Math 434 Homework 5 - Fall 2005

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    Statistical methods for survival data analysis., 3rd edition, by Lee and Wang

    HOMEWORK #5 due 12-6

    NOTES:   (THIS IS ALSO on the Math434 Web site.)

    1. ORGANIZE YOUR HOMEWORK in the following manner:
    (i) Your answers to all problems.
    (ii) Your SAS programs for any problems that require SAS
    (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.
    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).

  • Problem 1. -- A group of 50 individuals who were judged to have Property X were recruited for a study. The sex, family history of Property X, and another quantity called Wval was recorded for each individual. Since these individuals were thought to be too diverse to be able to estimate the effects of the three covariates on Property X in comparison with the general population, a similar group of 50 nonaffected individuals were chosen to match the first group by age, income, family status, and geographical location. (Thus, the entire sample of 100 individuals is not independent, since pairs of individuals, one with and one without Property X, were chosen to match on four other properties. However, the pairs can be assumed to be independent.) The data is given in Table 1 below. Hint added 12-04-05: See the file LGCaseCtrl.sas on the Math434 Web site, which does a similar paired-sample logistic regression. 
                     Table 1 - Data for 50 Matched Pairs
       Each numbered pair has Wval Sex Hist values for two matched individuals
        Property X   Not Property X          Property X   Not Property X
   1.    81  1  1      69  0  0         26.    84  1  1      78  0  1        
   2.    78  1  0      79  1  0         27.    88  1  1      85  0  0        
   3.    77  1  1      80  0  0         28.    79  1  1      78  0  0        
   4.    71  1  0      72  0  0         29.    88  0  1      74  1  0        
   5.    76  0  0      79  0  0         30.    72  1  1      80  1  0        
   6.    75  1  0      78  1  1         31.    79  0  1      77  0  1        
   7.    74  0  1      77  0  0         32.    73  1  1      74  0  1        
   8.    77  1  1      86  1  0         33.    76  1  1      77  0  1        
   9.    73  1  1      80  0  0         34.    78  0  0      72  1  0        
  10.    79  1  1      86  1  1         35.    76  1  0      73  0  1        
  11.    84  1  1      81  0  1         36.    81  1  1      78  0  0        
  12.    77  0  1      88  1  0         37.    79  1  0      94  1  0        
  13.    71  1  0      93  0  1         38.    67  0  1      82  0  0        
  14.    77  0  1      83  1  1         39.    83  1  1      90  0  0        
  15.    76  0  1      81  1  1         40.    71  1  1      76  0  0        
  16.    77  1  0      90  1  0         41.    78  0  0      90  0  0        
  17.    81  1  1      75  1  0         42.    83  0  1      83  1  0        
  18.    75  0  0      76  1  1         43.    82  1  0      80  0  0        
  19.    85  1  1      77  0  1         44.    83  0  1      87  1  1        
  20.    82  1  0      81  0  1         45.    67  0  0      84  1  0        
  21.    79  0  1      77  1  0         46.    79  0  0      76  1  0        
  22.    82  1  0      77  0  1         47.    79  1  1      81  0  0        
  23.    81  0  1      76  1  1         48.    72  1  0      84  0  0        
  24.    71  0  1      77  0  0         49.    78  1  1      74  0  1        
  25.    80  0  1      86  1  1         50.    77  0  1      75  1  0

    (i) Using the appropriate model to analyze the data, are the three covariates Wval, Sex, and Hist together significantly associated with Property X? What test procedure did you use?

    (ii) Individually, which of the three covariates is significantly associated with Property X? What is the P-value? For the covariates that are significant, is an increased value associated with a higher likelihood of Property X, or a decreased likelihood?

    (iii) If Sex or Hist is signicant in part (ii), by how much does Sex=1 or Hist=1 increase (or decrease) the odds ratio of having Property X for that individual in a matched pair with exactly one affected individual? If Wval is significant in part (ii), by how much does an increase of Wval by 5 units increase (or decrease) the same odds ratio?

  • Problem 2. Analyze the data in Table 1 in HomeWork 4 on the Math434 Web site using a Cox regression instead of an AFT Weibull or Exponential regression.
    (i) Using a Cox regression, do the failure times depend significantly on the three covariates together? What is the P-value? Which test did you use?
    (ii) Individually, do the survival times depend on the group? On DVAL? On FVAL? Find the P-values for the variables that are significant.
    (iii) If Group is significant, which group (Green or Blue) has the longer expected survival time? How can you tell from the Cox-regression output? If DVAL or FVAL is significant, do larger values of that variable lead to longer survival times or shorter survival times? How can you tell from the Cox-regression output?
    (iv) How do your results compare with the results that you obtained on Problem 1 on HomeWork 4? Are the Cox regression results more significant or less significant than for the Weibull AFT regression? for the Exponential AFT regression?
    (Hint: See ph3vars.sas or ph2samp.sas on the Math 434 Web site.)

  • Problem 3. Do any of the variables in the previous problem have effects that show a significant trend in time? Test each of the variables (Color, Dval, and Fval) in Problem 2 for time-dependent effects.
    (Hints: (a) Don't include time-dependent variables for all three variables in the model at the same time, since that may cause no variables to be significant. Test the covariates one at a time, with the other two covariates either in the model or not, as you choose.
    (b) See ph3vars.sas or ph2samp.sas on the Math 434 Web site.)

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