How to calculate confidence interval of C statistic by rcorr.cens

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How to calculate confidence interval of C statistic by rcorr.cens

khosoda
Hi,

I'm trying to calculate 95% confidence interval of C statistic of
logistic regression model using rcorr.cens in rms package. I wrote a
brief function for this purpose as the followings;

CstatisticCI <- function(x)   # x is object of rcorr.cens.
  {
    se <- x["S.D."]/sqrt(x["n"])
    Low95 <- x["C Index"] - 1.96*se
    Upper95 <- x["C Index"] + 1.96*se
    cbind(x["C Index"], Low95, Upper95)
  }

Then,

> MyModel.lrm.rcorr <- rcorr.cens(x=predict(MyModel.lrm), S=df$outcome)
> MyModel.lrm.rcorr
       C Index            Dxy           S.D.              n
missing     uncensored
     0.8222785      0.6445570      0.1047916    104.0000000
0.0000000    104.0000000
Relevant Pairs     Concordant      Uncertain
  3950.0000000   3248.0000000      0.0000000

> CstatisticCI(x5factor_final.lrm.pen.rcorr)
                      Low95   Upper95
C Index 0.8222785 0.8021382 0.8424188

I'm not sure what "S.D." in object of rcorr.cens means. Is this standard
deviation of "C Index" or standard deviation of "Dxy"?
I thought it is standard deviation of "C Index". Therefore, I wrote the
code above. Am I right?

I would appreciate any help in advance.

--
Kohkichi Hosoda M.D.

    Department of Neurosurgery,
    Kobe University Graduate School of Medicine,

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Re: How to calculate confidence interval of C statistic by rcorr.cens

Frank Harrell
S.D. is the standard deviation (standard error) of Dxy.  It already includes the effective sample size in its computation so the sqrt(n) terms is not needed.  The help file for rcorr.cens has an example where the confidence interval for C is computed.  Note that you are making the strong assumption that there is no overfitting in the model or that you are evaluating C on a sample not used in model development.
Frank

細田弘吉 wrote
Hi,

I'm trying to calculate 95% confidence interval of C statistic of
logistic regression model using rcorr.cens in rms package. I wrote a
brief function for this purpose as the followings;

CstatisticCI <- function(x)   # x is object of rcorr.cens.
  {
    se <- x["S.D."]/sqrt(x["n"])
    Low95 <- x["C Index"] - 1.96*se
    Upper95 <- x["C Index"] + 1.96*se
    cbind(x["C Index"], Low95, Upper95)
  }

Then,

> MyModel.lrm.rcorr <- rcorr.cens(x=predict(MyModel.lrm), S=df$outcome)
> MyModel.lrm.rcorr
       C Index            Dxy           S.D.              n
missing     uncensored
     0.8222785      0.6445570      0.1047916    104.0000000
0.0000000    104.0000000
Relevant Pairs     Concordant      Uncertain
  3950.0000000   3248.0000000      0.0000000

> CstatisticCI(x5factor_final.lrm.pen.rcorr)
                      Low95   Upper95
C Index 0.8222785 0.8021382 0.8424188

I'm not sure what "S.D." in object of rcorr.cens means. Is this standard
deviation of "C Index" or standard deviation of "Dxy"?
I thought it is standard deviation of "C Index". Therefore, I wrote the
code above. Am I right?

I would appreciate any help in advance.

--
Kohkichi Hosoda M.D.

    Department of Neurosurgery,
    Kobe University Graduate School of Medicine,

______________________________________________
[hidden email] 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.
Frank Harrell
Department of Biostatistics, Vanderbilt University
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Re: How to calculate confidence interval of C statistic by rcorr.cens

khosoda
Thank you for your comment, Prof Harrell.

I changed the function;

CstatisticCI <- function(x)   # x is object of rcorr.cens.
   {
     se <- x["S.D."]/2
     Low95 <- x["C Index"] - 1.96*se
     Upper95 <- x["C Index"] + 1.96*se

     cbind(x["C Index"], Low95, Upper95)
   }

 > CstatisticCI(MyModel.lrm.penalized.rcorr)
                       Low95   Upper95
C Index 0.8222785 0.7195828 0.9249742

I obtained wider CI than the previous incorrect one.
Regarding your comments on overfitting, this is a sample used in model
development. However, I performed penalization by pentrace and lrm in
rms package. The CI above is CI of penalized model. Results of
validation of each model are followings;

