Re: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata

From: Mike Marchywka <marchywka_at_hotmail.com>
Date: Sun, 22 May 2011 11:42:25 -0400



> From: marchywka_at_hotmail.com
> To: rvaradhan@jhmi.edu; pdalgd@gmail.com; alex.olssen@gmail.com
> Date: Sat, 21 May 2011 20:40:44 -0400
> CC: r-help_at_r-project.org
> Subject: Re: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata
>
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> ----------------------------------------
> > From: rvaradhan_at_jhmi.edu
> > To: marchywka_at_hotmail.com; pdalgd_at_gmail.com; alex.olssen_at_gmail.com
> > CC: r-help_at_r-project.org
> > Date: Sat, 21 May 2011 17:26:29 -0400
> > Subject: RE: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata
> >
> > Hi,
> >
> > I don't think the final verdict has been spoken. Peter's posts have hinted at ill-conditioning as the crux of the problem. So, I decided to try a couple of more things: (1) standardizing the covariates, (2) exact gradient, and (3) both (1) and (2).
>
> Cool, I thought everyone lost interest. I'll get back to this then.
> Before launching into this, I was curious however if the STATA
> solution ( or whatever other proudct was used) was thought
> to reprsent a good solution or the actual global optimum.
>
> IIRC, your earlier post along these lines,
>
> > # exact solution as the starting value
> > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
>
> was what got me started.
>
>
> I guess my point was that the problem would not obviously
> have a nice surface to optimize and IIRC the title of the paper
> suggested the system was a bit difficult ( I won't pay for med papers,
> not going to pay for econ LOL). The function to be minimized, and this
> is stated as fact but only for sake of eliciting criticism,
> is the determinant of 2 unrelated residuals vectors. And, from memory,
>
> > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
>
>
>
> this gives something like E=|e1|^2*|e2|^2*(1-Cos^2(d)) with
> d being angle between residual vectors ( in space with dimension of number
> of data points). Or, E=F*Sin^2(d) and depending on data is would seem
> possible to move in such a way that F increases but just more slowly than
> Sin^2(d). Any solution for which they are colinear would seem to be optimal.
>
> I guess if my starting point is not too far off I may see if
> I can find some diagnostics to determine is the data set creates
> a condition as I have outlined and optimizer.
>
> It might, for example, be interesting to see what happens to |e1||e2|
> and Sin(d) at various points.
> ( T1 is just theta components 1-4 and T2 is 5-8, I guess presuming "x0"=1 LOL),
> E1=Y1-T1*X , E2=Y2-T2*x
> E1.E2=Y1.Y2-Y1.(T2*X)-(T1*X).Y2+(T1*X).(T2*X)
>
> I guess I keep working on the above and see if it points
> to anything pathological or that doesn't play nice with
> optimizer.

I was too lazy to continue the above on paper but empirically this seems to be part of symptoms. For example,

e1m=sum(e1*e1);
e2m=sum(e2*e2);
e12=sum(e1*e2);

phit=e12/sqrt(e1m*e2m);
dete=det(sigma);
xxx=paste("e1m*e2m=",e1m*e2m," aphi=",phit, " dete=",dete,sep=""); print(xxx);

on this call,

print("doing BGFS from zero");
p <- optim(0*c(1:8), lnl, method="BFGS", hessian=TRUE, y1=data$y1, y2=data$y2, x 1=data$x1, x2=data$x2, x3=data$x3)

shows that the the product of e1 and e2 explodes with the error vectors become parallel, ( since I got an answer I like, I haven't bothered to check for typos or coding errors LOL note in particular things like sqrt() etc )

