[R] Naming a random effect in lmer
Douglas Bates
bates at stat.wisc.edu
Mon May 25 20:22:33 CEST 2009
On Sun, May 24, 2009 at 10:41 PM, Leigh Ann Starcevich <lah at peak.org> wrote:
> Thanks for your thoughts on this. I tried the approach for the grouping
> variable using the "within" function. I looked at a subset of my data for
> which I do not get the deparse error in lmer and compared the results. The
> approach using the "within" function to form the grouping variable
> underestimates that variance component.
But what you tried is not what I wrote. The paste function creates a
character string - that's all. You could create a numeric variable or
a factor by parsing and evaluating the resulting character string but
that is rarely a good way of doing things. See
library(fortunes)
fortune("rethink")
What I wrote was to evaluate the expression
Z29 <- Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z9 + Z9
within the data frame and that does work. See the attached.
I still don't really understand the approach. It is going about
things in an awkward way and I suspect it is the result of someone
with experience in SAS or SPSS trying to emulate their approach in R.
It is (or was, the last time I used either of those systems) common to
name variables like Z2, Z3, Z4, ... so you can access groups of
variables in those languages. In R that is unnecessary because data
can be packaged into arbitrary, self-describing structures, including
matrices. If the natural thing to do with the variables Z2, Z3, ...,
Z13 is to accumulate them in sums then it would be much simpler if you
create a matrix from them and use that.
> Strangely, the Zt matrix for the
> random effects in the proposed approach does not appear to contain rows for
> the Zgrouped variable, although it does report a variance estimate for the
> random effect. The results for the standard approach appear correct for
> this simulated data set. Any other ideas on how to make the grouping work?
>
> Thanks -- Leigh Ann
>
>
> # Assume that only mb=10 years of data are available to compare to results
> # that avoid deparse error
>
> # Group the grouping variable using within
> testsamp1<-testsamp
> Zs2<- paste("Z",2:9,sep="")
> Zsum2 <- paste(Zs2,collapse="+")
> testsamp1 <- within(testsamp1, Zgrouped<-Zsum2)
> fittest29.1<-lmer(LogY ~ WYear + (1+WYear|Site) + (1|Zgrouped), data =
> testsamp1[testsamp1$WYear<=10,])
>
> Linear mixed model fit by REML
> Formula: LogY ~ WYear + (1 + WYear | Site) + (1 | Zgrouped)
> Data: testsamp1[testsamp1$WYear <= 10, ]
> AIC BIC logLik deviance REMLdev
> -42.96 -26.55 28.48 -63.37 -56.96
> Random effects:
> Groups Name Variance Std.Dev. Corr
> Site (Intercept) 0.07351401 0.271135
> WYear 0.02620422 0.161877 0.057
> Zgrouped (Intercept) 0.00036891 0.019207
> Residual 0.01065189 0.103208
> Number of obs: 77, groups: Site, 7; Zgrouped, 1
>
> Fixed effects:
> Estimate Std. Error t value
> (Intercept) 0.9857992 0.1065573 9.251
> WYear -0.0002676 0.0612968 -0.004
>
> Correlation of Fixed Effects:
> (Intr)
> WYear 0.044
>
> # Use the standard formulation for the grouping variable with reduced number
> of
> # terms to avoid deparse error
>
> Trendformula2 <-as.formula(paste("LogY ~ WYear + (1+WYear|Site) + (1|",
> randommodel=paste(paste(Zs2,collapse="+"), ")")))
> fittest29.2<-lmer(Trendformula2, data = testsamp[testsamp$WYear<=10,])
> summary(fittest29.2)
>
> Linear mixed model fit by REML
> Formula: Trendformula2
> Data: testsamp[testsamp$WYear <= 10, ]
> AIC BIC logLik deviance REMLdev
> -147.5 -131.1 80.77 -167.7 -161.5
> Random effects:
> Groups Name Variance Std.Dev. Corr
> Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 (Intercept) 0.0095237 0.097589
> Site (Intercept) 0.0765445 0.276667
> WYear 0.0262909 0.162145 0.046
> Residual 0.0011282 0.033589
> Number of obs: 77, groups: Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9, 11; Site,
> 7
>
> Fixed effects:
> Estimate Std. Error t value
> (Intercept) 0.9857992 0.1183912 8.327
> WYear -0.0002676 0.0619989 -0.004
>
> Correlation of Fixed Effects:
> (Intr)
> WYear -0.020
>
>> fittest29.1 at Zt
> 15 x 77 sparse Matrix of class "dgCMatrix"
>> fittest29.2 at Zt
> 25 x 77 sparse Matrix of class "dgCMatrix"
>
>
>
>
>
>
> At 09:37 AM 5/24/2009 -0500, Douglas Bates wrote:
>
> Hi Bill,
>
> I'm about to take a look at this. If I understand the issue, very
> long expressions for what I call the "grouping factor" of a random
> effects term (the expressions on the right hand side of the vertical
> bar) are encountering problems with deparse. I should have realized
> that, any time one uses deparse, disaster looms.
>
> I can tell you the reason that the collection of random-effects terms
> is being named is partly for the printed form and partly so that terms
> with the same grouping factor can be associated.
>
> I guess my simplistic solution to the problem would be to precompute
> these sums and give them names, if it is the sum like
>
> Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9
>
> that is important, why not evaluate the sum
>
> testGroupSamp <- within(testGroupSamp, Z29 <- Z2 + Z3 + Z4 + Z5 + Z6 +
> Z7 + Z8 + Z9)
>
> and use Z29 as the grouping factor.
>
> Even the use of variables with names like Z1, Z2, ... and the use of
> expressions like paste("Z", 2:9, sep = "") is not idiomatic R/S code.
> It's an SPSS/SASism. (You know I never realized before how close the
> word "SASism", meaning a construction that is natural in SAS, is to
> "Sadism".) Why not create a matrix Z and evaluate these sums as
> matrix/vector products?
>
>
> Zs2<- paste("Z",2:9,sep="")
>
>
> On Fri, May 22, 2009 at 5:30 PM, William Dunlap <wdunlap at tibco.com> wrote:
>>> -----Original Message-----
>>> From: r-help-bounces at r-project.org
>>> [mailto:r-help-bounces at r-project.org] On Behalf Of spencerg
>>> Sent: Friday, May 22, 2009 3:01 PM
>>> To: Leigh Ann Starcevich
>>> Cc: r-help at r-project.org
>>> Subject: Re: [R] Naming a random effect in lmer
>>>
>>> [ ... elided statistical advice ... ]
>>> If you and your advisor still feel that what you are doing
>>> makes sense,
>>> I suggest you first get the source code via "svn checkout
>>> svn://svn.r-forge.r-project.org/svnroot/lme4" (or by downloading
>>> "lme4_0.999375-30.tar.gz" from
>>> "http://cran.fhcrc.org/web/packages/lme4/index.html"), then
>>> walk through
>>> the code line by line using the "debug" function (or "browser" or the
>>> "debug" package). From this, you will likely see either (a)
>>> how you can
>>> do what you want differently to achieve the same result or (b) how to
>>> modify the code so it does what you want.
>>
>> The coding error is right in the error message:
>> Error in names(bars) <- unlist(lapply(bars, function(x)
>> deparse(x[[3]])))
>> and I suspect that traceback() would tell you that came from a call
>> to lmerFactorList.
>>
>> That code implicitly assumes that deparse() will produce a scalar
>> character
>> vector, but it doesn't if the input expression is complicated enough.
