# Brown-Forsythe test of equality of MEANS

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## Brown-Forsythe test of equality of MEANS

 Dear friends, two years ago (as I found on the web) Paul sent the following message but I was not able to find if he got an answer. Today I have the same question and it would be great if I could find out that this test has been implemented (somehow) in R. Please do not confuse it with the Brown-Forsythe test of equality of variances. Thank you: I've been searching around for a function for computing the Brown-Forsythe F* statistic which is a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality. A couple of other people have asked this question, the responses I found have been:     ?oneway.test However, that function appears to use the Welch W statistic which, while good at handling unequal variances, is not as good as F* at handling non-normal distributions (or so my textbook tells me). So, two questions:    1. Is there a function ready to use for calculating the Brown-Forsythe F*?    2. If not, what do people use for checking the results of a (one-way) ANOVA when there is non-normality as well as non-constant variances? Thanks,       Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 Inline below. -- Bert On Mon, Aug 30, 2010 at 1:05 PM, Iasonas Lamprianou <[hidden email]>wrote: > > Dear friends, > two years ago (as I found on the web) Paul sent the following message but I > was not able to find if he got an answer. Today I have the same question and > it would be great if I could find out that this test has been implemented > (somehow) in R. Please do not confuse it with the Brown-Forsythe test of > equality of variances. Thank you: > > I've been searching around for a function for computing the Brown-Forsythe > F* statistic which is a substitute for the normal ANOVA F statistic for when > there are unequal variances, and when there is evidence of non-normality. False, I think, as I'm not entirely clear on your meaning. Brown-Fosythe is a test for the equality of spreads among groups. From Wikipedia: The transformed response variable is constructed to measure the spreadin each group. Let  [image: z_{ij}=\left\vert y_{ij} - \tilde{y}_j \right\vert] where [image: \tilde{y}_j] is the medianof group *j*. The BrownForsythe test statistic is the model *F* statistic from a one way ANOVA on *zij*: In particular, it is NOT " a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality." -- Bert Gunter Genentech Noclinical Statistics         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 On 30 Aug 2010 at 13:25, Bert Gunter wrote: > Inline below. > > -- Bert Wrong.  There *is* a Brown-Forsythe test of equality of means given heterogeneity of variance.   [Kirk's experimental design tst, 3rd Ed. p. 155 describes the test.] ---JRG John R. Gleason > > On Mon, Aug 30, 2010 at 1:05 PM, Iasonas Lamprianou <[hidden email]>wrote: > > > > > Dear friends, > > two years ago (as I found on the web) Paul sent the following message but I > > was not able to find if he got an answer. Today I have the same question and > > it would be great if I could find out that this test has been implemented > > (somehow) in R. Please do not confuse it with the Brown-Forsythe test of > > equality of variances. Thank you: > > > > I've been searching around for a function for computing the Brown-Forsythe > > F* statistic which is a substitute for the normal ANOVA F statistic for when > > there are unequal variances, and when there is evidence of non-normality. > > > False, I think, as I'm not entirely clear on your meaning. Brown-Fosythe is > a test for the equality of spreads among groups. From Wikipedia: > > > The transformed response variable is constructed to measure the > spreadin each > group. Let >  [image: z_{ij}=\left\vert y_{ij} - \tilde{y}_j \right\vert] > > where [image: \tilde{y}_j] is the > medianof group > *j*. The Brown-Forsythe test statistic is the model *F* statistic from a one > way ANOVA on *zij*: > > In particular, it is NOT " a substitute for the normal ANOVA F statistic for > when there are unequal variances, and when there is evidence of > non-normality." > > -- > > Bert Gunter > > Genentech Noclinical Statistics > > [[alternative HTML version deleted]] > > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 Thanks. I stand corrected, then. -- Bert On Mon, Aug 30, 2010 at 1:45 PM, JRG <[hidden email]> wrote: > On 30 Aug 2010 at 13:25, Bert Gunter wrote: > > > Inline below. > > > > -- Bert > > Wrong.  There *is* a Brown-Forsythe test of equality of means given > heterogeneity of variance. > [Kirk's experimental design tst, 3rd Ed. p. 155 describes the test.] > > ---JRG > > John R. Gleason > > > > > On Mon, Aug 30, 2010 at 1:05 PM, Iasonas Lamprianou < > [hidden email]>wrote: > > > > > > > > Dear friends, > > > two years ago (as I found on the web) Paul sent the following message > but I > > > was not able to find if he got an answer. Today I have the same > question and > > > it would be great if I could find out that this test has been > implemented > > > (somehow) in R. Please do not confuse it with the Brown-Forsythe test > of > > > equality of variances. Thank you: > > > > > > I've been searching around for a function for computing the > Brown-Forsythe > > > F* statistic which is a