Re: [R] sample(c(0, 1)...) vs. rbinom
the "something similar" is return a different result in two
situations where one might expect the same result, ie when
a probability vector with equal probabilities is supplied versus
the default of equal probabilities.
And, assuming that by "concerns me" you mean "worries me",
I have no clue why you think it does! It is a curiosity.
albyn
On Thu, May 23, 2013 at 04:38:18PM +, Nordlund, Dan (DSHS/RDA) wrote:
> > -Original Message-
> > From: r-help-boun...@r-project.org [mailto:r-help-bounces@r-
> > project.org] On Behalf Of Albyn Jones
> > Sent: Thursday, May 23, 2013 8:30 AM
> > To: r-help@r-project.org
> > Subject: Re: [R] sample(c(0, 1)...) vs. rbinom
> >
> > After a bit of playing around, I discovered that
> > sample() does something similar in other situations:
> >
> > > set.seed(105021)
> > > sample(1:5,1,prob=c(1,1,1,1,1))
> > [1] 3
> > > set.seed(105021)
> > > sample(1:5,1)
> > [1] 2
> >
> >
> > > set.seed(105021)
> > > sample(1:5,5,prob=c(1,1,1,1,1))
> > [1] 3 4 2 1 5
> > > set.seed(105021)
> > > sample(1:5,5)
> > [1] 2 5 1 4 3
> >
> > albyn
>
>
> What is the "something similar" you are referring to? And I guess I still
> don't understand what it is that concerns you about the sample function.
>
>
> Dan
>
> Daniel J. Nordlund
> Washington State Department of Social and Health Services
> Planning, Performance, and Accountability
> Research and Data Analysis Division
> Olympia, WA 98504-5204
>
>
>
> >
> >
> > On 2013-05-22 22:24, peter dalgaard wrote:
> > > On May 23, 2013, at 07:01 , Jeff Newmiller wrote:
> > >
> > >> You seem to be building an elaborate structure for testing the
> > >> reproducibility of the random number generator. I suspect that
> > rbinom
> > >> is calling the random number generator a different number of times
> > >> when you pass prob=0.5 than otherwise.
> > >
> > > Nope. It's switching 0 and 1:
> > >
> > >> set.seed(1); sample(0:1,10,replace=TRUE,prob=c(1-pp,pp));
> > >> set.seed(1); rbinom(10,1,pp)
> > > [1] 1 1 0 0 1 0 0 0 0 1
> > > [1] 0 0 1 1 0 1 1 1 1 0
> > >
> > > which is curious, but of course has no implication for the
> > > distributional properties. Curiouser, if you drop the prob= in
> > > sample.
> > >
> > >> set.seed(1); sample(0:1,10,replace=TRUE); set.seed(1);
> > >> rbinom(10,1,pp)
> > > [1] 0 0 1 1 0 1 1 1 1 0
> > > [1] 0 0 1 1 0 1 1 1 1 0
> > >
> > > However, it was never a design goal that two different random
> > > functions (or even two code paths within the same function) should
> > > give exactly the same values, even if they simulate the same
> > > distribution, so this is nothing more than a curiosity.
> > >
> > >
> > >>>
> > >>> Appendix A: some R code that exhibits the problem
> > >>> =
> > >>>
> > >>> ppp <- seq(0, 1, by = 0.01)
> > >>>
> > >>> result <- do.call(rbind, lapply(ppp, function(p) {
> > >>> set.seed(1)
> > >>> sampleRes <- sample(c(0, 1), size = 1, replace = TRUE,
> > >>> prob=c(1-p, p))
> > >>>
> > >>> set.seed(1)
> > >>> rbinomRes <- rbinom(1, size = 1, prob = p)
> > >>>
> > >>> data.frame(prob = p, equivalent = all(sampleRes == rbinomRes))
> > >>>
> > >>> }))
> > >>>
> > >>> result
> > >>>
> > >>>
> > >>> Appendix B: the output from the R code
> > >>> ==
> > >>>
> > >>> prob equivalent
> > >>> 1 0.00 TRUE
> > >>> 2 0.01 TRUE
> > >>> 3 0.02 TRUE
> > >>> 4 0.03 TRUE
> > >>> 5 0.04 TRUE
> > >>> 6 0.05 TRUE
> > >>> 7 0.06 TRUE
> > >>> 8 0.07 TRUE
> > >>> 9 0.08 TRUE
> > >>> 10 0.09 TRUE
> > >>> 11 0.10 TRUE
> > >>> 12 0.11 TRUE
> > >>> 13 0.12 TRUE
> > >>> 14 0.13 TRUE
> > >>> 15 0.1
Re: [R] sample(c(0, 1)...) vs. rbinom
> -Original Message-
> From: r-help-boun...@r-project.org [mailto:r-help-bounces@r-
> project.org] On Behalf Of Albyn Jones
> Sent: Thursday, May 23, 2013 8:30 AM
> To: r-help@r-project.org
> Subject: Re: [R] sample(c(0, 1)...) vs. rbinom
>
> After a bit of playing around, I discovered that
> sample() does something similar in other situations:
>
> > set.seed(105021)
> > sample(1:5,1,prob=c(1,1,1,1,1))
> [1] 3
> > set.seed(105021)
> > sample(1:5,1)
> [1] 2
>
>
> > set.seed(105021)
> > sample(1:5,5,prob=c(1,1,1,1,1))
> [1] 3 4 2 1 5
> > set.seed(105021)
> > sample(1:5,5)
> [1] 2 5 1 4 3
>
> albyn
What is the "something similar" you are referring to? And I guess I still
don't understand what it is that concerns you about the sample function.
