Predicted Values

The LIFEREG Procedure

Predicted Values

For a given set of covariates, x (including the intercept term), the pth quantile of the log response, y_p, is given by

$y_p = {x^' \beta} + \sigma w_p$

where w_p is the pth quantile of the baseline distribution. The estimated quantile is computed by replacing the unknown parameters with their estimates, including any shape parameters on which the baseline distribution might depend. The estimated quantile of the original response is obtained by taking the exponential of the estimated log quantile unless the NOLOG option is specified in the preceding MODEL statement.

The standard errors of the quantile estimates are computed using the estimated covariance matrix of the parameter estimates and a Taylor series expansion of the quantile estimate. The standard error is computed as

${STD} = \sqrt{{z^'Vz}}$

where V is the estimated covariance matrix of the parameter vector $(\beta^',\sigma,\delta)^'$ , and z is the vector

$z = [ x \ \hat{w}_p \ \hat{\sigma} \frac{\partial w_p}{\partial \delta} \ ]$

where $\delta$ is the vector of the shape parameters. Unless the NOLOG option is specified, this standard error estimate is converted into a standard error estimate for exp(y_p) as $\exp(\hat{y}_p)$ STD. It may be more desirable to compute confidence limits for the log response and convert them back to the original response variable than to use the standard error estimates for exp(y_p) directly. See Example 36.1 for a 90% confidence interval of the response constructed by exponentiating a confidence interval for the log response.

The variable, CDF, is computed as

CDF_i = F(w_i)

where the residual

$w_i=( \frac{ y_i-{x^'}_i{b} } { \hat{\sigma} } )$

and F is the baseline cumulative distribution function.

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