substitute for the normal ANOVA F statistic for > when > > > there are unequal variances, and when there is evidence of > non-normality. > > > > > > False, I think, as I'm not entirely clear on your meaning. Brown-Fosythe > is > > a test for the equality of spreads among groups. From Wikipedia: > > > > > > The transformed response variable is constructed to measure the > > spreadin each > > group. Let > >  [image: z_{ij}=\left\vert y_{ij} - \tilde{y}_j \right\vert] > > > > where [image: \tilde{y}_j] is the > > medianof group > > *j*. The Brown-Forsythe test statistic is the model *F* statistic from a > one > > way ANOVA on *zij*: > > > > In particular, it is NOT " a substitute for the normal ANOVA F statistic > for > > when there are unequal variances, and when there is evidence of > > non-normality." > > > > -- > > > > Bert Gunter > > > > Genentech Noclinical Statistics > > > >       [[alternative HTML version deleted]] > > > > > > > -- Bert Gunter Genentech Nonclinical Biostatistics 467-7374 http://devo.gene.com/groups/devo/depts/ncb/home.shtml        [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

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## Re: Brown-Forsythe test of equality of MEANS

 In reply to this post by Iasonas Lamprianou The following reference that contains a short Fortran program for the Brown-Forsythe ANOVA Reed, James F., I. & Stark, D. B. Robust alternatives to traditional analyses of variance: Welch $W^*$, James $J_I^*$, James $J_II^*$, and Brown-Forsythe $BF^*$ Computer Methods and Programs in Biomedicine, 1988, 26, 233-238 Iasonas Lamprianou <[hidden email]> Sent by: [hidden email] 08/30/2010 04:05 PM To [hidden email] cc Subject [R] Brown-Forsythe test of equality of MEANS Dear friends, two years ago (as I found on the web) Paul sent the following message but I was not able to find if he got an answer. Today I have the same question and it would be great if I could find out that this test has been implemented (somehow) in R. Please do not confuse it with the Brown-Forsythe test of equality of variances. Thank you: I've been searching around for a function for computing the Brown-Forsythe F* statistic which is a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality. A couple of other people have asked this question, the responses I found have been:     ?oneway.test However, that function appears to use the Welch W statistic which, while good at handling unequal variances, is not as good as F* at handling non-normal distributions (or so my textbook tells me). So, two questions:    1. Is there a function ready to use for calculating the Brown-Forsythe F*?    2. If not, what do people use for checking the results of a (one-way) ANOVA when there is non-normality as well as non-constant variances? Thanks,   Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 Thank you. It seems that nobody has implemented it in R yet. I'll have a look, thank you very much Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] --- On Tue, 31/8/10, [hidden email] <[hidden email]> wrote: From: [hidden email] <[hidden email]> Subject: Re: [R] Brown-Forsythe test of equality of MEANS To: "Iasonas Lamprianou" <[hidden email]> Cc: [hidden email], [hidden email] Date: Tuesday, 31 August, 2010, 20:26 The following reference that contains a short Fortran program for the Brown-Forsythe ANOVA Reed, James F., I. & Stark, D. B. Robust alternatives to traditional analyses of variance: Welch $W^*$, James $J_I^*$, James $J_II^*$, and Brown-Forsythe $BF^*$ Computer Methods and Programs in Biomedicine, 1988, 26, 233-238 Iasonas Lamprianou <[hidden email]> Sent by: [hidden email] 08/30/2010 04:05 PM To [hidden email] cc Subject [R] Brown-Forsythe test of equality of MEANS Dear friends, two years ago (as I found on the web) Paul sent the following message but I was not able to find if he got an answer. Today I have the same question and it would be great if I could find out that this test has been implemented (somehow) in R. Please do not confuse it with the Brown-Forsythe test of equality of variances. Thank you: I've been searching around for a function for computing the Brown-Forsythe F* statistic which is a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality. A couple of other people have asked this question, the responses I found have been:     ?oneway.test However, that function appears to use the Welch W statistic which, while good at handling unequal variances, is not as good as F* at handling non-normal distributions (or so my textbook tells me). So, two questions:    1. Is there a function ready to use for calculating the Brown-Forsythe F*?    2. If not, what do people use for checking the results of a (one-way) ANOVA when there is non-normality as well as non-constant variances? Thanks,        Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.               [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 In reply to this post by JLucke Learned Folks: Well, I've already advertised my ignorance about these matters, so I have nothing to lose by plunging ahead with further Questionable "advice." >From the references cited, Brown-Forsythe originated in the statistical medieval age -- that is, prior to large scale, cheap computing (to be honest, I have dim memories of it in BMD!). Then cameth Brad Efron and the enlightenment: If you are concerned about the distribution of this -- or indeed any reasonably smooth statistic (and some not so smooth: Hinkley - Davison's Bootstrap book has details) -- then bootstrap it. That is, get a confidence interval for the difference in means and see whether 0 falls within (or whatever Null you wish to test). If you are concerned about robustness (whatever that means in this context), well, gosheth! -- we have journeyed a long way since the 1970's. Indeed, there are several packages (e.g. robust, robustbase) with lots of robust alternatives. Most (maybe all?) of which can be bootstrapped, of course. So walketh with thy computers, my brethren, and enter the great age of enlightenment. (Thus endeth the lesson. Caveat Emptor!) Cheers to all, Bert On Tue, Aug 31, 2010 at 12:26 PM, <[hidden email]> wrote: > The following reference that contains a short Fortran program for the > Brown-Forsythe ANOVA > > Reed, James F., I. & Stark, D. B. > Robust alternatives to traditional analyses of variance: Welch $W^*$, > James $J_I^*$, James $J_II^*$, and Brown-Forsythe $BF^*$ > Computer Methods and Programs in Biomedicine, 1988, 26, 233-238 > > > > > Iasonas Lamprianou <[hidden email]> > Sent by: [hidden email] > 08/30/2010 04:05 PM > > To > [hidden email] > cc > > Subject > [R] Brown-Forsythe test of equality of MEANS > > > > > > > > Dear friends, > two years ago (as I found on the web) Paul sent the following message but > I was not able to find if he got an answer. Today I have the same question > and it would be great if I could find out that this test has been > implemented (somehow) in R. Please do not confuse it with the > Brown-Forsythe test of equality of variances. Thank you: > > I've been searching around for a function for computing the Brown-Forsythe > F* statistic which is a substitute for the normal ANOVA F statistic for > when there are unequal variances, and when there is evidence of > non-normality. A couple of other people have asked this question, the > responses I found have been: > >    ?oneway.test > > However, that function appears to use the Welch W statistic which, while > good at handling unequal variances, is not as good as F* at handling > non-normal distributions (or so my textbook tells me). So, two questions: > >   1. Is there a function ready to use for calculating the Brown-Forsythe > F*? >   2. If not, what do people use for checking the results of a (one-way) > ANOVA when there is non-normality as well as non-constant variances? > > Thanks, > > > Dr. Iasonas Lamprianou > > > Assistant Professor (Educational Research and Evaluation) > Department of Education Sciences > European University-Cyprus > P.O. Box 22006 > 1516 Nicosia > Cyprus > Tel.: +357-22-713178 > Fax: +357-22-590539 > > > Honorary Research Fellow > Department of Education > The University of Manchester > Oxford Road, Manchester M13 9PL, UK > Tel. 0044  161 275 3485 > [hidden email] > > > > > ______________________________________________ > [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. > > >        [[alternative HTML version deleted]] > > ______________________________________________ > [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. >         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Brown-Forsythe test of equality of MEANS

 Combining humor with useful information is nice! Thank you for the advice Bert. I'll try it and see what happens. Are you aware of any free! (on the internet) alternatives to the book you have recommended? Academics are not, usually, known for their deep pockets! Thank you again for the suggestions Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] --- On Tue, 31/8/10, Bert Gunter <[hidden email]> wrote: From: Bert Gunter <[hidden email]> Subject: Re: [R] Brown-Forsythe test of equality of MEANS To: [hidden email] Cc: "Iasonas Lamprianou" <[hidden email]>, [hidden email], [hidden email] Date: Tuesday, 31 August, 2010, 21:41 Learned Folks:   Well, I've already advertised my ignorance about these matters, so I have nothing to lose by plunging ahead with further Questionable "advice."   From the references cited, Brown-Forsythe originated in the statistical medieval age -- that is, prior to large scale, cheap computing (to be honest, I have dim memories of it in BMD!). Then cameth Brad Efron and the enlightenment: If you are concerned about the distribution of this -- or indeed any reasonably smooth statistic (and some not so smooth: Hinkley - Davison's Bootstrap book has details) -- then bootstrap it. That is, get a confidence interval for the difference in means and see whether 0 falls within (or whatever Null you wish to test). If you are concerned about robustness (whatever that means in this context), well, gosheth! -- we have journeyed a long way since the 1970's. Indeed, there are several packages (e.g. robust, robustbase) with lots of robust alternatives. Most (maybe all?) of which can be bootstrapped, of course.   So walketh with thy computers, my brethren, and enter the great age of enlightenment.   (Thus endeth the lesson. Caveat Emptor!)   