>> Changing the
>> deparse(x[[3]])
>> to
>> deparse(x[[3]])[1]
>> or
>> paste(collapse=" ", deparse(x[[3]])[1])
>> would fix it. The first truncates the name and the second my make a
>> very
>> long name.
>>
>> There is at least one other use of that idiom in the lme4 code and your
>> dataset and analysis may require that all of them be fixed.
>>
>>>
>>>
>>> Hope this helps.
>>> Spencer
>>>
>>>
>>> Leigh Ann Starcevich wrote:
>>> > Here is a test data set and code. I am including the data set after
>>> > the code and discussion to make reading easier. Apologies
>>> for the size
>>> > of the data set, but my problem occurs when there are a lot of Z
>>> > variables. Thanks for your time.
>>> >
>>> > # Enter data below
>>> >
>>> > # Sample code
>>> > library(lme4)
>>> > mb<- length(unique(testsamp$WYear))
>>> >
>>> > # Create the formula for the set of identically distributed random
>>> > effects
>>> > Zs<- paste("Z",2:(mb-1)),sep="")
>>> > Trendformula <-as.formula(paste("LogY ~ WYear +
>>> (1+WYear|Site) + (1|",
>>> > randommodel=paste(paste(Zs,collapse="+"), ")")))
>>> >
>>> > fittest<-lmer(Trendformula, data = testsamp)
>>> > summary(fittest)
>>> >
>>> > # Here I get an error because the name of the random effect is too
>>> > long to print
>>> > # in the random effects output (I think).
>>> > # The error message is: Error in names(bars) <- unlist(lapply(bars,
>>> > function(x)
>>> > # deparse(x[[3]]))) : 'names' attribute [3] must be the
>>> same length as
>>> > the vector [2]
>>> >
>>> > # However, when fewer Z variables are used in the random portion of
>>> > the model,
>>> > # there is no error.
>>> > # Using only Z2 + ... + Z9 for the random intercept
>>> >
>>> > Zs2<- paste("Z",2:9,sep="")
>>> > Trendformula2 <-as.formula(paste("LogY ~ WYear + (1+WYear|Site) +
>>> > (1|", randommodel=paste(paste(Zs2,collapse="+"), ")")))
>>> > fittest2<-lmer(Trendformula2, data = testsamp)
>>> > summary(fittest2)
>>> >
>>> >
>>> > # Is there a way to either name the set of iid random effects
>>> > something else or
>>> > # to define a random variable that could be used in the
>>> model to create a
>>> > # random intercept?
>>> >
>>> > # I have had some success in lme, but it would be helpful for my
>>> > simulation if I
>>> > # could conduct this analysis with lmer. My model in lme is
>>> not correctly
>>> > # estimating one of the variance components (random Site intercept).
>>> > # I am using:
>>> >
>>> > detach(package:lme4)
>>> > library(nlme)
>>> > random.model.lme
>>> >
>>> <-as.formula(paste("~-1+",paste(paste("Z",2:(mb-1),sep=""),col
>>> lapse="+")))
>>> >
>>> >
>>> > n<-dim(testsamp)[1]
>>> > testsampgroup <- rep(1,n)
>>> > testsamp.lme <-cbind(testsamp,testsampgroup)
>>> > testgroupSamp<- groupedData(LogY ~ WYearCen|testsampgroup,
>>> inner= ~Site,
>>> > data= data.frame(testsamp.lme))
>>> >
>>> > fittest3<-lme(LogY ~ WYearCen, random=
>>> > pdBlocked(list(pdIdent(~-1+WYearCen:as.factor(Site)),
>>> > pdIdent(~-1+as.factor(Site)), pdIdent(random.model.lme))),data=
>>> > testgroupSamp)
>>> > summary(fittest3)
>>> > VarCorr(fittest3)
>>> >
>>> >
>>> > # Data
>>> >
>>> > testsamp <-
>>> > structure(list(Year = c(2008, 2008, 2008, 2008, 2008, 2008, 2008,
>>> > 2009, 2009, 2009, 2009, 2009, 2009, 2009, 2010, 2010, 2010, 2010,
>>> > 2010, 2010, 2010, 2011, 2011, 2011, 2011, 2011, 2011, 2011, 2012,
>>> > 2012, 2012, 2012, 2012, 2012, 2012, 2013, 2013, 2013, 2013, 2013,
>>> > 2013, 2013, 2014, 2014, 2014, 2014, 2014, 2014, 2014, 2015, 2015,
>>> > 2015, 2015, 2015, 2015, 2015, 2016, 2016, 2016, 2016, 2016, 2016,
>>> > 2016, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2018, 2018, 2018,
>>> > 2018, 2018, 2018, 2018, 2019, 2019, 2019, 2019, 2019, 2019, 2019,
>>> > 2020, 2020, 2020, 2020, 2020, 2020, 2020, 2021, 2021, 2021, 2021,
>>> > 2021, 2021, 2021, 2022, 2022, 2022, 2022, 2022, 2022, 2022),
>>> > WYear = c(0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2,
>>> > 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4,
>>> > 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7,
>>> > 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10,
>>> > 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12,
>>> > 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14,
>>> > 14, 14), Site = c(4, 18, 26, 40, 67, 75, 94, 4, 18, 26, 40,
>>> > 67, 75, 94, 4, 18, 26, 40, 67, 75, 94, 4, 18, 26, 40, 67,
>>> > 75, 94, 4, 18, 26, 40, 67, 75, 94, 4, 18, 26, 40, 67, 75,
>>> > 94, 4, 18, 26, 40, 67, 75, 94, 4, 18, 26, 40, 67, 75, 94,
>>> > 4, 18, 26, 40, 67, 75, 94, 4, 18, 26, 40, 67, 75, 94, 4,
>>> > 18, 26, 40, 67, 75, 94, 4, 18, 26, 40, 67, 75, 94, 4, 18,
>>> > 26, 40, 67, 75, 94, 4, 18, 26, 40, 67, 75, 94, 4, 18, 26,
>>> > 40, 67, 75, 94), LogY = c(0.848648866298552, 0.809143925760456,
>>> > 0.734173952725014, 1.46749967704437, 0.716106254860468,
>>> > 0.843871512951468,
>>> > 1.09120092433378, 0.800893809796851, 0.996674977596997,
>>> > 0.917613604481207,
>>> > 1.71928772722884, 0.797853604855215, 0.922298691760041,
>>> > 0.964654422529188,
>>> > 0.782903180421921, 1.13457969553106, 1.21628917384868,
>>> 2.03084776647495,
>>> > 0.872954085910578, 1.02192794559856, 0.746307251774509,
>>> > 0.439812203188778,
>>> > 1.11164109549224, 1.18414357836729, 2.00157711459358,
>>> 0.66577753155877,
>>> > 0.856374433428581, 0.343862060001402, 0.278505653147057,
>>> > 1.20632152691478, 1.32289150679746, 2.19814430598707,
>>> 0.538363941164496,
>>> > 0.820038163290321, 0.070054765524828, 0.0479738684639024,
>>> > 1.29137364087568, 1.52436357249377, 2.32150525025777,
>>> 0.595507040392793,
>>> > 0.851417610550757, -0.115908144193410, 0.0118306018140099,
>>> > 1.39448009350962, 1.71677754106603, 2.59146662837284,
>>> 0.595750060671620,
>>> > 0.855387479311679, -0.430729785591898, 0.0178423104900579,
>>> > 1.60964246000316, 1.99184029256509, 2.86865842252168,
>>> 0.695124899993409,