[1] "e1m*e2m=41.1669479546532 aphi=0.960776458479452 dete=3.16609220303528"
[1] "e1m*e2m=41.0289397617336 aphi=0.960734505863182 dete=3.15878562836847"
[1] "e1m*e2m=41.3051898313859 aphi=0.960818247708638 dete=3.17340730403131"
[1] "e1m*e2m=39.7867410499191 aphi=0.960397122114274 dete=3.08893784983331"
[1] "e1m*e2m=42.5708634641545 aphi=0.961141202950479 dete=3.24422282329344"
[1] "e1m*e2m=39.952815544519 aphi=0.960383767149008 dete=3.10285630456706"
[1] "e1m*e2m=42.3994556719457 aphi=0.961155360764465 dete=3.23000632577225"
[1] "e1m*e2m=40.6077431942131 aphi=0.960502159746792 dete=3.14448500444145"
[1] "e1m*e2m=41.7300849780045 aphi=0.96104579811445 dete=3.18780183350788"
[1] "e1m*e2m=40.8461025929321 aphi=0.960565688832973 dete=3.15795749888956"
[1] "e1m*e2m=41.4890962339491 aphi=0.960985035909235 dete=3.17423784875441"
[1] "e1m*e2m=38.0026479775113 aphi=0.958424486959574 dete=3.09427071121483"
[1] "e1m*e2m=44.4634366247632 aphi=0.962889636822005 dete=3.23887444893151"
[1] "e1m*e2m=38.3679152097842 aphi=0.958864175571152 dete=3.09166714763783"
[1] "e1m*e2m=44.0684332841423 aphi=0.962518076859826 dete=3.24162775623887"
[1] "e1m*e2m=39.8519719942113 aphi=0.960141707858905 dete=3.11355091583515"
[1] "e1m*e2m=42.5038484736989 aphi=0.961384528172175 dete=3.21923251471042"
[1] "e1m*e2m=21375239704009490432 aphi=0.999977697068197 dete=953450394271031"
[1] "e1m*e2m=34177731168061536 aphi=0.999977905000934 dete=1510297191323.93"
[1] "e1m*e2m=54503403593545 aphi=0.999978923036089 dete=2297508328.68597"
[1] "e1m*e2m=85767789487.156 aphi=0.999983466344755 dete=2836086.67942723"
[1] "e1m*e2m=126124258.261637 aphi=0.999991704142303 dete=2092.60911721655"
[1] "e1m*e2m=127695.808241124 aphi=0.999539670386456 dete=117.537264947809"
[1] "e1m*e2m=1.79314748193414 aphi=0.458869014722204 dete=1.41558096262301"
[1] "e1m*e2m=1.82076399719734 aphi=0.469891415967073 dete=1.41874305229269"
[1] "e1m*e2m=1.76630916851626 aphi=0.447604317206231 dete=1.41242978935563"
[1] "e1m*e2m=2.10765446888618 aphi=0.559484707051066 dete=1.44790985442968"
[1] "e1m*e2m=1.55759367420566 aphi=0.33387034624138 dete=1.38396962928268"
[1] "e1m*e2m=2.07345917255291 aphi=0.547539248673881 dete=1.45183771159372"
[1] "e1m*e2m=1.57402829646691 aphi=0.351106401124671 dete=1.37998884867053"
[1] "e1m*e2m=1.92570387464855 aphi=0.500413807670452 dete=1.44348070520072"
[1] "e1m*e2m=1.67368610942138 aphi=0.413157386356889 dete=1.38798952087868"
[1] "e1m*e2m=1.80091875441907 aphi=0.458998972457491 dete=1.42150108909529"
[1] "e1m*e2m=1.78539325145298 aphi=0.458738462509367 dete=1.40967335131397"
[1] "e1m*e2m=1.87239816244392 aphi=0.46015903715983 dete=1.4759247054976"
[1] "e1m*e2m=1.71562581348667 aphi=0.457519283487653 dete=1.3565043362516"
[1] "e1m*e2m=1.86276444029005 aphi=0.460322529304816 dete=1.46805055851996"
[1] "e1m*e2m=1.72487059095196 aphi=0.457356617377108 dete=1.36407065493321"
[1] "e1m*e2m=1.82498046816382 aphi=0.460003447183359 dete=1.43880881331975"
[1] "e1m*e2m=1.76160126627035 aphi=0.457713483547811 dete=1.39254292425402"
[1] "e1m*e2m=1351553015129414 aphi=-0.999994920076885 dete=13731535927.7031"
[1] "e1m*e2m=2152851612889.67 aphi=-0.999994409734947 dete=24069954.9936224"
[1] "e1m*e2m=3367962636.07941 aphi=-0.999991168164692 dete=59490.3199496781"
[1] "e1m*e2m=4794712.10051354 aphi=-0.999956405381663 dete=418.038175816592"
[1] "e1m*e2m=3699.85578987196 aphi=-0.998908076827754 dete=8.07550521781124"
[1] "e1m*e2m=12.6686499822451 aphi=0.956985935569274 dete=1.06642059359798"
[1] "e1m*e2m=12.7330364607788 aphi=0.957056380194584 dete=1.07012366722186"
[1] "e1m*e2m=12.6044297074752 aphi=0.95691501854723 dete=1.0627254405059"

but what is interesting is that each time it emerges, it seems to get a better plateau.

