MA549 Chapter 4 Homework
Leyi Zhang
October 4, 2014
4.6 In section 1.6 a study is described to evaluate a protocol change in disinfectant practice in a large midwestern university medical center. Control of infection is the primary concern for the 155 patients entered into the burn unit with varying degrees of burns. The outcome variable is the time until infection from admission to the unit. Censoring variables are discharge from the hospital without an infection or death without an infection. Eighty-four patients were in the group which had chlorhexidine as the disinfectant and 72 patients received the routine disinfectant povidone-iodine.
(a) Estimate the survival (infection-free) functions and their standard errors for the chlorhexidine and povidone-iodine groups, provide an appropriate plot.
The survival function can be estimated by formula (4.2.1) in the textbook.
- routine bathing care method
## Loading required package: splines## Call: survfit(formula = Surv(bath$T3, bath$D3) ~ 1)
##
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 1 70 1 0.986 0.0142 0.958 1.000
## 3 69 3 0.943 0.0277 0.890 0.999
## 4 66 4 0.886 0.0380 0.814 0.963
## 5 62 1 0.871 0.0400 0.796 0.953
## 6 60 2 0.842 0.0436 0.761 0.932
## 7 58 3 0.799 0.0481 0.710 0.899
## 8 53 1 0.784 0.0495 0.693 0.887
## 9 50 2 0.752 0.0522 0.657 0.862
## 10 47 1 0.736 0.0535 0.639 0.849
## 11 44 2 0.703 0.0561 0.601 0.822
## 13 40 1 0.685 0.0574 0.582 0.808
## 19 33 1 0.665 0.0593 0.558 0.791
## 21 30 1 0.642 0.0613 0.533 0.774
## 23 27 1 0.619 0.0635 0.506 0.756
## 32 18 1 0.584 0.0686 0.464 0.735
## 44 8 1 0.511 0.0910 0.361 0.725
## 47 6 1 0.426 0.1086 0.258 0.702
## 51 4 1 0.320 0.1231 0.150 0.680- new body-cleansing bathing care method
## Call: survfit(formula = Surv(cleaning$T3, cleaning$D3) ~ 1)
##
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 2 84 3 0.964 0.0202 0.925 1.000
## 3 81 1 0.952 0.0232 0.908 0.999
## 4 80 1 0.940 0.0258 0.891 0.992
## 5 79 5 0.881 0.0353 0.814 0.953
## 8 73 1 0.869 0.0369 0.800 0.944
## 10 70 1 0.856 0.0384 0.784 0.935
## 11 68 1 0.844 0.0398 0.769 0.926
## 14 57 1 0.829 0.0418 0.751 0.915
## 16 50 1 0.812 0.0441 0.730 0.904
## 17 47 2 0.778 0.0485 0.688 0.879
## 18 41 2 0.740 0.0531 0.643 0.852
## 42 9 1 0.658 0.0907 0.502 0.862- plot
(b) Estimate the cumulative hazard rates and their standard errors for the chlorhexidine and povidone-iodine groups. Plot these estimates. Does it appear that the two cumulative hazard rates are proportional to each other?
The cumulative hazard function H(t) can be estimated by -ln(S(t))
- routine bathing care method
## time event risk ht1 standardError1
## 1 1 1 70 0.01439 0.01429
## 2 3 3 69 0.05884 0.02888
## 3 4 4 66 0.12136 0.04186
## 4 5 1 62 0.13762 0.04486
## 5 6 2 60 0.17152 0.05068
## 6 7 3 58 0.22463 0.05882
## 7 8 1 53 0.24368 0.06177
## 8 9 2 50 0.28450 0.06794
## 9 10 1 47 0.30601 0.07119
## 10 11 2 44 0.35253 0.07811
## 11 13 1 40 0.37785 0.08202
## 12 19 1 33 0.40862 0.08744
## 13 21 1 30 0.44252 0.09357
## 14 23 1 27 0.48026 0.10064
## 15 32 1 18 0.53742 0.11495
## 16 44 1 8 0.67095 0.16982
## 17 47 1 6 0.85327 0.23794
## 18 51 1 4 1.14095 0.34513- new body-cleansing bathing care method
## time event risk ht2 standardError2
## 1 2 3 84 0.03637 0.02062
## 2 3 1 81 0.04879 0.02403
## 3 4 1 80 0.06137 0.02709
## 4 5 5 79 0.12675 0.03918
## 5 8 1 73 0.14055 0.04150
## 6 10 1 70 0.15493 0.04389
## 7 11 1 68 0.16975 0.04629
## 8 14 1 57 0.18745 0.04951
## 9 16 1 50 0.20765 0.05339
## 10 17 2 47 0.25114 0.06129
## 11 18 2 41 0.30115 0.07033
## 12 42 1 9 0.41893 0.13150- plot
It seems two cumulative hazard rates are NOT proportional to each other.
(c) Provide estimates of the median time to infection and find 95% confidence intervals for the median time to infection for both the chlorhexidine and povidone-iodine groups using the linear, log-transformed, and arcsine formulas.