Cheers to all, Bert On Tue, Aug 31, 2010 at 12:26 PM, <[hidden email]> wrote: The following reference that contains a short Fortran program for the Brown-Forsythe ANOVA Reed, James F., I. & Stark, D. B. Robust alternatives to traditional analyses of variance: Welch $W^*$, James $J_I^*$, James $J_II^*$, and Brown-Forsythe $BF^*$ Computer Methods and Programs in Biomedicine, 1988, 26, 233-238 Iasonas Lamprianou <[hidden email]> Sent by: [hidden email] 08/30/2010 04:05 PM To [hidden email] cc Subject [R] Brown-Forsythe test of equality of MEANS Dear friends, two years ago (as I found on the web) Paul sent the following message but I was not able to find if he got an answer. Today I have the same question and it would be great if I could find out that this test has been implemented (somehow) in R. Please do not confuse it with the Brown-Forsythe test of equality of variances. Thank you: I've been searching around for a function for computing the Brown-Forsythe F* statistic which is a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality. A couple of other people have asked this question, the responses I found have been:    ?oneway.test However, that function appears to use the Welch W statistic which, while good at handling unequal variances, is not as good as F* at handling non-normal distributions (or so my textbook tells me). So, two questions:   1. Is there a function ready to use for calculating the Brown-Forsythe F*?   2. If not, what do people use for checking the results of a (one-way) ANOVA when there is non-normality as well as non-constant variances? Thanks, Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.        [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.               [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
 In reply to this post by Bert Gunter Yes. I too remember BMD, BMDP, punch cards, computer printouts, and 24-hour turn-around.  However, I am also old enough that when someone asks for an antique method, I will gladly supply it if I can, if only for historical reasons. Bert Gunter <[hidden email]> 08/31/2010 04:41 PM To [hidden email] cc Iasonas Lamprianou <[hidden email]>, [hidden email], [hidden email] Subject Re: [R] Brown-Forsythe test of equality of MEANS Learned Folks:   Well, I've already advertised my ignorance about these matters, so I have nothing to lose by plunging ahead with further Questionable "advice."   >From the references cited, Brown-Forsythe originated in the statistical medieval age -- that is, prior to large scale, cheap computing (to be honest, I have dim memories of it in BMD!). Then cameth Brad Efron and the enlightenment: If you are concerned about the distribution of this -- or indeed any reasonably smooth statistic (and some not so smooth: Hinkley - Davison's Bootstrap book has details) -- then bootstrap it. That is, get a confidence interval for the difference in means and see whether 0 falls within (or whatever Null you wish to test). If you are concerned about robustness (whatever that means in this context), well, gosheth! -- we have journeyed a long way since the 1970's. Indeed, there are several packages (e.g. robust, robustbase) with lots of robust alternatives. Most (maybe all?) of which can be bootstrapped, of course.   So walketh with thy computers, my brethren, and enter the great age of enlightenment.   (Thus endeth the lesson. Caveat Emptor!)   Cheers to all, Bert On Tue, Aug 31, 2010 at 12:26 PM, <[hidden email]> wrote: The following reference that contains a short Fortran program for the Brown-Forsythe ANOVA Reed, James F., I. & Stark, D. B. Robust alternatives to traditional analyses of variance: Welch $W^*$, James $J_I^*$, James $J_II^*$, and Brown-Forsythe $BF^*$ Computer Methods and Programs in Biomedicine, 1988, 26, 233-238 Iasonas Lamprianou <[hidden email]> Sent by: [hidden email] 08/30/2010 04:05 PM To [hidden email] cc Subject [R] Brown-Forsythe test of equality of MEANS Dear friends, two years ago (as I found on the web) Paul sent the following message but I was not able to find if he got an answer. Today I have the same question and it would be great if I could find out that this test has been implemented (somehow) in R. Please do not confuse it with the Brown-Forsythe test of equality of variances. Thank you: I've been searching around for a function for computing the Brown-Forsythe F* statistic which is a substitute for the normal ANOVA F statistic for when there are unequal variances, and when there is evidence of non-normality. A couple of other people have asked this question, the responses I found have been:    ?oneway.test However, that function appears to use the Welch W statistic which, while good at handling unequal variances, is not as good as F* at handling non-normal distributions (or so my textbook tells me). So, two questions:   1. Is there a function ready to use for calculating the Brown-Forsythe F*?   2. If not, what do people use for checking the results of a (one-way) ANOVA when there is non-normality as well as non-constant variances? Thanks, Dr. Iasonas Lamprianou Assistant Professor (Educational Research and Evaluation) Department of Education Sciences European University-Cyprus P.O. Box 22006 1516 Nicosia Cyprus Tel.: +357-22-713178 Fax: +357-22-590539 Honorary Research Fellow Department of Education The University of Manchester Oxford Road, Manchester M13 9PL, UK Tel. 0044  161 275 3485 [hidden email] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.        [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.