The RELIABILITY Procedure

Regression Model Statistics Computed for Each Observation for Recurrent Events Data

Subsections:

This section describes statistics that are computed for each observation when you fit a model for recurrent events data. For regression models that are fit using the MODEL statement, you can specify a variety of statistics to be computed for each observation in the input data set. This section describes the method of computation for each statistic. See Table 16.32 and Table 16.34 for the syntax to request these statistics.

Let $t_ i$ be the event time in the ith observation in the input data set. The following statistics use the definitions of the mean function $\mr{M}(t; \eta , \beta )$ and intensity function $\lambda (t; \eta , \beta )$ in Table 16.72, where $\eta $ and $\beta $ are replaced by their maximum likelihood estimates. The shape parameter $\beta $ is assumed to be constant for all observations. For regression models, the scale parameter $\eta $ in Table 16.72 for the ith observation is

\[ \eta _{i} = \beta _{0} + \beta _{1}x_{i1} + \ldots \]

where $x_{i1}, x_{i2}, \ldots $ are regression coefficients and $\beta _0, \beta _1, \ldots $ are the maximum likelihood estimates of the regression parameters.

Predicted Values of Scale Parameter

The scale parameter that is predicted by the model for the ith observation is

\[ \hat{\eta }_{i}=\mb{x}_{i}^\prime \hat{\bbeta } \]

where $\mb{x}_{i}$ is the vector of explanatory variables for the ith observation and $\bbeta $ is the vector of maximum likelihood estimates of the regression parameters.

Mean Function

The predicted mean function is computed as $\mr{M}(t_ i, \hat{\eta }_ i, \hat{\beta }) $.

Confidence Limits for the Mean Function

Confidence limits for the estimated $\mr{M}(t_ i)$ are computed as described in the section Table 16.74, using $t_ i, \hat{\eta }_ i$, and $\hat{\beta }$.

Standard Error of the Mean Function

The standard error of the estimated $\mr{M}(t_ i)$ is computed as described in the section Table 16.74, using $t_ i, \hat{\eta }_ i$, and $\hat{\beta }$.

Intensity Function

The predicted intensity function is computed as $\lambda (t_ i, \hat{\eta }_ i, \hat{\beta })$.

Confidence Limits for the Intensity Function

Confidence limits for the estimated $\lambda (t_ i)$ are computed as described in the section Table 16.74, using $t_ i, \hat{\eta }_ i$, and $\hat{\beta }$.

Standard Error of the Intensity Function

The standard error of the estimated $\lambda (t_ i)$ is computed as described in the section Table 16.74, using $t_ i, \hat{\eta }_ i$, and $\hat{\beta }$.