>>> > 0.96175860396451, -0.600991172113926, -0.174420349224615,
>>> > 1.73794158380868, 2.06718359946362, 3.04502112974038,
>>> 0.730730638403177,
>>> > 0.961110819398807, -0.856693722990918, -0.549458074126028,
>>> > 1.52302453916110, 2.05923417821491, 2.95885487422314,
>>> 0.302432275785407,
>>> > 0.718537282886428, -1.41807557701934, -0.474811468779057,
>>> > 1.88486023048201, 2.47921637905194, 3.31076232685622,
>>> 0.600331824659821,
>>> > 0.997014319132077, -1.36760375228162, -0.491828047247449,
>>> > 2.09353379020743, 2.74721582678157, 3.58818112810645,
>>> 0.65742631492698,
>>> > 1.06027358323846, -1.58298244571648, -0.934472752299038,
>>> > 1.83456803880667, 2.62936515021684, 3.56971226362308,
>>> 0.389323036788794,
>>> > 0.748060984171013, -2.13928096660883, -1.26100797232926,
>>> > 1.83430218924218, 2.65380472944419, 3.62529901196750,
>>> 0.271765247781891,
>>> > 0.639379726021376, -2.50138714617066, -1.36434864536697,
>>> > 1.95417376162485, 2.84537426171405, 3.76029816545047,
>>> 0.171262422338166,
>>> > 0.63595411426061, -2.72648123827278), Y = c(2.33648781083889,
>>> > 2.24598443402562, 2.08375999711081, 4.33837423214822,
>>> 2.04644932501053,
>>> > 2.32535220351989, 2.97784809630864, 2.22753102803039,
>>> 2.70925849019601,
>>> > 2.50330936997246, 5.5805521729813, 2.22076916113014,
>>> 2.51506510968675,
>>> > 2.62388074570782, 2.18781467519546, 3.10986617752641,
>>> 3.37464176115807,
>>> > 7.62054406285807, 2.39397241834575, 2.77854648986394,
>>> 2.10919688492552,
>>> > 1.55241565242538, 3.03934215461328, 3.26788693363161,
>>> 7.40071868133414,
>>> > 1.94600301201132, 2.35460841009671, 1.41038407371191,
>>> 1.32115407405314,
>>> > 3.34117160999488, 3.75426116973332, 9.0082813655785,
>>> 1.71320166935217,
>>> > 2.27058648893032, 1.07256691933604, 1.04914323924183,
>>> 3.63778012876121,
>>> > 4.59222002395514, 10.1910027761430, 1.81395045797806,
>>> 2.34296591199978,
>>> > 0.89055702237416, 1.01190086017652, 4.03287730840125,
>>> 5.56656151506009,
>>> > 13.3493357676049, 1.81439133829339, 2.35228566604212,
>>> 0.650034535742804,
>>> > 1.01800243542715, 5.00102284245171, 7.32900887997844,
>>> 17.6133726381587,
>>> > 2.00395935282089, 2.61629345414488, 0.548267938797843,
>>> 0.839943753736141,
>>> > 5.68562798048777, 7.90253504257589, 21.0104751440743,
>>> 2.07659729526341,
>>> > 2.61459920843830, 0.424563488824068, 0.577262559146385,
>>> 4.58607500338397,
>>> > 7.83996349588931, 19.2758857986172, 1.35314603256567,
>>> 2.05143035272921,
>>> > 0.242179624775457, 0.622002312266844, 6.58543393476032,
>>> 11.9319106582846,
>>> > 27.4060098478603, 1.82272352466727, 2.71017801013783,
>>> 0.254716592899401,
>>> > 0.611507507191544, 8.11353609874542, 15.5991406590765,
>>> 36.1682306843681,
>>> > 1.92981919064962, 2.8871607600197, 0.20536170381795,
>>> 0.392792910103641,
>>> > 6.26242843503692, 13.8649649375266, 35.5063752059022,
>>> 1.47598127057995,
>>> > 2.11289909722918, 0.117739471204143, 0.283368255138982,
>>> 6.26076379244682,
>>> > 14.2079935097071, 37.5359451785784, 1.31227890476568,
>>> 1.89530490631499,
>>> > 0.0819712136688162, 0.2555470735606, 7.0580849532232,
>>> 17.2079977586343,
>>> > 42.961233624846, 1.18680215155599, 1.88882343537745,
>>> 0.065449185044807
>>> > ), Z2 = c(0.47227027821606, 0.47227027821606, 0.47227027821606,
>>> > 0.47227027821606, 0.47227027821606, 0.47227027821606,
>>> 0.47227027821606,
>>> > 0.269868730409177, 0.269868730409177, 0.269868730409177,
>>> > 0.269868730409177, 0.269868730409177, 0.269868730409177,
>>> > 0.269868730409177, 0.0986058822648916, 0.0986058822648916,
>>> > 0.0986058822648916, 0.0986058822648916, 0.0986058822648916,
>>> > 0.0986058822648916, 0.0986058822648916, -0.0415182662167964,
>>> > -0.0415182662167964, -0.0415182662167964, -0.0415182662167964,
>>> > -0.0415182662167964, -0.0415182662167964, -0.0415182662167964,
>>> > -0.150503715035887, -0.150503715035887, -0.150503715035887,
>>> > -0.150503715035887, -0.150503715035887, -0.150503715035887,
>>> > -0.150503715035887, -0.228350464192381, -0.228350464192381,
>>> > -0.228350464192381, -0.228350464192381, -0.228350464192381,
>>> > -0.228350464192381, -0.228350464192381, -0.275058513686277,
>>> > -0.275058513686277, -0.275058513686277, -0.275058513686277,
>>> > -0.275058513686277, -0.275058513686277, -0.275058513686277,
>>> > -0.290627863517575, -0.290627863517575, -0.290627863517575,
>>> > -0.290627863517575, -0.290627863517575, -0.290627863517575,
>>> > -0.290627863517575, -0.275058513686277, -0.275058513686277,
>>> > -0.275058513686277, -0.275058513686277, -0.275058513686277,
>>> > -0.275058513686277, -0.275058513686277, -0.228350464192381,
>>> > -0.228350464192381, -0.228350464192381, -0.228350464192381,
>>> > -0.228350464192381, -0.228350464192381, -0.228350464192381,
>>> > -0.150503715035887, -0.150503715035887, -0.150503715035887,
>>> > -0.150503715035887, -0.150503715035887, -0.150503715035887,
>>> > -0.150503715035887, -0.0415182662167965, -0.0415182662167965,
>>> > -0.0415182662167965, -0.0415182662167965, -0.0415182662167965,
>>> > -0.0415182662167965, -0.0415182662167965, 0.0986058822648916,
>>> > 0.0986058822648916, 0.0986058822648916, 0.0986058822648916,
>>> > 0.0986058822648916, 0.0986058822648916, 0.0986058822648916,
>>> > 0.269868730409177, 0.269868730409177, 0.269868730409177,
>>> > 0.269868730409177, 0.269868730409177, 0.269868730409177,
>>> > 0.269868730409177, 0.47227027821606, 0.47227027821606,
>>> 0.47227027821606,
>>> > 0.47227027821606, 0.47227027821606, 0.47227027821606,
>>> 0.47227027821606
>>> > ), Z3 = c(-0.456256435177108, -0.456256435177108,
>>> -0.456256435177108,
>>> > -0.456256435177108, -0.456256435177108, -0.456256435177108,
>>> > -0.456256435177108, -0.0651794907395872, -0.0651794907395872,
>>> > -0.0651794907395872, -0.0651794907395872, -0.0651794907395872,
>>> > -0.0651794907395872, -0.0651794907395872, 0.175483244298888,