Dan
Daniel J. Nordlund
Washington State Department of Social and Health Services
Planning, Performance, and Accountability
Research and Data Analysis Division
Olympia, WA 98504-5204
>
>
> On 2013-05-22 22:24, peter dalgaard wrote:
> > On May 23, 2013, at 07:01 , Jeff Newmiller wrote:
> >
> >> You seem to be building an elaborate structure for testing the
> >> reproducibility of the random number generator. I suspect that
> rbinom
> >> is calling the random number generator a different number of times
> >> when you pass prob=0.5 than otherwise.
> >
> > Nope. It's switching 0 and 1:
> >
> >> set.seed(1); sample(0:1,10,replace=TRUE,prob=c(1-pp,pp));
> >> set.seed(1); rbinom(10,1,pp)
> > [1] 1 1 0 0 1 0 0 0 0 1
> > [1] 0 0 1 1 0 1 1 1 1 0
> >
> > which is curious, but of course has no implication for the
> > distributional properties. Curiouser, if you drop the prob= in
> > sample.
> >
> >> set.seed(1); sample(0:1,10,replace=TRUE); set.seed(1);
> >> rbinom(10,1,pp)
> > [1] 0 0 1 1 0 1 1 1 1 0
> > [1] 0 0 1 1 0 1 1 1 1 0
> >
> > However, it was never a design goal that two different random
> > functions (or even two code paths within the same function) should
> > give exactly the same values, even if they simulate the same
> > distribution, so this is nothing more than a curiosity.
> >
> >
> >>>
> >>> Appendix A: some R code that exhibits the problem
> >>> =
> >>>
> >>> ppp <- seq(0, 1, by = 0.01)
> >>>
> >>> result <- do.call(rbind, lapply(ppp, function(p) {
> >>> set.seed(1)
> >>> sampleRes <- sample(c(0, 1), size = 1, replace = TRUE,
> >>> prob=c(1-p, p))
> >>>
> >>> set.seed(1)
> >>> rbinomRes <- rbinom(1, size = 1, prob = p)
> >>>
> >>> data.frame(prob = p, equivalent = all(sampleRes == rbinomRes))
> >>>
> >>> }))
> >>>
> >>> result
> >>>
> >>>
> >>> Appendix B: the output from the R code
> >>> ==
> >>>
> >>> prob equivalent
> >>> 1 0.00 TRUE
> >>> 2 0.01 TRUE
> >>> 3 0.02 TRUE
> >>> 4 0.03 TRUE
> >>> 5 0.04 TRUE
> >>> 6 0.05 TRUE
> >>> 7 0.06 TRUE
> >>> 8 0.07 TRUE
> >>> 9 0.08 TRUE
> >>> 10 0.09 TRUE
> >>> 11 0.10 TRUE
> >>> 12 0.11 TRUE
> >>> 13 0.12 TRUE
> >>> 14 0.13 TRUE
> >>> 15 0.14 TRUE
> >>> 16 0.15 TRUE
> >>> 17 0.16 TRUE
> >>> 18 0.17 TRUE
> >>> 19 0.18 TRUE
> >>> 20 0.19 TRUE
> >>> 21 0.20 TRUE
> >>> 22 0.21 TRUE
> >>> 23 0.22 TRUE
> >>> 24 0.23 TRUE
> >>> 25 0.24 TRUE
> >>> 26 0.25 TRUE
> >>> 27 0.26 TRUE
> >>> 28 0.27 TRUE
> >>> 29 0.28 TRUE
> >>> 30 0.29 TRUE
> >>> 31 0.30 TRUE
> >>> 32 0.31 TRUE
> >>> 33 0.32 TRUE
> >>> 34 0.33 TRUE
> >>> 35 0.34 TRUE
> >>> 36 0.35 TRUE
> >>> 37 0.36 TRUE
> >>> 38 0.37 TRUE
> >>> 39 0.38 TRUE
> >>> 40 0.39 TRUE
> >>> 41 0.40 TRUE
> >>> 42
Re: [R] sample(c(0, 1)...) vs. rbinom
