The LIFETEST Procedure

Kernel-Smoothed Hazard Estimate

Kernel-smoothed estimators of the hazard function h left-parenthesis t right-parenthesis are based on the Nelson-Aalen estimator ModifyingAbove upper H With tilde left-parenthesis t right-parenthesis and its variance ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t right-parenthesis right-parenthesis. Consider the jumps of ModifyingAbove upper H With tilde left-parenthesis t right-parenthesis and ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t right-parenthesis right-parenthesis at the event times t 1 less-than t 2 less-than midline-horizontal-ellipsis less-than t Subscript upper D as follows:

StartLayout 1st Row 1st Column normal upper Delta ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis 2nd Column equals 3rd Column ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis minus ModifyingAbove upper H With tilde left-parenthesis t Subscript i minus 1 Baseline right-parenthesis 2nd Row 1st Column ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis right-parenthesis 2nd Column equals 3rd Column ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis right-parenthesis minus ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t Subscript i minus 1 Baseline right-parenthesis right-parenthesis EndLayout

where t 0=0.

The kernel-smoothed estimator of h left-parenthesis t right-parenthesis is a weighted average of normal upper Delta ModifyingAbove upper H With tilde left-parenthesis t right-parenthesis over event times that are within a bandwidth distance b of t. The weights are controlled by the choice of kernel function, upper K left-parenthesis right-parenthesis, defined on the interval [–1,1]. The choices are as follows:

  • uniform kernel:

    upper K Subscript upper U Baseline left-parenthesis x right-parenthesis equals one-half comma negative 1 less-than-or-equal-to x less-than-or-equal-to 1

  • Epanechnikov kernel:

    upper K Subscript upper E Baseline left-parenthesis x right-parenthesis equals three-fourths left-parenthesis 1 minus x squared right-parenthesis comma negative 1 less-than-or-equal-to x less-than-or-equal-to 1

  • biweight kernel:

    upper K Subscript upper B upper W Baseline left-parenthesis x right-parenthesis equals StartFraction 15 Over 16 EndFraction left-parenthesis 1 minus x squared right-parenthesis squared comma negative 1 less-than-or-equal-to x less-than-or-equal-to 1

The kernel-smoothed hazard rate estimator is defined for all time points on left-parenthesis 0 comma t Subscript upper D Baseline right-parenthesis. For time points t for which b less-than-or-equal-to t less-than-or-equal-to t Subscript upper D Baseline minus b, the kernel-smoothed estimated of h left-parenthesis t right-parenthesis based on the kernel upper K left-parenthesis right-parenthesis is given by

ModifyingAbove h With caret left-parenthesis t right-parenthesis equals StartFraction 1 Over b EndFraction sigma-summation Underscript i equals 1 Overscript upper D Endscripts upper K left-parenthesis StartFraction t minus t Subscript i Baseline Over b EndFraction right-parenthesis normal upper Delta ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis

The variance of ModifyingAbove h With caret left-parenthesis t right-parenthesis is estimated by

ModifyingAbove sigma squared With caret left-parenthesis ModifyingAbove h With caret left-parenthesis t right-parenthesis right-parenthesis equals StartFraction 1 Over b squared EndFraction sigma-summation Underscript i equals 1 Overscript upper D Endscripts upper K left-parenthesis StartFraction t minus t Subscript i Baseline Over b EndFraction right-parenthesis squared normal upper Delta ModifyingAbove upper V With caret left-parenthesis ModifyingAbove upper H With tilde left-parenthesis t Subscript i Baseline right-parenthesis right-parenthesis

For t < b, the symmetric kernels upper K left-parenthesis right-parenthesis are replaced by the corresponding asymmetric kernels of Gasser and Müller (1979). Let q equals StartFraction t Over b EndFraction. The modified kernels are as follows:

  • uniform kernel:

    upper K Subscript upper U comma q Baseline left-parenthesis x right-parenthesis equals StartFraction 4 left-parenthesis 1 plus q cubed right-parenthesis Over left-parenthesis 1 plus q right-parenthesis Superscript 4 Baseline EndFraction plus StartFraction 6 left-parenthesis 1 minus q right-parenthesis Over left-parenthesis 1 plus q right-parenthesis cubed EndFraction x comma negative 1 less-than-or-equal-to x less-than-or-equal-to q

  • Epanechnikov kernel:

    upper K Subscript upper E comma q Baseline left-parenthesis x right-parenthesis equals upper K Subscript upper E Baseline left-parenthesis x right-parenthesis StartFraction 64 left-parenthesis 2 minus 4 q plus 6 q squared minus 3 q cubed right-parenthesis plus 240 left-parenthesis 1 minus q right-parenthesis squared x Over left-parenthesis 1 plus q right-parenthesis Superscript 4 Baseline left-parenthesis 19 minus 18 q plus 3 q squared right-parenthesis EndFraction comma negative 1 less-than-or-equal-to x less-than-or-equal-to q