First model
 > validate(MyModel.lrm, bw=F, B=1000)
           index.orig training    test optimism index.corrected    n
Dxy           0.6385   0.6859  0.6198   0.0661          0.5724 1000
R2            0.3745   0.4222  0.3388   0.0834          0.2912 1000
Intercept     0.0000   0.0000 -0.1446   0.1446         -0.1446 1000
Slope         1.0000   1.0000  0.8266   0.1734          0.8266 1000
Emax          0.0000   0.0000  0.0688   0.0688          0.0688 1000
D             0.2784   0.3248  0.2474   0.0774          0.2010 1000
U            -0.0192  -0.0192  0.0200  -0.0392          0.0200 1000
Q             0.2976   0.3440  0.2274   0.1166          0.1810 1000
B             0.1265   0.1180  0.1346  -0.0167          0.1431 1000
g             1.7010   2.0247  1.5763   0.4484          1.2526 1000
gp            0.2414   0.2512  0.2287   0.0225          0.2189 1000

penalized model
 > validate(MyModel.lrm.penalized, bw=F, B=1000)
           index.orig training    test optimism index.corrected    n
Dxy           0.6446   0.6898  0.6256   0.0642          0.5804 1000
R2            0.3335   0.3691  0.3428   0.0264          0.3072 1000
Intercept     0.0000   0.0000  0.0752  -0.0752          0.0752 1000
Slope         1.0000   1.0000  1.0547  -0.0547          1.0547 1000
Emax          0.0000   0.0000  0.0249   0.0249          0.0249 1000
D             0.2718   0.2744  0.2507   0.0236          0.2481 1000
U            -0.0192  -0.0192 -0.0027  -0.0165         -0.0027 1000
Q             0.2910   0.2936  0.2534   0.0402          0.2508 1000
B             0.1279   0.1192  0.1336  -0.0144          0.1423 1000
g             1.3942   1.5259  1.5799  -0.0540          1.4482 1000
gp            0.2141   0.2188  0.2298  -0.0110          0.2251 1000

Optimism of slope and intercept were improved from 0.1446 and 0.1734 to
-0.0752 and -0.0547, respectively. Emax was improved from 0.0688 to
0.0249. Therefore, I thought overfitting was improved at least to some
extent. Well, I'm not sure whether this is enough improvement though.

--
Kohkichi

(11/05/22 23:27), Frank Harrell wrote:

> S.D. is the standard deviation (standard error) of Dxy.  It already includes
> the effective sample size in its computation so the sqrt(n) terms is not
> needed.  The help file for rcorr.cens has an example where the confidence
> interval for C is computed.  Note that you are making the strong assumption
> that there is no overfitting in the model or that you are evaluating C on a
> sample not used in model development.
> Frank
>
>
> Kohkichi wrote:
>>
>> Hi,
>>
>> I'm trying to calculate 95% confidence interval of C statistic of
>> logistic regression model using rcorr.cens in rms package. I wrote a
>> brief function for this purpose as the followings;
>>
>> CstatisticCI<- function(x)   # x is object of rcorr.cens.
>>    {
>>      se<- x["S.D."]/sqrt(x["n"])
>>      Low95<- x["C Index"] - 1.96*se
>>      Upper95<- x["C Index"] + 1.96*se
>>      cbind(x["C Index"], Low95, Upper95)
>>    }
>>
>> Then,
>>
>>> MyModel.lrm.rcorr<- rcorr.cens(x=predict(MyModel.lrm), S=df$outcome)
>>> MyModel.lrm.rcorr
>>         C Index            Dxy           S.D.              n
>> missing     uncensored
>>       0.8222785      0.6445570      0.1047916    104.0000000
>> 0.0000000    104.0000000
>> Relevant Pairs     Concordant      Uncertain
>>    3950.0000000   3248.0000000      0.0000000
>>
>>> CstatisticCI(x5factor_final.lrm.pen.rcorr)
>>                        Low95   Upper95
>> C Index 0.8222785 0.8021382 0.8424188
>>
>> I'm not sure what "S.D." in object of rcorr.cens means. Is this standard
>> deviation of "C Index" or standard deviation of "Dxy"?
>> I thought it is standard deviation of "C Index". Therefore, I wrote the
>> code above. Am I right?
>>
>> I would appreciate any help in advance.
>>
>> --
>> Kohkichi Hosoda M.D.
>>
>>      Department of Neurosurgery,
>>      Kobe University Graduate School of Medicine,
>>
>> ______________________________________________
>> [hidden email] 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.
>>
>
>
> -----
> Frank Harrell
> Department of Biostatistics, Vanderbilt University
> --
> View this message in context: http://r.789695.n4.nabble.com/How-to-calculate-confidence-interval-of-C-statistic-by-rcorr-cens-tp3541709p3542163.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> [hidden email] 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.