After a bit of playing around, I discovered that
sample() does something similar in other situations:
set.seed(105021)
sample(1:5,1,prob=c(1,1,1,1,1))
[1] 3
set.seed(105021)
sample(1:5,1)
[1] 2
set.seed(105021)
sample(1:5,5,prob=c(1,1,1,1,1))
[1] 3 4 2 1 5
set.seed(105021)
sample(1:5,5)
[1] 2 5 1 4 3
albyn
On 2013-05-22 22:24, peter dalgaard wrote:
On May 23, 2013, at 07:01 , Jeff Newmiller wrote:
You seem to be building an elaborate structure for testing the
reproducibility of the random number generator. I suspect that rbinom
is calling the random number generator a different number of times
when you pass prob=0.5 than otherwise.
Nope. It's switching 0 and 1:
set.seed(1); sample(0:1,10,replace=TRUE,prob=c(1-pp,pp));
set.seed(1); rbinom(10,1,pp)
[1] 1 1 0 0 1 0 0 0 0 1
[1] 0 0 1 1 0 1 1 1 1 0
which is curious, but of course has no implication for the
distributional properties. Curiouser, if you drop the prob= in
sample.
set.seed(1); sample(0:1,10,replace=TRUE); set.seed(1);
rbinom(10,1,pp)
[1] 0 0 1 1 0 1 1 1 1 0
[1] 0 0 1 1 0 1 1 1 1 0
However, it was never a design goal that two different random
functions (or even two code paths within the same function) should
give exactly the same values, even if they simulate the same
distribution, so this is nothing more than a curiosity.
Appendix A: some R code that exhibits the problem
=
ppp <- seq(0, 1, by = 0.01)
result <- do.call(rbind, lapply(ppp, function(p) {
set.seed(1)
sampleRes <- sample(c(0, 1), size = 1, replace = TRUE,
prob=c(1-p, p))
set.seed(1)
rbinomRes <- rbinom(1, size = 1, prob = p)
data.frame(prob = p, equivalent = all(sampleRes == rbinomRes))
}))
result
Appendix B: the output from the R code
==
prob equivalent
1 0.00 TRUE
2 0.01 TRUE
3 0.02 TRUE
4 0.03 TRUE
5 0.04 TRUE
6 0.05 TRUE
7 0.06 TRUE
8 0.07 TRUE
9 0.08 TRUE
10 0.09 TRUE
11 0.10 TRUE
12 0.11 TRUE
13 0.12 TRUE
14 0.13 TRUE
15 0.14 TRUE
16 0.15 TRUE
17 0.16 TRUE
18 0.17 TRUE
19 0.18 TRUE
20 0.19 TRUE
21 0.20 TRUE
22 0.21 TRUE
23 0.22 TRUE
24 0.23 TRUE
25 0.24 TRUE
26 0.25 TRUE
27 0.26 TRUE
28 0.27 TRUE
29 0.28 TRUE
30 0.29 TRUE
31 0.30 TRUE
32 0.31 TRUE
33 0.32 TRUE
34 0.33 TRUE
35 0.34 TRUE
36 0.35 TRUE
37 0.36 TRUE
38 0.37 TRUE
39 0.38 TRUE
40 0.39 TRUE
41 0.40 TRUE
42 0.41 TRUE
43 0.42 TRUE
44 0.43 TRUE
45 0.44 TRUE
46 0.45 TRUE
47 0.46 TRUE
48 0.47 TRUE
49 0.48 TRUE
50 0.49 TRUE
51 0.50 FALSE
52 0.51 TRUE
53 0.52 TRUE
54 0.53 TRUE
55 0.54 TRUE
56 0.55 TRUE
57 0.56 TRUE
58 0.57 TRUE
59 0.58 TRUE
60 0.59 TRUE
61 0.60 TRUE
62 0.61 TRUE
63 0.62 TRUE
64 0.63 TRUE
65 0.64 TRUE
66 0.65 TRUE
67 0.66 TRUE
68 0.67 TRUE
69 0.68 TRUE
70 0.69 TRUE
71 0.70 TRUE
72 0.71 TRUE
73 0.72 TRUE
74 0.73 TRUE
75 0.74 TRUE
76 0.75 TRUE
77 0.76 TRUE
78 0.77 TRUE
