E. Appendix E

See Table 5.

� log ð Þ U <

When K<sup>2</sup> ≤ t < K3, we calculate the cumulative hazard function HiðÞ¼ t

K ð1

8 ><

>:

where U is a value of u � Uni½ � 0; 1 . The condition K<sup>2</sup> ≤ t < K<sup>3</sup> is equal to

K ð1

0

hið Þs ds. Survival time t is the solution of the equation

K ð2

K<sup>1</sup>

hið Þs ds þ

When K<sup>3</sup> ≤ t, the cumulative hazard function has the form HiðÞ¼ t

0

hið Þs ds þ

hið Þs ds þ

K ð2

K<sup>1</sup>

K ð2

K<sup>1</sup>

hið Þs ds þ

In particular, Ruppert et al. [9] introduced a default choices for knot location and number of knots. The idea is to choose sufficient knots to resolve the essential structure in the underlying regression function. But for more complicated penalized spline models, there are computational advantages to keeping the number of knots relatively low. A reasonable default is to choose the knots to ensure that there are a fixed number of unique observations, say 4–5, between each knot. For large data sets, this can lead to an excessive numbers of knots;

therefore, a maximum number of allowable knots (say, 20–40 total) are recommended.

K ð3

K<sup>2</sup>

hið Þs ds. The survival time t is the solution of the equation

2 6 4

� log ð Þ U <

K ð1

8 ><

2 6 4

>:

0

U ¼ exp �

Ðt K<sup>2</sup>

126 Topics in Splines and Applications

K Ð 3 K<sup>2</sup>

hið Þ<sup>s</sup> ds <sup>þ</sup> <sup>Ð</sup><sup>t</sup>

D. Appendix D

K<sup>3</sup>

U ¼ exp �

K ð1

0

hið Þs ds þ

K ð2

K<sup>1</sup>

hið Þs ds þ

hið Þs ds þ

hið Þs ds:

ðt

hið Þs ds

hið Þs ds:

ðt

hið Þs ds

K<sup>3</sup>

9 >=

3 7 5,

> K ð1

> > 0

9 >=

3 7 5:

>;

hið Þs ds þ

K ð2

K<sup>1</sup>

hið Þs dsþ

>;

K<sup>2</sup>

K ð3

K<sup>2</sup>

hið Þs ds þ

K ð1

0

hið Þs ds þ

K ð2

K<sup>1</sup>

hið Þs dsþ


Table 5. Summary statistics for parameter estimation of the simulated data of the model in (22) for different censoring rates.
