Math 434 Homework 2 - Fall 2005

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  • Text references are to
    Statistical methods for survival data analysis., by Lee and Wang, 3rd edition

    HOMEWORK #2 due 10-11

    NOTES:   (This is also on the Math434 Web site.)

    1. ORGANIZE YOUR HOMEWORK in the following three parts:
    (i) Your answers to all problems.
    (ii) Your SAS programs for any problems that require SAS, here Problems #4-5, and
    (iii) the SAS output that you got.
    For problems involving SAS, add page numbers to your homework so that you can make references from part (i) to part (iii). (For example, in part (i), you can say things like, ``The answer to part (a) is 7. This answer is highly significant (P=0.007). The plot for part (b) is on page #Y in the SAS output.'')
    Include your name in title statements in your SAS programs so that your name will appear at the top of each SAS output page.
    (The reason for the (i,ii,iii) order is to have your conclusions first, then any SAS programs, then the SAS output on which you based your SAS conclusions. This will make the organization of your homework much clearer for problems that involve SAS, particularly when the problems become more complicated.)
    2. If a problem asks you to do a statistical test, EXPLAIN CLEARLY what the null hypothesis H_0 is, what the alternative H_1 is, what test you used, what the P-value is, and whether the data is significant, highly significant, or neither. Include this as part of your answer in part (i).
    3. Make sure that you keep a copy of your homework!   It may not be handed back in time if you need it for the next homework or test.

  • Problem 1.   (See Problems #2.4 and #2.5, p17)
    (i) Given the hazard function   h(t)=c   for t>=0, derive the probability density function of survival times and the survival function.
    (ii) Given the survival function for survival times T
        S(t) = Pr(T > t) = exp(-ta)   for t >= 0,  a>0
    derive the probability density function of T and the hazard function.

  • Problem 2. -- (Text p104) Problem #4-11.
    (This problem asks about direct standardization of death rates in Oklahoma and Montana.)

  • Problem 3. -- (Text p104) Problem #4-12.
    (This problem asks about indirect standardization of death rates in Japan and Chile.)

  • Problem 4.   (i) Use SAS to plot the two survival functions for the remission times for the two groups of patients in Table 1 below. This is the same data set that you analyzed by hand in Problems 4 and 6 in HW#1. Make sure that the two survival plots appear on the same graph.
    (Hint: Your SAS program may be similar to ltcmf.sas or lthepat.sas on the Math434 Web site. Either of the two ways of entering the data in lthepat.sas may be easier than that in the first data step in ltcmf.sas.)
    (ii) Compare the results of the Wilcoxon test in your SAS output with your results in HW#1. (Did you save a copy of HW#1?) Are they similar? (SAS uses the counting-process variance for the Wilcoxon score instead of the permutation variance, but the scores themselves should be exactly the same.) 

         Table 1 - Survival distributions for two samples
     (Example 3.3 from p29 of text)
     6-MP (n1=21)    6,6,6,6+, 7, 9+, 10,10+, 11+, 13, 16, 17+,
          19+,20+, 22, 23, 25+, 32+,32+, 34+, 35+
     Placebo (n2=21) 1,1, 2,2, 3, 4,4, 5,5, 8,8,8,8, 11,11,
          12,12, 15, 17, 22, 23

  • Problem 5.   A clinical study was made of 10,000 individuals over 12 years who were diagnosed with a rare disease. The number of observed deaths and the number of individuals who dropped out of the study were noted at the end of each year. The observations were: 
      Years    Begin   Deaths   Dropped   Yr.End 
  0-1      10000      134    953       8913 
  1-2       8913      366    221       8326 
  2-3       8326      574    148       7604 
  3-4       7604      720     74       6810 
  4-5       6810     1334     74       5402 
  5-6       5402      770     48       4584 
  6-7       4584      775     33       3776 
  7-8       3776      788     25       2963 
  8-9       2963      587     20       2356 
  9-10      2356      876     14       1466 
 10-11      1466      418     12       1036 
 11-12      1036      332      3        701
 12-13       701        0    701          0

    (i) Use proc lifetest in SAS to plot the survival and hazard functions for the 10,000 individuals over this time span and make sure that the plots are in your output. Use the actuarial method to estimate the hazard rate for each year. (That is, use the appropriate options in proc lifetest.  See page 13 in the text. Note also ltangina.sas on the Math434 Web site.)

    (ii) What are the estimated hazard rates during the twelve years? (Or else say where they can be found in your output.) Does the hazard rate appear high in any of the years?

    NOTE:   SAS's proc lifetest only allows you to enter numbers of deaths and censored individuals for each year. There is no way to enter the initial sample size, here 10,000. Thus if you do not include the 701 12-th-year survivors somewhere in the input, then SAS will assume that your starting sample was 10,000-701=9299 instead of 10,000. That is one of the reasons for the final line in the table above.

  • Problem 6. -- Use SAS to do Problem #5-12 on page 133 of the text:
    (a) Do survival plots of the four treatments appear separated?
    (b) What are the P-values for the hypothesis H_0 that the treatments are equally effective, for SAS's version of both the log rank and Wilcoxon tests?

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