  • biweight kernel:

    upper K Subscript upper B upper W comma q Baseline left-parenthesis x right-parenthesis equals upper K Subscript upper B upper W Baseline left-parenthesis x right-parenthesis StartFraction 64 left-parenthesis 8 minus 24 q plus 48 q squared minus 45 q cubed plus 15 q Superscript 4 Baseline right-parenthesis plus 1120 left-parenthesis 1 minus q right-parenthesis cubed x Over left-parenthesis 1 plus q right-parenthesis Superscript 5 Baseline left-parenthesis 81 minus 168 q plus 126 q squared minus 40 q cubed plus 5 q Superscript 4 Baseline right-parenthesis EndFraction comma negative 1 less-than-or-equal-to x less-than-or-equal-to q

For t Subscript upper D Baseline minus b less-than-or-equal-to t less-than-or-equal-to t Subscript upper D, let q equals StartFraction t Subscript upper D Baseline minus t Over b EndFraction. The asymmetric kernels for t less-than b are used with x replaced by –x.

Using the log transform on the smoothed hazard rate, the 100(1–alpha)% pointwise confidence interval for the smoothed hazard rate h left-parenthesis t right-parenthesis is given by

ModifyingAbove h With caret left-parenthesis t right-parenthesis equals ModifyingAbove h With caret left-parenthesis t right-parenthesis exp left-bracket plus-or-minus StartFraction z Subscript 1 minus alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove h With caret left-parenthesis t right-parenthesis right-parenthesis Over ModifyingAbove h With caret left-parenthesis t right-parenthesis EndFraction right-bracket

where z Subscript 1 minus StartFraction alpha Over 2 EndFraction is the (100(1–StartFraction alpha Over 2 EndFraction))th percentile of the standard normal distribution.

Optimal Bandwidth

The following mean integrated squared error (MISE) over the range tau Subscript upper L and tau Subscript upper U is used as a measure of the global performance of the kernel function estimator:

StartLayout 1st Row 1st Column MISE left-parenthesis b right-parenthesis 2nd Column equals 3rd Column upper E integral Subscript tau Subscript upper L Baseline Superscript tau Subscript upper U Baseline left-parenthesis ModifyingAbove h With caret left-parenthesis i right-parenthesis minus h left-parenthesis u right-parenthesis right-parenthesis squared d u 2nd Row 1st Column Blank 2nd Column equals 3rd Column upper E integral Subscript tau Subscript upper L Baseline Superscript tau Subscript upper U Baseline ModifyingAbove h With caret squared left-parenthesis u right-parenthesis d u minus 2 upper E integral Subscript tau Subscript upper L Baseline Superscript tau Subscript upper U Baseline ModifyingAbove h With caret left-parenthesis u right-parenthesis h left-parenthesis u right-parenthesis d u plus upper E integral Subscript tau Subscript upper L Baseline Superscript tau Subscript upper U Baseline h squared left-parenthesis u right-parenthesis d u EndLayout

The last term is independent of the choice of the kernel and bandwidth and can be ignored when you are looking for the best value of b. The first integral can be approximated by using the trapezoid rule by evaluating ModifyingAbove h With caret left-parenthesis t right-parenthesis at a grid of points tau Subscript upper L Baseline equals u 1 less-than midline-horizontal-ellipsis less-than u Subscript upper M Baseline equals tau Subscript upper U. You can specify tau Subscript upper L Baseline comma tau Subscript upper R Baseline, and M by using the options GRIDL=, GRIDU=, and NMINGRID=, respectively, of the HAZARD plot. The second integral can be estimated by the Ramlau-Hansen (1983a, 1983b) cross-validation estimate:

StartFraction 1 Over b EndFraction sigma-summation Underscript i not-equals j Endscripts upper K left-parenthesis StartFraction t Subscript i Baseline minus t Subscript j Baseline Over b EndFraction right-parenthesis normal upper Delta ModifyingAbove upper H With caret left-parenthesis t Subscript i Baseline right-parenthesis normal upper Delta ModifyingAbove upper H With caret left-parenthesis t Subscript j Baseline right-parenthesis

Therefore, for a fixed kernel, the optimal bandwidth is the quantity b that minimizes

g left-parenthesis b right-parenthesis equals sigma-summation Underscript i equals 1 Overscript upper M minus 1 Endscripts left-bracket StartFraction u Subscript i plus 1 Baseline minus u Subscript k Baseline Over 2 EndFraction left-parenthesis ModifyingAbove h With caret squared left-parenthesis u Subscript i Baseline right-parenthesis plus ModifyingAbove h With caret squared left-parenthesis u Subscript i plus 1 Baseline right-parenthesis right-parenthesis right-bracket minus StartFraction 2 Over b EndFraction sigma-summation Underscript i not-equals j Endscripts upper K left-parenthesis StartFraction t Subscript i Baseline minus t Subscript j Baseline Over b EndFraction right-parenthesis normal upper Delta ModifyingAbove upper H With caret left-parenthesis t Subscript i Baseline right-parenthesis normal upper Delta ModifyingAbove upper H With caret left-parenthesis t Subscript j Baseline right-parenthesis

The minimization is carried out by the golden section search algorithm.

Last updated: December 09, 2022