Homework #2
Name:
1.
_
Date:
_
The cumulative hazard function, H(t), is found through the following equation:
t
H (t )   h(u )du
0
It is not necessary to understand calculus to understand this equation. Basically, it
is the area under the curve of the function, h(u), on the interval from 0 to t. A
given cumulative hazard will remove a certain proportion of objects (or be
associated with a probability of surviving beyond t), see Eq. 2.6. For example, a
cumulative hazard of 0 (i.e., H(t)=0) has 100% associated survival (i.e. S(t)=1).
(A) Find the associated survival when the cumulative hazard is 0.5, 1, 2, and 10.
(B) How much cumulative hazard is required so that S(t)=0.90, 0.75, 0.50, 0.25, 0.10.
(C) The hazard rate of an exponential survival distribution is always a constant. Show
that h(t)= when f(t)=*exp(-*t) and S(t)= exp(-*t) using Eq. 2.4. Use
geometry to explain why H(t)= *t.
(D) Intuitively, we feel that changing the units of time should not have a fundamental
impact on the survival analysis. The proportion of individuals who die after one
week or 7 days is the same because the time period is the same. Likewise, the
cumulative hazard should be the same. Using this intuition, find the daily hazard
rate, hD(t), for an individual with exponential survival whose cumulative weekly
hazard after 2 weeks is 1, i.e. HW(2)=1. (Hint: HW(2)=HD(14) and HD(t)= D*t)
2.
Use SAS to plot the hazard and survival function whose distribution is
characterized as follows:
h(t | Intensity, unit) = Intensity * ( t / unit – 5 )2
3

 t
 
 5 
125  
 unit
 

where unit is 1 and Intensity is equal to 0.01, 0.05, and 0.20.
Intensity  unit
H (t ) 
3
Hint: Use the following SAS program and modify as needed. It may be easier to see
the survival curve if the “overlay” option is removed for some cases.
data twoA;
insy=0.01;
do t=0 to 15 by 0.1;
h=insy*(t-5)**2;
ch=insy/3*(125+(t-5)**3);
s=2.7182818**(-ch);
output;
Homework #2
end;
run;
proc gplot data=twoA;
symbol1 v=none i=join c=black;
symbol2 v=none i=join c=black;
plot (h s)*t /overlay;
run;
(B) Describe the relationship you see between the hazard function and survival for
each case. Describe any striking features in any of these graphs.
(C) What is the ratio of h(t | 0.20, 1) and h(t | 0.05, 1) to h(t | 0.01, 1)? What is the
ratio of h(t | 0.20, 1) to h(t | 0.05, 1)?
(D) Use SAS to plot the hazard and survival functions when (a) Intensity is 0.01 and
unit is 1, and (b) Intensity is 0.005 and unit is 2. Extra Credit: What is the
relationship between the two graphs for the hazard and survival functions? (Hint:
Compare the values of h(t) and S(t) when t=0, t=5 and 10, and t=15 and 30.)