>>> > 0.175483244298888, 0.175483244298888, 0.175483244298888,
>>> > 0.175483244298888, 0.175483244298888, 0.175483244298888,
>>> > 0.290800804838157, 0.290800804838157, 0.290800804838157,
>>> > 0.290800804838157, 0.290800804838157, 0.290800804838157,
>>> > 0.290800804838157, 0.305842225778062, 0.305842225778062,
>>> > 0.305842225778062, 0.305842225778062, 0.305842225778062,
>>> > 0.305842225778062, 0.305842225778062, 0.245676542018443,
>>> > 0.245676542018443, 0.245676542018443, 0.245676542018443,
>>> > 0.245676542018443, 0.245676542018443, 0.245676542018443,
>>> > 0.135372788459142, 0.135372788459142, 0.135372788459142,
>>> > 0.135372788459142, 0.135372788459142, 0.135372788459142,
>>> > 0.135372788459142, 4.80758108354045e-17, 4.80758108354045e-17,
>>> > 4.80758108354045e-17, 4.80758108354045e-17, 4.80758108354045e-17,
>>> > 4.80758108354045e-17, 4.80758108354045e-17, -0.135372788459142,
>>> > -0.135372788459142, -0.135372788459142, -0.135372788459142,
>>> > -0.135372788459142, -0.135372788459142, -0.135372788459142,
>>> > -0.245676542018443, -0.245676542018443, -0.245676542018443,
>>> > -0.245676542018443, -0.245676542018443, -0.245676542018443,
>>> > -0.245676542018443, -0.305842225778062, -0.305842225778062,
>>> > -0.305842225778062, -0.305842225778062, -0.305842225778062,
>>> > -0.305842225778062, -0.305842225778062, -0.290800804838157,
>>> > -0.290800804838157, -0.290800804838157, -0.290800804838157,
>>> > -0.290800804838157, -0.290800804838157, -0.290800804838157,
>>> > -0.175483244298888, -0.175483244298888, -0.175483244298888,
>>> > -0.175483244298888, -0.175483244298888, -0.175483244298888,
>>> > -0.175483244298888, 0.0651794907395869, 0.0651794907395869,
>>> > 0.0651794907395869, 0.0651794907395869, 0.0651794907395869,
>>> > 0.0651794907395869, 0.0651794907395869, 0.456256435177108,
>>> > 0.456256435177108, 0.456256435177108, 0.456256435177108,
>>> > 0.456256435177108, 0.456256435177108, 0.456256435177108),
>>> > Z4 = c(0.393641410895141, 0.393641410895141, 0.393641410895141,
>>> > 0.393641410895141, 0.393641410895141, 0.393641410895141,
>>> > 0.393641410895141, -0.168703461812203, -0.168703461812203,
>>> > -0.168703461812203, -0.168703461812203, -0.168703461812203,
>>> > -0.168703461812203, -0.168703461812203, -0.341732653414463,
>>> > -0.341732653414463, -0.341732653414463, -0.341732653414463,
>>> > -0.341732653414463, -0.341732653414463, -0.341732653414463,
>>> > -0.276846706563615, -0.276846706563615, -0.276846706563615,
>>> > -0.276846706563615, -0.276846706563615, -0.276846706563615,
>>> > -0.276846706563615, -0.0979187925203697, -0.0979187925203697,
>>> > -0.0979187925203697, -0.0979187925203697, -0.0979187925203697,
>>> > -0.0979187925203697, -0.0979187925203697, 0.0987052888458345,
>>> > 0.0987052888458345, 0.0987052888458345, 0.0987052888458345,
>>> > 0.0987052888458345, 0.0987052888458345, 0.0987052888458345,
>>> > 0.244207109056825, 0.244207109056825, 0.244207109056825,
>>> > 0.244207109056825, 0.244207109056825, 0.244207109056825,
>>> > 0.244207109056825, 0.297295611025701, 0.297295611025701,
>>> > 0.297295611025701, 0.297295611025701, 0.297295611025701,
>>> > 0.297295611025701, 0.297295611025701, 0.244207109056826,
>>> > 0.244207109056826, 0.244207109056826, 0.244207109056826,
>>> > 0.244207109056826, 0.244207109056826, 0.244207109056826,
>>> > 0.0987052888458344, 0.0987052888458344, 0.0987052888458344,
>>> > 0.0987052888458344, 0.0987052888458344, 0.0987052888458344,
>>> > 0.0987052888458344, -0.0979187925203697, -0.0979187925203697,
>>> > -0.0979187925203697, -0.0979187925203697, -0.0979187925203697,
>>> > -0.0979187925203697, -0.0979187925203697, -0.276846706563615,
>>> > -0.276846706563615, -0.276846706563615, -0.276846706563615,
>>> > -0.276846706563615, -0.276846706563615, -0.276846706563615,
>>> > -0.341732653414463, -0.341732653414463, -0.341732653414463,
>>> > -0.341732653414463, -0.341732653414463, -0.341732653414463,
>>> > -0.341732653414463, -0.168703461812203, -0.168703461812203,
>>> > -0.168703461812203, -0.168703461812203, -0.168703461812203,
>>> > -0.168703461812203, -0.168703461812203, 0.393641410895141,
>>> > 0.393641410895141, 0.393641410895141, 0.393641410895141,
>>> > 0.393641410895141, 0.393641410895141, 0.393641410895141),
>>> > Z5 = c(-0.307723646230283, -0.307723646230283, -0.307723646230283,
>>> > -0.307723646230283, -0.307723646230283, -0.307723646230283,
>>> > -0.307723646230283, 0.351684167120324, 0.351684167120324,
>>> > 0.351684167120324, 0.351684167120324, 0.351684167120324,
>>> > 0.351684167120324, 0.351684167120324, 0.300960489170276,
>>> > 0.300960489170276, 0.300960489170276, 0.300960489170276,
>>> > 0.300960489170276, 0.300960489170276, 0.300960489170276,
>>> > 0.0135263141200131, 0.0135263141200131, 0.0135263141200131,
>>> > 0.0135263141200131, 0.0135263141200131, 0.0135263141200131,
>>> > 0.0135263141200131, -0.230869588730213, -0.230869588730213,
>>> > -0.230869588730213, -0.230869588730213, -0.230869588730213,
>>> > -0.230869588730213, -0.230869588730213, -0.307416230000283,
>>> > -0.307416230000283, -0.307416230000283, -0.307416230000283,
>>> > -0.307416230000283, -0.307416230000283, -0.307416230000283,
>>> > -0.207505955250191, -0.207505955250191, -0.207505955250191,
>>> > -0.207505955250191, -0.207505955250191, -0.207505955250191,
>>> > -0.207505955250191, 2.73029795708800e-17, 2.73029795708800e-17,
>>> > 2.73029795708800e-17, 2.73029795708800e-17, 2.73029795708800e-17,
>>> > 2.73029795708800e-17, 2.73029795708800e-17, 0.207505955250191,
>>> > 0.207505955250191, 0.207505955250191, 0.207505955250191,
>>> > 0.207505955250191, 0.207505955250191, 0.207505955250191,
>>> > 0.307416230000283, 0.307416230000283, 0.307416230000283,
>>> > 0.307416230000283, 0.307416230000283, 0.307416230000283,
>>> > 0.307416230000283, 0.230869588730212, 0.230869588730212,