79 0.78 TRUE
80 0.79 TRUE
81 0.80 TRUE
82 0.81 TRUE
83 0.82 TRUE
84 0.83 TRUE
85 0.84 TRUE
86 0.85 TRUE
87 0.86 TRUE
88 0.87 TRUE
89 0.88 TRUE
90 0.89 TRUE
91 0.90 TRUE
92 0.91 TRUE
93 0.92 TRUE
94 0.93 TRUE
95 0.94 TRUE
96 0.95 TRUE
97 0.96 TRUE
98 0.97 TRUE
99 0.98 TRUE
100 0.99 TRUE
101 1.00 TRUE
Appendix C: Session information
===
sessionInfo()
R version 3.0.0 (2013-04-03)
Platform: x86_64-redhat-linux-gnu (64-bit)
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=C LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods
base
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducibl
Re: [R] sample(c(0, 1)...) vs. rbinom
On May 23, 2013, at 07:01 , Jeff Newmiller wrote:
> You seem to be building an elaborate structure for testing the
> reproducibility of the random number generator. I suspect that rbinom is
> calling the random number generator a different number of times when you pass
> prob=0.5 than otherwise.
Nope. It's switching 0 and 1:
> set.seed(1); sample(0:1,10,replace=TRUE,prob=c(1-pp,pp)); set.seed(1);
> rbinom(10,1,pp)
[1] 1 1 0 0 1 0 0 0 0 1
[1] 0 0 1 1 0 1 1 1 1 0
which is curious, but of course has no implication for the distributional
properties. Curiouser, if you drop the prob= in sample.
> set.seed(1); sample(0:1,10,replace=TRUE); set.seed(1); rbinom(10,1,pp)
[1] 0 0 1 1 0 1 1 1 1 0
[1] 0 0 1 1 0 1 1 1 1 0
However, it was never a design goal that two different random functions (or
even two code paths within the same function) should give exactly the same
values, even if they simulate the same distribution, so this is nothing more
than a curiosity.
>>
>> Appendix A: some R code that exhibits the problem
>> =
>>
>> ppp <- seq(0, 1, by = 0.01)
>>
>> result <- do.call(rbind, lapply(ppp, function(p) {
>> set.seed(1)
>> sampleRes <- sample(c(0, 1), size = 1, replace = TRUE,
>> prob=c(1-p, p))
>>
>> set.seed(1)
>> rbinomRes <- rbinom(1, size = 1, prob = p)
>>
>> data.frame(prob = p, equivalent = all(sampleRes == rbinomRes))
>>
>> }))
>>
>> result
>>
>>
>> Appendix B: the output from the R code
>> ==
>>
>> prob equivalent
>> 1 0.00 TRUE
>> 2 0.01 TRUE
>> 3 0.02 TRUE
>> 4 0.03 TRUE
>> 5 0.04 TRUE
>> 6 0.05 TRUE
>> 7 0.06 TRUE
>> 8 0.07 TRUE
>> 9 0.08 TRUE
>> 10 0.09 TRUE
>> 11 0.10 TRUE
>> 12 0.11 TRUE
>> 13 0.12 TRUE
>> 14 0.13 TRUE
>> 15 0.14 TRUE
>> 16 0.15 TRUE
>> 17 0.16 TRUE
>> 18 0.17 TRUE
>> 19 0.18 TRUE
>> 20 0.19 TRUE
>> 21 0.20 TRUE
>> 22 0.21 TRUE
>> 23 0.22 TRUE
>> 24 0.23 TRUE
>> 25 0.24 TRUE
>> 26 0.25 TRUE
>> 27 0.26 TRUE
>> 28 0.27 TRUE
>> 29 0.28 TRUE
>> 30 0.29 TRUE
>> 31 0.30 TRUE
>> 32 0.31 TRUE
>> 33 0.32 TRUE
>> 34 0.33 TRUE
>> 35 0.34 TRUE
>> 36 0.35 TRUE
>> 37 0.36 TRUE
>> 38 0.37 TRUE