If you actually look at something like Poincare map, it is interesting that aphi heads to +/-1 as the error magnitude product gets large which is about what I motivated. There seem to be a cluster of points around aphi=-.5 with minimal product. Using scatterplot3d on the above with log for magnitudes there seem to be "paths" but interpreation is not immediately clear. I admit at this point an interactive viewer would be helpful to rotate the thing, and in fact rgl.surface may be useful ...

>
>
>
>
> >
> > I compute the "exact" gradient using a complex-step derivative approach. This works just like the standard first-order, forward differencing. The only (but, essential) difference is that an imaginary increment, i*dx, is used. This, incredibly, gives exact gradients (up to machine precision).
> >
> > Here are the code and the results of my experiments:
> >
> > data <- read.table("h:/computations/optimx_example.dat", header=TRUE, sep=",")
> > attach(data)
> > require(optimx)
> >
> > ## model 18
> > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > n <- length(y1)
> > beta <- theta[1:8]
> > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > e <- cbind(e1, e2)
> > sigma <- t(e)%*%e
> > det.sigma <- sigma[1,1] * sigma[2,2] - sigma[1,2] * sigma[2,1]
> > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det.sigma))
> > return(-logl)
> > }
> >
> > csd <- function(fn, x, ...) {
> > # Complex step derivatives; yields exact derivatives
> > h <- .Machine$double.eps
> > n <- length(x)
> > h0 <- g <- rep(0, n)
> > for (i in 1:n) {
> > h0[i] <- h * 1i
> > g[i] <- Im(fn(x+h0, ...))/h
> > h0[i] <- 0
> > }
> > g
> > }
> >
> > gr.csd <- function(theta,y1, y2, x1, x2, x3) {
> > csd(lnl, theta, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3)
> > }
> >
> > # exact solution as the starting value
> > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > p1 <- optimx(start, lnl, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> >
> > # numerical gradient
> > p2 <- optimx(rep(0,8), lnl, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> >
> > # exact gradient
> > p2g <- optimx(rep(0,8), lnl, gr.csd, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> >
> > # comparing p2 and p2g, we see the dramatic improvement in BFGS when exact gradient is used, we also see a major difference for L-BFGS-B
> > # Exact gradient did not affect the gradient methods, CG and spg, much. However, convergence of Rcgmin improved when exact gradient was used
> >
> > x1s <- scale(x1)
> > x2s <- scale(x2)
> > x3s <- scale(x3)
> >
> > p3 <- optimx(rep(0,8),lnl, y1=y1, y2=y2, x1=x1s, x2=x2s, x3=x3s, control=list(all.methods=TRUE, maxit=1500))
> >
> > # both scaling and exact gradient
> > p3g <- optimx(rep(0,8),lnl, gr.csd, y1=y1, y2=y2, x1=x1s, x2=x2s, x3=x3s, control=list(all.methods=TRUE, maxit=1500))
> >
> > # Comparing p3 and p3g, use of exact gradient improved spg dramatically[[elided Hotmail spam]]
> >
> > # Of course, derivative-free methods newuoa, and Nelder-Mead are improved by scaling, but not by the availability of exact gradients. I don't know what is wrong with bobyqa in this example.
> >
> > In short, even with scaling and exact gradients, this optimization problem is recalcitrant.
> >
> > Best,
> > Ravi.
> >
> > ________________________________________
> > From: Mike Marchywka [marchywka_at_hotmail.com]
> > Sent: Thursday, May 12, 2011 8:30 AM
> > To: Ravi Varadhan; pdalgd_at_gmail.com; alex.olssen_at_gmail.com
> > Cc: r-help_at_r-project.org
> > Subject: RE: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata
> >
> > So what was the final verdict on this discussion? I kind of
> > lost track if anyone has a minute to summarize and critique my summary below.
> >
> >
> > Apparently there were two issues, the comparison between R and Stata
> > was one issue and the "optimum" solution another. As I understand it,
> > there was some question about R numerical gradient calculation. This would
> > suggest some features of the function may be of interest to consider.
> >
> > The function to be optimized appears to be, as OP stated,