>>> > 0.230869588730212, 0.230869588730212, 0.230869588730212,
>>> > 0.230869588730212, 0.230869588730212, -0.0135263141200123,
>>> > -0.0135263141200123, -0.0135263141200123, -0.0135263141200123,
>>> > -0.0135263141200123, -0.0135263141200123, -0.0135263141200123,
>>> > -0.300960489170277, -0.300960489170277, -0.300960489170277,
>>> > -0.300960489170277, -0.300960489170277, -0.300960489170277,
>>> > -0.300960489170277, -0.351684167120323, -0.351684167120323,
>>> > -0.351684167120323, -0.351684167120323, -0.351684167120323,
>>> > -0.351684167120323, -0.351684167120323, 0.307723646230283,
>>> > 0.307723646230283, 0.307723646230283, 0.307723646230283,
>>> > 0.307723646230283, 0.307723646230283, 0.307723646230283),
>>> > Z6 = c(0.219001863703187, 0.219001863703187, 0.219001863703187,
>>> > 0.219001863703187, 0.219001863703187, 0.219001863703187,
>>> > 0.219001863703187, -0.438003727406375, -0.438003727406375,
>>> > -0.438003727406375, -0.438003727406375, -0.438003727406375,
>>> > -0.438003727406375, -0.438003727406375, -0.084231486039687,
>>> > -0.084231486039687, -0.084231486039687, -0.084231486039687,
>>> > -0.084231486039687, -0.084231486039687, -0.084231486039687,
>>> > 0.269540755326999, 0.269540755326999, 0.269540755326999,
>>> > 0.269540755326999, 0.269540755326999, 0.269540755326999,
>>> > 0.269540755326999, 0.301701868178519, 0.301701868178519,
>>> > 0.301701868178519, 0.301701868178519, 0.301701868178519,
>>> > 0.301701868178519, 0.301701868178519, 0.076574078217897,
>>> > 0.076574078217897, 0.076574078217897, 0.076574078217897,
>>> > 0.076574078217897, 0.076574078217897, 0.076574078217897,
>>> > -0.191435195544744, -0.191435195544744, -0.191435195544744,
>>> > -0.191435195544744, -0.191435195544744, -0.191435195544744,
>>> > -0.191435195544744, -0.306296312871592, -0.306296312871592,
>>> > -0.306296312871592, -0.306296312871592, -0.306296312871592,
>>> > -0.306296312871592, -0.306296312871592, -0.191435195544745,
>>> > -0.191435195544745, -0.191435195544745, -0.191435195544745,
>>> > -0.191435195544745, -0.191435195544745, -0.191435195544745,
>>> > 0.0765740782178983, 0.0765740782178983, 0.0765740782178983,
>>> > 0.0765740782178983, 0.0765740782178983, 0.0765740782178983,
>>> > 0.0765740782178983, 0.301701868178517, 0.301701868178517,
>>> > 0.301701868178517, 0.301701868178517, 0.301701868178517,
>>> > 0.301701868178517, 0.301701868178517, 0.269540755327,
>>> 0.269540755327,
>>> > 0.269540755327, 0.269540755327, 0.269540755327, 0.269540755327,
>>> > 0.269540755327, -0.0842314860396875, -0.0842314860396875,
>>> > -0.0842314860396875, -0.0842314860396875, -0.0842314860396875,
>>> > -0.0842314860396875, -0.0842314860396875, -0.438003727406375,
>>> > -0.438003727406375, -0.438003727406375, -0.438003727406375,
>>> > -0.438003727406375, -0.438003727406375, -0.438003727406375,
>>> > 0.219001863703188, 0.219001863703188, 0.219001863703188,
>>> > 0.219001863703188, 0.219001863703188, 0.219001863703188,
>>> > 0.219001863703188), Z7 = c(-0.141858517577507, -0.141858517577507,
>>> > -0.141858517577507, -0.141858517577507, -0.141858517577507,
>>> > -0.141858517577507, -0.141858517577507, 0.42557555273252,
>>> > 0.42557555273252, 0.42557555273252, 0.42557555273252,
>>> 0.42557555273252,
>>> > 0.42557555273252, 0.42557555273252, -0.185507292216737,
>>> > -0.185507292216737,
>>> > -0.185507292216737, -0.185507292216737, -0.185507292216737,
>>> > -0.185507292216737, -0.185507292216737, -0.338278003454058,
>>> > -0.338278003454058, -0.338278003454058, -0.338278003454058,
>>> > -0.338278003454058, -0.338278003454058, -0.338278003454058,
>>> > -0.0327365809794237, -0.0327365809794237, -0.0327365809794237,
>>> > -0.0327365809794237, -0.0327365809794237, -0.0327365809794237,
>>> > -0.0327365809794237, 0.272804841495205, 0.272804841495205,
>>> > 0.272804841495205, 0.272804841495205, 0.272804841495205,
>>> > 0.272804841495205, 0.272804841495205, 0.272804841495207,
>>> > 0.272804841495207, 0.272804841495207, 0.272804841495207,
>>> > 0.272804841495207, 0.272804841495207, 0.272804841495207,
>>> > 4.25663509926017e-16, 4.25663509926017e-16, 4.25663509926017e-16,
>>> > 4.25663509926017e-16, 4.25663509926017e-16, 4.25663509926017e-16,
>>> > 4.25663509926017e-16, -0.272804841495207, -0.272804841495207,
>>> > -0.272804841495207, -0.272804841495207, -0.272804841495207,
>>> > -0.272804841495207, -0.272804841495207, -0.272804841495206,
>>> > -0.272804841495206, -0.272804841495206, -0.272804841495206,
>>> > -0.272804841495206, -0.272804841495206, -0.272804841495206,
>>> > 0.0327365809794244, 0.0327365809794244, 0.0327365809794244,
>>> > 0.0327365809794244, 0.0327365809794244, 0.0327365809794244,
>>> > 0.0327365809794244, 0.338278003454055, 0.338278003454055,
>>> > 0.338278003454055, 0.338278003454055, 0.338278003454055,
>>> > 0.338278003454055, 0.338278003454055, 0.185507292216741,
>>> > 0.185507292216741, 0.185507292216741, 0.185507292216741,
>>> > 0.185507292216741, 0.185507292216741, 0.185507292216741,
>>> > -0.425575552732522, -0.425575552732522, -0.425575552732522,
>>> > -0.425575552732522, -0.425575552732522, -0.425575552732522,
>>> > -0.425575552732522, 0.141858517577507, 0.141858517577507,
>>> > 0.141858517577507, 0.141858517577507, 0.141858517577507,
>>> > 0.141858517577507, 0.141858517577507), Z8 = c(0.083314261738317,
>>> > 0.083314261738317, 0.083314261738317, 0.083314261738317,
>>> > 0.083314261738317, 0.083314261738317, 0.083314261738317,
>>> > -0.345159084344455, -0.345159084344455, -0.345159084344455,
>>> > -0.345159084344455, -0.345159084344455, -0.345159084344455,
>>> > -0.345159084344455, 0.379949655180232, 0.379949655180232,
>>> > 0.379949655180232, 0.379949655180232, 0.379949655180232,
>>> > 0.379949655180232, 0.379949655180232, 0.143739990032049,
>>> > 0.143739990032049, 0.143739990032049, 0.143739990032049,
>>> > 0.143739990032049, 0.143739990032049, 0.143739990032049,