>> 39 0.38 TRUE
>> 40 0.39 TRUE
>> 41 0.40 TRUE
>> 42 0.41 TRUE
>> 43 0.42 TRUE
>> 44 0.43 TRUE
>> 45 0.44 TRUE
>> 46 0.45 TRUE
>> 47 0.46 TRUE
>> 48 0.47 TRUE
>> 49 0.48 TRUE
>> 50 0.49 TRUE
>> 51 0.50 FALSE
>> 52 0.51 TRUE
>> 53 0.52 TRUE
>> 54 0.53 TRUE
>> 55 0.54 TRUE
>> 56 0.55 TRUE
>> 57 0.56 TRUE
>> 58 0.57 TRUE
>> 59 0.58 TRUE
>> 60 0.59 TRUE
>> 61 0.60 TRUE
>> 62 0.61 TRUE
>> 63 0.62 TRUE
>> 64 0.63 TRUE
>> 65 0.64 TRUE
>> 66 0.65 TRUE
>> 67 0.66 TRUE
>> 68 0.67 TRUE
>> 69 0.68 TRUE
>> 70 0.69 TRUE
>> 71 0.70 TRUE
>> 72 0.71 TRUE
>> 73 0.72 TRUE
>> 74 0.73 TRUE
>> 75 0.74 TRUE
>> 76 0.75 TRUE
>> 77 0.76 TRUE
>> 78 0.77 TRUE
>> 79 0.78 TRUE
>> 80 0.79 TRUE
>> 81 0.80 TRUE
>> 82 0.81 TRUE
>> 83 0.82 TRUE
>> 84 0.83 TRUE
>> 85 0.84 TRUE
>> 86 0.85 TRUE
>> 87 0.86 TRUE
>> 88 0.87 TRUE
>> 89 0.88 TRUE
>> 90 0.89 TRUE
>> 91 0.90 TRUE
>> 92 0.91 TRUE
>> 93 0.92 TRUE
>> 94 0.93 TRUE
>> 95 0.94 TRUE
>> 96 0.95 TRUE
>> 97 0.96 TRUE
>> 98 0.97 TRUE
>> 99 0.98 TRUE
>> 100 0.99 TRUE
>> 101 1.00 TRUE
>>
>> Appendix C: Session information
>> ===
>>
>>> sessionInfo()
>> R version 3.0.0 (2013-04-03)
>> Platform: x86_64-redhat-linux-gnu (64-bit)
>>
>> locale:
>> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
>> [3] LC_TIME=en_US.UTF-8LC_COLLATE=en_US.UTF-8
>> [5] LC_MONETARY=en_US.UTF-8LC_MESSAGES=en_US.UTF-8
>> [7] LC_PAPER=C LC_NAME=C
>> [9] LC_ADDRESS=C LC_TELEPHONE=C
>> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
>>
>> attached base packages:
>> [1] stats graphics grDevices utils datasets methods base
>>
>>>
>>
>> __
>> R-help@r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> __
> R-help@r-project.org mailing list
> ht
Re: [R] sample(c(0, 1)...) vs. rbinom
You seem to be building an elaborate structure for testing the reproducibility
of the random number generator. I suspect that rbinom is calling the random
number generator a different number of times when you pass prob=0.5 than
otherwise.
---
Jeff NewmillerThe . . Go Live...
DCN:Basics: ##.#. ##.#. Live Go...
Live: OO#.. Dead: OO#.. Playing
Research Engineer (Solar/BatteriesO.O#. #.O#. with
/Software/Embedded Controllers) .OO#. .OO#. rocks...1k
---
Sent from my phone. Please excuse my brevity.
Michael Hannon wrote:
>Greetings. My wife is teaching an introductory stat class at UC
>Davis. The
>class emphasizes the use of simulations, rather than mathematics, to
>get
>insight into statistics, and R is the mandated tool. A student in the
>class
>recently inquired about different approaches to sampling from a
>binomial
>distribution. I've appended some code that exhibits the idea, the gist
>of
>which is that using sample(c(0, 1), ...) and rbinom(...) should give
>equivalent results.