> > some function of residuals of two ( unrelated ) fits.
> > The residual vectors e1 and e2 are dotted in various combinations
> > creating a matrix whose determinant is (e1.e1)(e2.e2)-(e1.e2)^2 which
> > is the result to be minimized by choice of theta. Theta it seems is
> > an 8 component vector, 4 components determine e1 and the other 4 e2.
> > Presumably a unique solution would require that e1 and e2, both n-component vectors,
> > point in different directions or else both could become aribtarily large
> > while keeping the error signal at zero. For fixed magnitudes, colinearity
> > would reduce the "Error." The intent would appear to be to
> > keep the residuals distributed similarly in the two ( unrelated) fits.
> > I guess my question is,
> > " did anyone determine that there is a unique solution?" or
> > am I totally wrong here ( I haven't used these myself to any
> > extent and just try to run some simple teaching examples, asking
> > for my own clarification as much as anything).
> >
> > Thanks.
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > ----------------------------------------
> > > From: rvaradhan_at_jhmi.edu
> > > To: pdalgd_at_gmail.com; alex.olssen_at_gmail.com
> > > Date: Sat, 7 May 2011 11:51:56 -0400
> > > CC: r-help_at_r-project.org
> > > Subject: Re: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata
> > >
> > > There is something strange in this problem. I think the log-likelihood is incorrect. See the results below from "optimx". You can get much larger log-likelihood values than for the exact solution that Peter provided.
> > >
> > > ## model 18
> > > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > > n <- length(y1)
> > > beta <- theta[1:8]
> > > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > > e <- cbind(e1, e2)
> > > sigma <- t(e)%*%e
> > > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det(sigma))) # it looks like there is something wrong here
> > > return(-logl)
> > > }
> > >
> > > data <- read.table("e:/computing/optimx_example.dat", header=TRUE, sep=",")
> > >
> > > attach(data)
> > >
> > > require(optimx)
> > >
> > > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > >
> > > # the warnings can be safely ignored in the "optimx" calls
> > > p1 <- optimx(start, lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > p2 <- optimx(rep(0,8), lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > p3 <- optimx(rep(0.5,8), lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > Ravi.
> > > ________________________________________
> > > From: r-help-bounces_at_r-project.org [r-help-bounces_at_r-project.org] On Behalf Of peter dalgaard [pdalgd_at_gmail.com]
> > > Sent: Saturday, May 07, 2011 4:46 AM
> > > To: Alex Olssen
> > > Cc: r-help_at_r-project.org
> > > Subject: Re: [R] maximum likelihood convergence reproducing Anderson Blundell 1982 Econometrica R vs Stata
> > >
> > > On May 6, 2011, at 14:29 , Alex Olssen wrote:
> > >
> > > > Dear R-help,
> > > >
> > > > I am trying to reproduce some results presented in a paper by Anderson
> > > > and Blundell in 1982 in Econometrica using R.
> > > > The estimation I want to reproduce concerns maximum likelihood
> > > > estimation of a singular equation system.
> > > > I can estimate the static model successfully in Stata but for the
> > > > dynamic models I have difficulty getting convergence.
> > > > My R program which uses the same likelihood function as in Stata has
> > > > convergence properties even for the static case.
> > > >
> > > > I have copied my R program and the data below. I realise the code
> > > > could be made more elegant - but it is short enough.
> > > >
> > > > Any ideas would be highly appreciated.
> > >
> > > Better starting values would help. In this case, almost too good values are available:
> > >
> > > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > >
> > > which appears to be the _exact_ solution.
> > >
> > > Apart from that, it seems that the conjugate gradient methods have difficulties with this likelihood, for some less than obvious reason. Increasing the maxit gets you closer but still not satisfactory.