>>> > -0.284733356050736, -0.284733356050736, -0.284733356050736,
>>> > -0.284733356050736, -0.284733356050736, -0.284733356050736,
>>> > -0.284733356050736, -0.251773867890516, -0.251773867890516,
>>> > -0.251773867890516, -0.251773867890516, -0.251773867890516,
>>> > -0.251773867890516, -0.251773867890516, 0.114442667222960,
>>> > 0.114442667222960, 0.114442667222960, 0.114442667222960,
>>> > 0.114442667222960, 0.114442667222960, 0.114442667222960,
>>> > 0.320439468224296, 0.320439468224296, 0.320439468224296,
>>> > 0.320439468224296, 0.320439468224296, 0.320439468224296,
>>> > 0.320439468224296, 0.114442667222965, 0.114442667222965,
>>> > 0.114442667222965, 0.114442667222965, 0.114442667222965,
>>> > 0.114442667222965, 0.114442667222965, -0.251773867890520,
>>> > -0.251773867890520, -0.251773867890520, -0.251773867890520,
>>> > -0.251773867890520, -0.251773867890520, -0.251773867890520,
>>> > -0.284733356050731, -0.284733356050731, -0.284733356050731,
>>> > -0.284733356050731, -0.284733356050731, -0.284733356050731,
>>> > -0.284733356050731, 0.143739990032042, 0.143739990032042,
>>> > 0.143739990032042, 0.143739990032042, 0.143739990032042,
>>> > 0.143739990032042, 0.143739990032042, 0.379949655180234,
>>> > 0.379949655180234, 0.379949655180234, 0.379949655180234,
>>> > 0.379949655180234, 0.379949655180234, 0.379949655180234,
>>> > -0.345159084344455, -0.345159084344455, -0.345159084344455,
>>> > -0.345159084344455, -0.345159084344455, -0.345159084344455,
>>> > -0.345159084344455, 0.0833142617383167, 0.0833142617383167,
>>> > 0.0833142617383167, 0.0833142617383167, 0.0833142617383167,
>>> > 0.0833142617383167, 0.0833142617383167), Z9 = c(-0.0440394305225472,
>>> > -0.0440394305225472, -0.0440394305225472, -0.0440394305225472,
>>> > -0.0440394305225472, -0.0440394305225472, -0.0440394305225472,
>>> > 0.239071194265257, 0.239071194265257, 0.239071194265257,
>>> > 0.239071194265257, 0.239071194265257, 0.239071194265257,
>>> > 0.239071194265257, -0.436038757151818, -0.436038757151818,
>>> > -0.436038757151818, -0.436038757151818, -0.436038757151818,
>>> > -0.436038757151818, -0.436038757151818, 0.16647872637096,
>>> > 0.16647872637096, 0.16647872637096, 0.16647872637096,
>>> 0.16647872637096,
>>> > 0.16647872637096, 0.16647872637096, 0.318922908948979,
>>> 0.318922908948979,
>>> > 0.318922908948979, 0.318922908948979, 0.318922908948979,
>>> > 0.318922908948979, 0.318922908948979, -0.120987446490507,
>>> > -0.120987446490507, -0.120987446490507, -0.120987446490507,
>>> > -0.120987446490507, -0.120987446490507, -0.120987446490507,
>>> > -0.326666105524386, -0.326666105524386, -0.326666105524386,
>>> > -0.326666105524386, -0.326666105524386, -0.326666105524386,
>>> > -0.326666105524386, 1.6560876488147e-15, 1.6560876488147e-15,
>>> > 1.6560876488147e-15, 1.6560876488147e-15, 1.6560876488147e-15,
>>> > 1.6560876488147e-15, 1.6560876488147e-15, 0.326666105524385,
>>> > 0.326666105524385, 0.326666105524385, 0.326666105524385,
>>> > 0.326666105524385, 0.326666105524385, 0.326666105524385,
>>> > 0.120987446490508, 0.120987446490508, 0.120987446490508,
>>> > 0.120987446490508, 0.120987446490508, 0.120987446490508,
>>> > 0.120987446490508, -0.318922908948989, -0.318922908948989,
>>> > -0.318922908948989, -0.318922908948989, -0.318922908948989,
>>> > -0.318922908948989, -0.318922908948989, -0.166478726370943,
>>> > -0.166478726370943, -0.166478726370943, -0.166478726370943,
>>> > -0.166478726370943, -0.166478726370943, -0.166478726370943,
>>> > 0.436038757151805, 0.436038757151805, 0.436038757151805,
>>> > 0.436038757151805, 0.436038757151805, 0.436038757151805,
>>> > 0.436038757151805, -0.239071194265252, -0.239071194265252,
>>> > -0.239071194265252, -0.239071194265252, -0.239071194265252,
>>> > -0.239071194265252, -0.239071194265252, 0.0440394305225465,
>>> > 0.0440394305225465, 0.0440394305225465, 0.0440394305225465,
>>> > 0.0440394305225465, 0.0440394305225465, 0.0440394305225465
>>> > ), Z10 = c(0.0207056819701757, 0.0207056819701757,
>>> 0.0207056819701757,
>>> > 0.0207056819701757, 0.0207056819701757, 0.0207056819701757,
>>> > 0.0207056819701757, -0.141981819224062, -0.141981819224062,
>>> > -0.141981819224062, -0.141981819224062, -0.141981819224062,
>>> > -0.141981819224062, -0.141981819224062, 0.371109530696229,
>>> > 0.371109530696229, 0.371109530696229, 0.371109530696229,
>>> > 0.371109530696229, 0.371109530696229, 0.371109530696229,
>>> > -0.392270282599825, -0.392270282599825, -0.392270282599825,
>>> > -0.392270282599825, -0.392270282599825, -0.392270282599825,
>>> > -0.392270282599825, -0.0361780597061064, -0.0361780597061064,
>>> > -0.0361780597061064, -0.0361780597061064, -0.0361780597061064,
>>> > -0.0361780597061064, -0.0361780597061064, 0.356774827793781,
>>> > 0.356774827793781, 0.356774827793781, 0.356774827793781,
>>> > 0.356774827793781, 0.356774827793781, 0.356774827793781,
>>> > -0.0061434441010547, -0.0061434441010547, -0.0061434441010547,
>>> > -0.0061434441010547, -0.0061434441010547, -0.0061434441010547,
>>> > -0.0061434441010547, -0.344032869658285, -0.344032869658285,
>>> > -0.344032869658285, -0.344032869658285, -0.344032869658285,
>>> > -0.344032869658285, -0.344032869658285, -0.00614344410104096,
>>> > -0.00614344410104096, -0.00614344410104096, -0.00614344410104096,
>>> > -0.00614344410104096, -0.00614344410104096, -0.00614344410104096,
>>> > 0.356774827793798, 0.356774827793798, 0.356774827793798,
>>> > 0.356774827793798, 0.356774827793798, 0.356774827793798,
>>> > 0.356774827793798, -0.0361780597061478, -0.0361780597061478,
>>> > -0.0361780597061478, -0.0361780597061478, -0.0361780597061478,
>>> > -0.0361780597061478, -0.0361780597061478, -0.392270282599797,
>>> > -0.392270282599797, -0.392270282599797, -0.392270282599797,
>>> > -0.392270282599797, -0.392270282599797, -0.392270282599797,
>>> > 0.371109530696221, 0.371109530696221, 0.371109530696221,
