sum over uncensored observations, the second sum over right-censored observations, the third sum over left-censored observations, the last sum over interval-censored observations, and

$v_i = \frac{1}{\sigma} (z_i-x_i^'{\beta})$

where z_i is the lower end of a censoring interval.

If the response is specified in the binomial format, events/trials, then the log-likelihood function is

$L = \sum r_i \log(P_i) + (n_i - r_i) \log(1-P_i)$

where r_i is the number of events and n_i is the number of trials for the ith observation. In this case, $P_i = 1 - F(-x^'_i{\beta})$ . For the symmetric distributions, logistic and normal, this is the same as $F(x^'_i{\beta})$ . Additional information on censored and limited dependent variable models can be found in Kalbfleisch and Prentice (1980) and Maddala (1983).

The estimated covariance matrix of the parameter estimates is computed as the negative inverse of I, which is the information matrix of second derivatives of L with respect to the parameters evaluated at the final parameter estimates. If I is not positive definite, a positive definite submatrix of I is inverted, and the remaining rows and columns of the inverse are set to zero. If some of the parameters, such as the scale and intercept, are restricted, the corresponding elements of the estimated covariance matrix are set to zero. The standard error estimates for the parameter estimates are taken as the square roots of the corresponding diagonal elements.

For restrictions placed on the intercept, scale, and shape parameters, one-degree-of-freedom Lagrange multiplier test statistics are computed. These statistics are computed as

$\chi^2 = \frac{g^2}V$

where g is the derivative of the log likelihood with respect to the restricted parameter at the restricted maximum and

V = I₁₁ - I₁₂I^-1₂₂I₂₁

where the 1 subscripts refer to the restricted parameter and the 2 subscripts refer to the unrestricted parameters. The information matrix is evaluated at the restricted maximum. These statistics are asymptotically distributed as chi-squares with one degree of freedom under the null hypothesis that the restrictions are valid, provided that some regularity conditions are satisfied. See Rao (1973, p. 418) for a more complete discussion. It is possible for these statistics to be missing if the observed information matrix is not positive definite. Higher degree-of-freedom tests for multiple restrictions are not currently computed.

A Lagrange multiplier test statistic is computed to test this constraint. Notice that this test statistic is comparable to the Wald test statistic for testing that the scale is one. The Wald statistic is the result of squaring the difference of the estimate of the scale parameter from one and dividing this by the square of its estimated standard error.

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