______________________________________________
[hidden email] 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.
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Re: How to calculate confidence interval of C statistic by rcorr.cens

Frank Harrell
Hi Kohkichi,
What we really need to figure out is how to make validate give you confidence intervals for Dxy or C while it is penalizing for overfitting.  Some people have ad hoc solutions for that but nothing is nailed down yet.
Frank
 
khosoda wrote
Thank you for your comment, Prof Harrell.

I changed the function;

CstatisticCI <- function(x)   # x is object of rcorr.cens.
   {
     se <- x["S.D."]/2
     Low95 <- x["C Index"] - 1.96*se
     Upper95 <- x["C Index"] + 1.96*se

     cbind(x["C Index"], Low95, Upper95)
   }

 > CstatisticCI(MyModel.lrm.penalized.rcorr)
                       Low95   Upper95
C Index 0.8222785 0.7195828 0.9249742

I obtained wider CI than the previous incorrect one.
Regarding your comments on overfitting, this is a sample used in model
development. However, I performed penalization by pentrace and lrm in
rms package. The CI above is CI of penalized model. Results of
validation of each model are followings;

First model
 > validate(MyModel.lrm, bw=F, B=1000)
           index.orig training    test optimism index.corrected    n
Dxy           0.6385   0.6859  0.6198   0.0661          0.5724 1000
R2            0.3745   0.4222  0.3388   0.0834          0.2912 1000
Intercept     0.0000   0.0000 -0.1446   0.1446         -0.1446 1000
Slope         1.0000   1.0000  0.8266   0.1734          0.8266 1000
Emax          0.0000   0.0000  0.0688   0.0688          0.0688 1000
D             0.2784   0.3248  0.2474   0.0774          0.2010 1000
U            -0.0192  -0.0192  0.0200  -0.0392          0.0200 1000
Q             0.2976   0.3440  0.2274   0.1166          0.1810 1000
B             0.1265   0.1180  0.1346  -0.0167          0.1431 1000
g             1.7010   2.0247  1.5763   0.4484          1.2526 1000
gp            0.2414   0.2512  0.2287   0.0225          0.2189 1000

penalized model
 > validate(MyModel.lrm.penalized, bw=F, B=1000)
           index.orig training    test optimism index.corrected    n
Dxy           0.6446   0.6898  0.6256   0.0642          0.5804 1000
R2            0.3335   0.3691  0.3428   0.0264          0.3072 1000
Intercept     0.0000   0.0000  0.0752  -0.0752          0.0752 1000
Slope         1.0000   1.0000  1.0547  -0.0547          1.0547 1000
Emax          0.0000   0.0000  0.0249   0.0249          0.0249 1000
D             0.2718   0.2744  0.2507   0.0236          0.2481 1000
U            -0.0192  -0.0192 -0.0027  -0.0165         -0.0027 1000
Q             0.2910   0.2936  0.2534   0.0402          0.2508 1000
B             0.1279   0.1192  0.1336  -0.0144          0.1423 1000
g             1.3942   1.5259  1.5799  -0.0540          1.4482 1000
gp            0.2141   0.2188  0.2298  -0.0110          0.2251 1000

Optimism of slope and intercept were improved from 0.1446 and 0.1734 to
-0.0752 and -0.0547, respectively. Emax was improved from 0.0688 to
0.0249. Therefore, I thought overfitting was improved at least to some
extent. Well, I'm not sure whether this is enough improvement though.