- routine bathing care method
## time S Linear Log Arcsin
## 1 1 0.9857 34.2455 3.8748 11.1374
## 2 3 0.9429 15.9628 4.9321 9.1031
## 3 4 0.8857 10.1431 4.9254 7.3717
## 4 5 0.8714 9.2840 4.8464 7.0056
## 5 6 0.8424 7.8476 4.6249 6.2996
## 6 7 0.7988 6.2148 4.2052 5.3406
## 7 8 0.7837 5.7345 4.0350 5.0209
## 8 9 0.7524 4.8322 3.6495 4.3726
## 9 10 0.7364 4.4170 3.4428 4.0541
## 10 11 0.7029 3.6186 2.9878 3.4055
## 11 13 0.6853 3.2311 2.7392 3.0743
## 12 19 0.6646 2.7770 2.4216 2.6721
## 13 21 0.6424 2.3238 2.0816 2.2588
## 14 23 0.6186 1.8690 1.7176 1.8331
## 15 32 0.5843 1.2279 1.1644 1.2161
## 16 44 0.5112 0.1234 0.1227 0.1234
## 17 47 0.4260 -0.6812 -0.6957 -0.6762
## 18 51 0.3195 -1.4668 -1.4765 -1.3995We can find the 95% confidence interval for the median by selecting value between [-1.96, 1.96] from above frame, so the CI for both three methods are x >23. The estimated median is x = 47, because surv(x=47)=0.426<0.5.
- new body-cleansing bathing care method
## time S Linear Log Arcsin
## 1 2 0.9643 22.930 5.105 10.912
## 2 3 0.9524 19.469 5.307 10.364
## 3 4 0.9405 17.063 5.420 9.879
## 4 5 0.8810 10.781 5.369 7.939
## 5 8 0.8689 10.009 5.288 7.599
## 6 10 0.8565 9.293 5.183 7.254
## 7 11 0.8439 8.638 5.062 6.915
## 8 14 0.8291 7.877 4.865 6.473
## 9 16 0.8125 7.085 4.611 5.974
## 10 17 0.7779 5.726 4.087 5.047
## 11 18 0.7400 4.522 3.501 4.138
## 12 42 0.6578 1.738 1.529 1.678We can find the 95% confidence interval for the median by selecting value between [-1.96, 1.96] from above frame, so the CI for both three methods are x > 42. The estimated median is larger than 47, we cannot find the exact value.
(d) Find 95% confidence intervals for the survival (infection-free) functions at 10 days postadmission for both the chlorhexidine and povidone-iodine groups using the log transformed and arcsine transformed formulas.
- routine bathing care method The 95% confidence intervals for the survival (infection-free) functions using the log transformed is :
## [1] 0.6142## [1] 0.8252The 95% confidence intervals for the survival (infection-free) functions using the arcsine transformed is :
## [1] 0.6258## [1] 0.8336- new body-cleansing bathing care method
The 95% confidence intervals for the survival (infection-free) functions using the log transformed is :
## [1] 0.7611## [1] 0.9158The 95% confidence intervals for the survival (infection-free) functions using the arcsine transformed is :
## [1] 0.7737## [1] 0.9229(e) Find 95% confidence bands for the infection-free functions over the range 8–20 days postinfection for both the chlorhexidine and povidone-iodine groups using the arcsin-square root transform only.
- routine bathing care method
## time lower upper
## 7 8 0.6100 0.9166
## 8 9 0.5722 0.8958
## 9 10 0.5533 0.8848
## 10 11 0.5141 0.8613
## 11 13 0.4938 0.8488
## 12 19 0.4684 0.8349
- new body-cleansing bathing care method
## time lower upper
## 5 8 0.7409 0.9577
## 6 10 0.7247 0.9503
## 7 11 0.7084 0.9426
## 8 14 0.6883 0.9339
## 9 16 0.6652 0.9243
## 10 17 0.6187 0.9035
## 11 18 0.5689 0.8799
(f) Find 95% HW confidence bands for the infection-free functions over the range 8–20 days postinfection for both the chlorhexidine and povidone-iodine, use the log transform only.
- routine bathing care method
## time lower upper
## 7 8 0.6034 0.9204
## 8 9 0.5751 0.8940
## 9 10 0.5603 0.8802
## 10 11 0.5285 0.8512
## 11 13 0.5114 0.8359
## 12 19 0.4899 0.8187
- new body-cleansing bathing care method
## time lower upper
## 5 8 0.6949 0.9759
## 6 10 0.6855 0.9673
## 7 11 0.6755 0.9580
## 8 14 0.6626 0.9468
## 9 16 0.6472 0.9340
## 10 17 0.6138 0.9065
## 11 18 0.5750 0.8758
(g) Based on the results above, how does the infection experience of the chlorhexidine and povidone-iodine groups compare? Supply an infomative plot or plots, based on the results in (f), to help answer the question.
We can combine the graphs in (f) into one picture. 
Although we can see the survival line of new bathing method is slightly better than that of routine bathing method, we cannot decide which one is better because two confidence intervals are highly overlapped.