> > >
> > > I would suggest trying out the experimental optimx package. Apparently, some of the algorithms in there are much better at handling this likelihood, notably "nlm" and "nlminb".
> > >
> > >
> > >
> > >
> > > >
> > > > ## model 18
> > > > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > > > n <- length(y1)
> > > > beta <- theta[1:8]
> > > > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > > > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > > > e <- cbind(e1, e2)
> > > > sigma <- t(e)%*%e
> > > > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det(sigma)))
> > > > return(-logl)
> > > > }
> > > > p <- optim(0*c(1:8), lnl, method="BFGS", hessian=TRUE, y1=y1, y2=y2,
> > > > x1=x1, x2=x2, x3=x3)
> > > >
> > > > "year","y1","y2","x1","x2","x3"
> > > > 1929,0.554779,0.266051,9.87415,8.60371,3.75673
> > > > 1930,0.516336,0.297473,9.68621,8.50492,3.80692
> > > > 1931,0.508201,0.324199,9.4701,8.27596,3.80437
> > > > 1932,0.500482,0.33958,9.24692,7.99221,3.76251
> > > > 1933,0.501695,0.276974,9.35356,7.98968,3.69071
> > > > 1934,0.591426,0.287008,9.42084,8.0362,3.63564
> > > > 1935,0.565047,0.244096,9.53972,8.15803,3.59285
> > > > 1936,0.605954,0.239187,9.6914,8.32009,3.56678
> > > > 1937,0.620161,0.218232,9.76817,8.42001,3.57381
> > > > 1938,0.592091,0.243161,9.51295,8.19771,3.6024
> > > > 1939,0.613115,0.217042,9.68047,8.30987,3.58147
> > > > 1940,0.632455,0.215269,9.78417,8.49624,3.57744
> > > > 1941,0.663139,0.184409,10.0606,8.69868,3.6095
> > > > 1942,0.698179,0.164348,10.2892,8.84523,3.66664
> > > > 1943,0.70459,0.146865,10.4731,8.93024,3.65388
> > > > 1944,0.694067,0.161722,10.4465,8.96044,3.62434
> > > > 1945,0.674668,0.197231,10.279,8.82522,3.61489
> > > > 1946,0.635916,0.204232,10.1536,8.77547,3.67562
> > > > 1947,0.642855,0.187224,10.2053,8.77481,3.82632
> > > > 1948,0.641063,0.186566,10.2227,8.83821,3.96038
> > > > 1949,0.646317,0.203646,10.1127,8.82364,4.0447
> > > > 1950,0.645476,0.187497,10.2067,8.84161,4.08128
> > > > 1951,0.63803,0.197361,10.2773,8.9401,4.10951
> > > > 1952,0.634626,0.209992,10.283,9.01603,4.1693
> > > > 1953,0.631144,0.219287,10.3217,9.06317,4.21727
> > > > 1954,0.593088,0.235335,10.2101,9.05664,4.2567
> > > > 1955,0.60736,0.227035,10.272,9.07566,4.29193
> > > > 1956,0.607204,0.246631,10.2743,9.12407,4.32252
> > > > 1957,0.586994,0.256784,10.2396,9.1588,4.37792
> > > > 1958,0.548281,0.271022,10.1248,9.14025,4.42641
> > > > 1959,0.553401,0.261815,10.2012,9.1598,4.4346
> > > > 1960,0.552105,0.275137,10.1846,9.19297,4.43173
> > > > 1961,0.544133,0.280783,10.1479,9.19533,4.44407
> > > > 1962,0.55382,0.281286,10.197,9.21544,4.45074
> > > > 1963,0.549951,0.28303,10.2036,9.22841,4.46403
> > > > 1964,0.547204,0.291287,10.2271,9.23954,4.48447
> > > > 1965,0.55511,0.281313,10.2882,9.26531,4.52057
> > > > 1966,0.558182,0.280151,10.353,9.31675,4.58156
> > > > 1967,0.545735,0.294385,10.3351,9.35382,4.65983
> > > > 1968,0.538964,0.294593,10.3525,9.38361,4.71804
> > > > 1969,0.542764,0.299927,10.3676,9.40725,4.76329
> > > > 1970,0.534595,0.315319,10.2968,9.39139,4.81136
> > > > 1971,0.545591,0.315828,10.2592,9.34121,4.84082
> > > >
> > > > ______________________________________________
> > > > R-help_at_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.
> > >
> > > --
> > > Peter Dalgaard
> > > Center for Statistics, Copenhagen Business School
> > > Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> > > Phone: (+45)38153501
> > > Email: pd.mes_at_cbs.dk Priv: PDalgd_at_gmail.com
> > >
> > > ______________________________________________
> > > R-help_at_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_at_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_at_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.
                                               



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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. Received on Sun 22 May 2011 - 15:44:34 GMT

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