--
Kohkichi

(11/05/22 23:27), Frank Harrell wrote:
> S.D. is the standard deviation (standard error) of Dxy.  It already includes
> the effective sample size in its computation so the sqrt(n) terms is not
> needed.  The help file for rcorr.cens has an example where the confidence
> interval for C is computed.  Note that you are making the strong assumption
> that there is no overfitting in the model or that you are evaluating C on a
> sample not used in model development.
> Frank
>
>
> Kohkichi wrote:
>>
>> Hi,
>>
>> I'm trying to calculate 95% confidence interval of C statistic of
>> logistic regression model using rcorr.cens in rms package. I wrote a
>> brief function for this purpose as the followings;
>>
>> CstatisticCI<- function(x)   # x is object of rcorr.cens.
>>    {
>>      se<- x["S.D."]/sqrt(x["n"])
>>      Low95<- x["C Index"] - 1.96*se
>>      Upper95<- x["C Index"] + 1.96*se
>>      cbind(x["C Index"], Low95, Upper95)
>>    }
>>
>> Then,
>>
>>> MyModel.lrm.rcorr<- rcorr.cens(x=predict(MyModel.lrm), S=df$outcome)
>>> MyModel.lrm.rcorr
>>         C Index            Dxy           S.D.              n
>> missing     uncensored
>>       0.8222785      0.6445570      0.1047916    104.0000000
>> 0.0000000    104.0000000
>> Relevant Pairs     Concordant      Uncertain
>>    3950.0000000   3248.0000000      0.0000000
>>
>>> CstatisticCI(x5factor_final.lrm.pen.rcorr)
>>                        Low95   Upper95
>> C Index 0.8222785 0.8021382 0.8424188
>>
>> I'm not sure what "S.D." in object of rcorr.cens means. Is this standard
>> deviation of "C Index" or standard deviation of "Dxy"?
>> I thought it is standard deviation of "C Index". Therefore, I wrote the
>> code above. Am I right?
>>
>> I would appreciate any help in advance.
>>
>> --
>> Kohkichi Hosoda M.D.
>>
>>      Department of Neurosurgery,
>>      Kobe University Graduate School of Medicine,
>>
>> ______________________________________________
>> [hidden email] 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.
>>
>
>
> -----
> Frank Harrell
> Department of Biostatistics, Vanderbilt University
> --
> View this message in context: http://r.789695.n4.nabble.com/How-to-calculate-confidence-interval-of-C-statistic-by-rcorr-cens-tp3541709p3542163.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> [hidden email] 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.

______________________________________________
[hidden email] 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.
Frank Harrell
Department of Biostatistics, Vanderbilt University
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Re: How to calculate confidence interval of C statistic by rcorr.cens

khosoda
Dear Prof. Harrell,

I'm sorry to say this, but I'm afraid I cannot understand what you write
very well. Do you mean that the method to calculate confidence intervals
for Dxy or C statistics in logistic model penalized for overfitting has
not been established yet and what I did is wrong?
Could you elaborate it or teach me some reference point?

Kohkichi

(11/05/23 4:22), Frank Harrell wrote:

> Hi Kohkichi,
> What we really need to figure out is how to make validate give you
> confidence intervals for Dxy or C while it is penalizing for overfitting.
> Some people have ad hoc solutions for that but nothing is nailed down yet.
> Frank
>
> khosoda wrote:
>>
>> Thank you for your comment, Prof Harrell.
>>
>> I changed the function;
>>
>> CstatisticCI<- function(x)   # x is object of rcorr.cens.
>>     {
>>       se<- x["S.D."]/2
>>       Low95<- x["C Index"] - 1.96*se
>>       Upper95<- x["C Index"] + 1.96*se
>>
>>       cbind(x["C Index"], Low95, Upper95)
>>     }
>>
>>   >  CstatisticCI(MyModel.lrm.penalized.rcorr)
>>                         Low95   Upper95
>> C Index 0.8222785 0.7195828 0.9249742
>>
>> I obtained wider CI than the previous incorrect one.
>> Regarding your comments on overfitting, this is a sample used in model
>> development. However, I performed penalization by pentrace and lrm in
>> rms package. The CI above is CI of penalized model. Results of
>> validation of each model are followings;
>>
>> First model
>>   >  validate(MyModel.lrm, bw=F, B=1000)
>>             index.orig training    test optimism index.corrected    n
>> Dxy           0.6385   0.6859  0.6198   0.0661          0.5724 1000
>> R2            0.3745   0.4222  0.3388   0.0834          0.2912 1000
>> Intercept     0.0000   0.0000 -0.1446   0.1446         -0.1446 1000
>> Slope         1.0000   1.0000  0.8266   0.1734          0.8266 1000
>> Emax          0.0000   0.0000  0.0688   0.0688          0.0688 1000
>> D             0.2784   0.3248  0.2474   0.0774          0.2010 1000
>> U            -0.0192  -0.0192  0.0200  -0.0392          0.0200 1000
>> Q             0.2976   0.3440  0.2274   0.1166          0.1810 1000
>> B             0.1265   0.1180  0.1346  -0.0167          0.1431 1000
>> g             1.7010   2.0247  1.5763   0.4484          1.2526 1000
>> gp            0.2414   0.2512  0.2287   0.0225          0.2189 1000
>>
>> penalized model
>>   >  validate(MyModel.lrm.penalized, bw=F, B=1000)
>>             index.orig training    test optimism index.corrected    n
>> Dxy           0.6446   0.6898  0.6256   0.0642          0.5804 1000
>> R2            0.3335   0.3691  0.3428   0.0264          0.3072 1000
>> Intercept     0.0000   0.0000  0.0752  -0.0752          0.0752 1000
>> Slope         1.0000   1.0000  1.0547  -0.0547          1.0547 1000
>> Emax          0.0000   0.0000  0.0249   0.0249          0.0249 1000
>> D             0.2718   0.2744  0.2507   0.0236          0.2481 1000
>> U            -0.0192  -0.0192 -0.0027  -0.0165         -0.0027 1000
>> Q             0.2910   0.2936  0.2534   0.0402          0.2508 1000
>> B             0.1279   0.1192  0.1336  -0.0144          0.1423 1000
>> g             1.3942   1.5259  1.5799  -0.0540          1.4482 1000
>> gp            0.2141   0.2188  0.2298  -0.0110          0.2251 1000
>>
>> Optimism of slope and intercept were improved from 0.1446 and 0.1734 to
>> -0.0752 and -0.0547, respectively. Emax was improved from 0.0688 to
>> 0.0249. Therefore, I thought overfitting was improved at least to some
>> extent. Well, I'm not sure whether this is enough improvement though.
>>
>> --
>> Kohkichi
>>
>> (11/05/22 23:27), Frank Harrell wrote:
>>> S.D. is the standard deviation (standard error) of Dxy.  It already
>>> includes
>>> the effective sample size in its computation so the sqrt(n) terms is not
>>> needed.  The help file for rcorr.cens has an example where the confidence
>>> interval for C is computed.  Note that you are making the strong
>>> assumption
>>> that there is no overfitting in the model or that you are evaluating C on
>>> a
>>> sample not used in model development.
>>> Frank
>>>
>>>
>>> Kohkichi wrote:
>>>>
>>>> Hi,
>>>>
>>>> I'm trying to calculate 95% confidence interval of C statistic of
>>>> logistic regression model using rcorr.cens in rms package. I wrote a
>>>> brief function for this purpose as the followings;
>>>>
>>>> CstatisticCI<- function(x)   # x is object of rcorr.cens.
>>>>     {
>>>>       se<- x["S.D."]/sqrt(x["n"])
>>>>       Low95<- x["C Index"] - 1.96*se
>>>>       Upper95<- x["C Index"] + 1.96*se
>>>>       cbind(x["C Index"], Low95, Upper95)
>>>>     }
>>>>
>>>> Then,
>>>>
>>>>> MyModel.lrm.rcorr<- rcorr.cens(x=predict(MyModel.lrm), S=df$outcome)
>>>>> MyModel.lrm.rcorr
>>>>          C Index            Dxy           S.D.              n
>>>> missing     uncensored
>>>>        0.8222785      0.6445570      0.1047916    104.0000000
>>>> 0.0000000    104.0000000
>>>> Relevant Pairs     Concordant      Uncertain
>>>>     3950.0000000   3248.0000000      0.0000000
>>>>
>>>>> CstatisticCI(x5factor_final.lrm.pen.rcorr)
>>>>                         Low95   Upper95
>>>> C Index 0.8222785 0.8021382 0.8424188
>>>>
>>>> I'm not sure what "S.D." in object of rcorr.cens means. Is this standard
>>>> deviation of "C Index" or standard deviation of "Dxy"?
>>>> I thought it is standard deviation of "C Index". Therefore, I wrote the
>>>> code above. Am I right?
>>>>
>>>> I would appreciate any help in advance.
>>>>
>>>> --
>>>> Kohkichi Hosoda M.D.
>>>>
>>>>       Department of Neurosurgery,
>>>>       Kobe University Graduate School of Medicine,
>>>>
>>>> ______________________________________________
>>>> [hidden email] mailing list
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>>>> PLEASE do read the posting guide
>>>> http://www.R-project.org/posting-guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>
>>>
>>> -----
>>> Frank Harrell
>>> Department of Biostatistics, Vanderbilt University
>>> --
>>> View this message in context:
>>> http://r.789695.n4.nabble.com/How-to-calculate-confidence-interval-of-C-statistic-by-rcorr-cens-tp3541709p3542163.html
>>> Sent from the R help mailing list archive at Nabble.com.
>>>
>>> ______________________________________________
>>> [hidden email] 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.
>>
>> ______________________________________________
>> [hidden email] 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.
>>
>
>
> -----
> Frank Harrell
> Department of Biostatistics, Vanderbilt University
> --
> View this message in context: http://r.789695.n4.nabble.com/How-to-calculate-confidence-interval-of-C-statistic-by-rcorr-cens-tp3541709p3542654.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> [hidden email] mailing list
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> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>


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