>>> > 0.371109530696221, 0.371109530696221, 0.371109530696221,
>>> > 0.371109530696221, -0.141981819224062, -0.141981819224062,
>>> > -0.141981819224062, -0.141981819224062, -0.141981819224062,
>>> > -0.141981819224062, -0.141981819224062, 0.0207056819701758,
>>> > 0.0207056819701758, 0.0207056819701758, 0.0207056819701758,
>>> > 0.0207056819701758, 0.0207056819701758, 0.0207056819701758
>>> > ), Z11 = c(-0.00849937774690609, -0.00849937774690609,
>>> > -0.00849937774690609,
>>> > -0.00849937774690609, -0.00849937774690609, -0.00849937774690609,
>>> > -0.00849937774690609, 0.0716376124382085, 0.0716376124382085,
>>> > 0.0716376124382085, 0.0716376124382085, 0.0716376124382085,
>>> > 0.0716376124382085, 0.0716376124382085, -0.248910348302249,
>>> > -0.248910348302249, -0.248910348302249, -0.248910348302249,
>>> > -0.248910348302249, -0.248910348302249, -0.248910348302249,
>>> > 0.43225406827121, 0.43225406827121, 0.43225406827121,
>>> 0.43225406827121,
>>> > 0.43225406827121, 0.43225406827121, 0.43225406827121,
>>> -0.307191795709549,
>>> > -0.307191795709549, -0.307191795709549, -0.307191795709549,
>>> > -0.307191795709549, -0.307191795709549, -0.307191795709549,
>>> > -0.146917815339487, -0.146917815339487, -0.146917815339487,
>>> > -0.146917815339487, -0.146917815339487, -0.146917815339487,
>>> > -0.146917815339487, 0.360616455833108, 0.360616455833108,
>>> > 0.360616455833108, 0.360616455833108, 0.360616455833108,
>>> > 0.360616455833108, 0.360616455833108, 2.37442501121216e-14,
>>> > 2.37442501121216e-14, 2.37442501121216e-14, 2.37442501121216e-14,
>>> > 2.37442501121216e-14, 2.37442501121216e-14, 2.37442501121216e-14,
>>> > -0.360616455833123, -0.360616455833123, -0.360616455833123,
>>> > -0.360616455833123, -0.360616455833123, -0.360616455833123,
>>> > -0.360616455833123, 0.146917815339424, 0.146917815339424,
>>> > 0.146917815339424, 0.146917815339424, 0.146917815339424,
>>> > 0.146917815339424, 0.146917815339424, 0.307191795709669,
>>> > 0.307191795709669, 0.307191795709669, 0.307191795709669,
>>> > 0.307191795709669, 0.307191795709669, 0.307191795709669,
>>> > -0.432254068271316, -0.432254068271316, -0.432254068271316,
>>> > -0.432254068271316, -0.432254068271316, -0.432254068271316,
>>> > -0.432254068271316, 0.248910348302301, 0.248910348302301,
>>> > 0.248910348302301, 0.248910348302301, 0.248910348302301,
>>> > 0.248910348302301, 0.248910348302301, -0.0716376124382226,
>>> > -0.0716376124382226, -0.0716376124382226, -0.0716376124382226,
>>> > -0.0716376124382226, -0.0716376124382226, -0.0716376124382226,
>>> > 0.00849937774690764, 0.00849937774690764, 0.00849937774690764,
>>> > 0.00849937774690764, 0.00849937774690764, 0.00849937774690764,
>>> > 0.00849937774690764), Z12 = c(0.00295373795124409,
>>> 0.00295373795124409,
>>> > 0.00295373795124409, 0.00295373795124409, 0.00295373795124409,
>>> > 0.00295373795124409, 0.00295373795124409, -0.0299593420769036,
>>> > -0.0299593420769036, -0.0299593420769036, -0.0299593420769036,
>>> > -0.0299593420769036, -0.0299593420769036, -0.0299593420769036,
>>> > 0.132074282677049, 0.132074282677049, 0.132074282677049,
>>> > 0.132074282677049, 0.132074282677049, 0.132074282677049,
>>> > 0.132074282677049, -0.323223324378965, -0.323223324378965,
>>> > -0.323223324378965, -0.323223324378965, -0.323223324378965,
>>> > -0.323223324378965, -0.323223324378965, 0.450234056282428,
>>> > 0.450234056282428, 0.450234056282428, 0.450234056282428,
>>> > 0.450234056282428, 0.450234056282428, 0.450234056282428,
>>> > -0.27385370433672, -0.27385370433672, -0.27385370433672,
>>> > -0.27385370433672, -0.27385370433672, -0.27385370433672,
>>> > -0.27385370433672, -0.15317241090017, -0.15317241090017,
>>> > -0.15317241090017, -0.15317241090017, -0.15317241090017,
>>> > -0.15317241090017, -0.15317241090017, 0.389893409563986,
>>> > 0.389893409563986, 0.389893409563986, 0.389893409563986,
>>> > 0.389893409563986, 0.389893409563986, 0.389893409563986,
>>> > -0.153172410899934, -0.153172410899934, -0.153172410899934,
>>> > -0.153172410899934, -0.153172410899934, -0.153172410899934,
>>> > -0.153172410899934, -0.273853704336934, -0.273853704336934,
>>> > -0.273853704336934, -0.273853704336934, -0.273853704336934,
>>> > -0.273853704336934, -0.273853704336934, 0.450234056282481,
>>> > 0.450234056282481, 0.450234056282481, 0.450234056282481,
>>> > 0.450234056282481, 0.450234056282481, 0.450234056282481,
>>> > -0.323223324378928, -0.323223324378928, -0.323223324378928,
>>> > -0.323223324378928, -0.323223324378928, -0.323223324378928,
>>> > -0.323223324378928, 0.132074282677017, 0.132074282677017,
>>> > 0.132074282677017, 0.132074282677017, 0.132074282677017,
>>> > 0.132074282677017, 0.132074282677017, -0.0299593420768935,
>>> > -0.0299593420768935, -0.0299593420768935, -0.0299593420768935,
>>> > -0.0299593420768935, -0.0299593420768935, -0.0299593420768935,
>>> > 0.00295373795124293, 0.00295373795124293, 0.00295373795124293,
>>> > 0.00295373795124293, 0.00295373795124293, 0.00295373795124293,
>>> > 0.00295373795124293), Z13 = c(-0.000820388989417742,
>>> > -0.000820388989417742,
>>> > -0.000820388989417742, -0.000820388989417742, -0.000820388989417742,
>>> > -0.000820388989417742, -0.000820388989417742, 0.00984466787301128,
>>> > 0.00984466787301128, 0.00984466787301128, 0.00984466787301128,
>>> > 0.00984466787301128, 0.00984466787301128, 0.00984466787301128,
>>> > -0.0533252843121323, -0.0533252843121323, -0.0533252843121323,
>>> > -0.0533252843121323, -0.0533252843121323, -0.0533252843121323,
>>> > -0.0533252843121323, 0.170640909798784, 0.170640909798784,
>>> > 0.170640909798784, 0.170640909798784, 0.170640909798784,
>>> > 0.170640909798784, 0.170640909798784, -0.351946876459913,
>>> > -0.351946876459913, -0.351946876459913, -0.351946876459913,
>>> > -0.351946876459913, -0.351946876459913, -0.351946876459913,