>
>The surprising (to me) result is that the two approaches DO give the
>same
>result, EXCEPT when the probability is exactly 0.5. See Appendix A for
>the
>code and Appendix B for the output. I don't think this issue is
>system-dependent, but I've put my session information in Appendix C.
>
>Another wrinkle in this is that if I omit the "prob" parameter from the
>call
>to sample, meaning to take the default value of 0.5, the two methods DO
>give
>the same result.
>
>Any thoughts about this? Thanks.
>
>--Mike
>
>Appendix A: some R code that exhibits the problem
>=
>
>ppp <- seq(0, 1, by = 0.01)
>
>result <- do.call(rbind, lapply(ppp, function(p) {
> set.seed(1)
> sampleRes <- sample(c(0, 1), size = 1, replace = TRUE,
> prob=c(1-p, p))
>
> set.seed(1)
> rbinomRes <- rbinom(1, size = 1, prob = p)
>
> data.frame(prob = p, equivalent = all(sampleRes == rbinomRes))
>
>}))
>
>result
>
>
>Appendix B: the output from the R code
>==
>
> prob equivalent
>1 0.00 TRUE
>2 0.01 TRUE
>3 0.02 TRUE
>4 0.03 TRUE
>5 0.04 TRUE
>6 0.05 TRUE
>7 0.06 TRUE
>8 0.07 TRUE
>9 0.08 TRUE
>10 0.09 TRUE
>11 0.10 TRUE
>12 0.11 TRUE
>13 0.12 TRUE
>14 0.13 TRUE
>15 0.14 TRUE
>16 0.15 TRUE
>17 0.16 TRUE
>18 0.17 TRUE
>19 0.18 TRUE
>20 0.19 TRUE
>21 0.20 TRUE
>22 0.21 TRUE
>23 0.22 TRUE
>24 0.23 TRUE
>25 0.24 TRUE
>26 0.25 TRUE
>27 0.26 TRUE
>28 0.27 TRUE
>29 0.28 TRUE
>30 0.29 TRUE
>31 0.30 TRUE
>32 0.31 TRUE
>33 0.32 TRUE
>34 0.33 TRUE
>35 0.34 TRUE
>36 0.35 TRUE
>37 0.36 TRUE
>38 0.37 TRUE
>39 0.38 TRUE
>40 0.39 TRUE
>41 0.40 TRUE
>42 0.41 TRUE
>43 0.42 TRUE
>44 0.43 TRUE
>45 0.44 TRUE
>46 0.45 TRUE
>47 0.46 TRUE
>48 0.47 TRUE
>49 0.48 TRUE
>50 0.49 TRUE
>51 0.50 FALSE
>52 0.51 TRUE
>53 0.52 TRUE
>54 0.53 TRUE
>55 0.54 TRUE
>56 0.55 TRUE
>57 0.56 TRUE
>58 0.57 TRUE
>59 0.58 TRUE
>60 0.59 TRUE
>61 0.60 TRUE
>62 0.61 TRUE
>63 0.62 TRUE
>64 0.63 TRUE
>65 0.64 TRUE
>66 0.65 TRUE
>67 0.66 TRUE
>68 0.67 TRUE
>69 0.68 TRUE
>70 0.69 TRUE
>71 0.70 TRUE
>72 0.71 TRUE
>73 0.72 TRUE
>74 0.73 TRUE
>75 0.74 TRUE
>76 0.75 TRUE
>77 0.76 TRUE
>78 0.77 TRUE
>79 0.78 TRUE
>80 0.79 TRUE
>81 0.80 TRUE
>82 0.81 TRUE
>83 0.82 TRUE
>84 0.83 TRUE
>85 0.84 TRUE
>86 0.85 TRUE
>87 0.86 TRUE
>88 0.87 TRUE
>89 0.88 TRUE
>90 0.89 TRUE
>91 0.90 TRUE
>92 0.91 TRUE
>93 0.92 TRUE
>94 0.93 TRUE
>95 0.94 TRUE
>96 0.95 TRUE
>97 0.96 TRUE
>98 0.97 TRUE
>99 0.98 TRUE
>100 0.99 TRUE
>101 1.00 TRUE
>
>Appendix C: Session information
>===
>
>> sessionInfo()
>R version 3.0.0 (2013-04-03)
>Platform: x86_64-redhat-linux-gnu (64-bit)
>
>locale:
> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
> [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
> [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
> [7] LC_PAPER=C LC_NAME=C
> [9] LC_ADDRESS=C LC_TELEPHONE=C
>[11] LC_MEASUREMENT=