>>> > 0.469262501946444, 0.469262501946444, 0.469262501946444,
>>> > 0.469262501946444, 0.469262501946444, 0.469262501946444,
>>> > 0.469262501946444, -0.351946876459763, -0.351946876459763,
>>> > -0.351946876459763, -0.351946876459763, -0.351946876459763,
>>> > -0.351946876459763, -0.351946876459763, 1.30897668096353e-14,
>>> > 1.30897668096353e-14, 1.30897668096353e-14, 1.30897668096353e-14,
>>> > 1.30897668096353e-14, 1.30897668096353e-14, 1.30897668096353e-14,
>>> > 0.351946876459646, 0.351946876459646, 0.351946876459646,
>>> > 0.351946876459646, 0.351946876459646, 0.351946876459646,
>>> > 0.351946876459646, -0.469262501946203, -0.469262501946203,
>>> > -0.469262501946203, -0.469262501946203, -0.469262501946203,
>>> > -0.469262501946203, -0.469262501946203, 0.351946876459683,
>>> > 0.351946876459683, 0.351946876459683, 0.351946876459683,
>>> > 0.351946876459683, 0.351946876459683, 0.351946876459683,
>>> > -0.170640909798657, -0.170640909798657, -0.170640909798657,
>>> > -0.170640909798657, -0.170640909798657, -0.170640909798657,
>>> > -0.170640909798657, 0.0533252843120879, 0.0533252843120879,
>>> > 0.0533252843120879, 0.0533252843120879, 0.0533252843120879,
>>> > 0.0533252843120879, 0.0533252843120879, -0.009844667873002,
>>> > -0.009844667873002, -0.009844667873002, -0.009844667873002,
>>> > -0.009844667873002, -0.009844667873002, -0.009844667873002,
>>> > 0.000820388989416903, 0.000820388989416903, 0.000820388989416903,
>>> > 0.000820388989416903, 0.000820388989416903, 0.000820388989416903,
>>> > 0.000820388989416903), Z14 = c(0.000157883934625923,
>>> > 0.000157883934625923,
>>> > 0.000157883934625923, 0.000157883934625923, 0.000157883934625923,
>>> > 0.000157883934625923, 0.000157883934625923, -0.00221037508476475,
>>> > -0.00221037508476475, -0.00221037508476475, -0.00221037508476475,
>>> > -0.00221037508476475, -0.00221037508476475, -0.00221037508476475,
>>> > 0.0143674380509829, 0.0143674380509829, 0.0143674380509829,
>>> > 0.0143674380509829, 0.0143674380509829, 0.0143674380509829,
>>> > 0.0143674380509829, -0.0574697522039801, -0.0574697522039801,
>>> > -0.0574697522039801, -0.0574697522039801, -0.0574697522039801,
>>> > -0.0574697522039801, -0.0574697522039801, 0.158041818561072,
>>> > 0.158041818561072, 0.158041818561072, 0.158041818561072,
>>> > 0.158041818561072, 0.158041818561072, 0.158041818561072,
>>> > -0.316083637122359, -0.316083637122359, -0.316083637122359,
>>> > -0.316083637122359, -0.316083637122359, -0.316083637122359,
>>> > -0.316083637122359, 0.474125455683776, 0.474125455683776,
>>> > 0.474125455683776, 0.474125455683776, 0.474125455683776,
>>> > 0.474125455683776, 0.474125455683776, -0.541857663638757,
>>> > -0.541857663638757, -0.541857663638757, -0.541857663638757,
>>> > -0.541857663638757, -0.541857663638757, -0.541857663638757,
>>> > 0.474125455683927, 0.474125455683927, 0.474125455683927,
>>> > 0.474125455683927, 0.474125455683927, 0.474125455683927,
>>> > 0.474125455683927, -0.316083637122536, -0.316083637122536,
>>> > -0.316083637122536, -0.316083637122536, -0.316083637122536,
>>> > -0.316083637122536, -0.316083637122536, 0.158041818561182,
>>> > 0.158041818561182, 0.158041818561182, 0.158041818561182,
>>> > 0.158041818561182, 0.158041818561182, 0.158041818561182,
>>> > -0.0574697522040225, -0.0574697522040225, -0.0574697522040225,
>>> > -0.0574697522040225, -0.0574697522040225, -0.0574697522040225,
>>> > -0.0574697522040225, 0.0143674380509932, 0.0143674380509932,
>>> > 0.0143674380509932, 0.0143674380509932, 0.0143674380509932,
>>> > 0.0143674380509932, 0.0143674380509932, -0.00221037508476627,
>>> > -0.00221037508476627, -0.00221037508476627, -0.00221037508476627,
>>> > -0.00221037508476627, -0.00221037508476627, -0.00221037508476627,
>>> > 0.00015788393462604, 0.00015788393462604, 0.00015788393462604,
>>> > 0.00015788393462604, 0.00015788393462604, 0.00015788393462604,
>>> > 0.00015788393462604), WYearCen = c(-7, -7, -7, -7, -7, -7,
>>> > -7, -6, -6, -6, -6, -6, -6, -6, -5, -5, -5, -5, -5, -5, -5,
>>> > -4, -4, -4, -4, -4, -4, -4, -3, -3, -3, -3, -3, -3, -3, -2,
>>> > -2, -2, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, 0, 0,
>>> > 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2,
>>> > 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5,
>>> > 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7)), .Names = c("Year",
>>> > "WYear", "Site", "LogY", "Y", "Z2", "Z3", "Z4", "Z5", "Z6", "Z7",
>>> > "Z8", "Z9", "Z10", "Z11", "Z12", "Z13", "Z14", "WYearCen"),
>>> row.names
>>> > = c(4L,
>>> > 18L, 26L, 40L, 67L, 75L, 94L, 104L, 118L, 126L, 140L, 167L, 175L,
>>> > 194L, 204L, 218L, 226L, 240L, 267L, 275L, 294L, 304L, 318L, 326L,
>>> > 340L, 367L, 375L, 394L, 404L, 418L, 426L, 440L, 467L, 475L, 494L,
>>> > 504L, 518L, 526L, 540L, 567L, 575L, 594L, 604L, 618L, 626L, 640L,
>>> > 667L, 675L, 694L, 704L, 718L, 726L, 740L, 767L, 775L, 794L, 804L,
>>> > 818L, 826L, 840L, 867L, 875L, 894L, 904L, 918L, 926L, 940L, 967L,
>>> > 975L, 994L, 1004L, 1018L, 1026L, 1040L, 1067L, 1075L, 1094L,
>>> > 1104L, 1118L, 1126L, 1140L, 1167L, 1175L, 1194L, 1204L, 1218L,
>>> > 1226L, 1240L, 1267L, 1275L, 1294L, 1304L, 1318L, 1326L, 1340L,
>>> > 1367L, 1375L, 1394L, 1404L, 1418L, 1426L, 1440L, 1467L, 1475L,
>>> > 1494L), class = "data.frame")
>>> >
>>> >
>>> >
>>> >
>>> >
>>> >
>>> > At 11:17 AM 5/22/2009 -0700, spencerg wrote:
>>> >> The first exaample on the "lmer" help page uses a formula
>>> "Reaction ~
>>> >> Days + (Days|Subject)". Here, "Subject" is the name of a column in
>>> >> the data.frame "sleepstudy", with levels "308", "309", ... .
>>> >>
>>> >> Does this answer your question? If no, please provide commented,
>>> >> minimal, self-contained, reproducible code, as requested in the
>>> >> posting guide "http://www.R-project.org/posting-guide.html". Your
>>> >> example is not "self-contained, reproducible".
>>> >>
>>> >> Hope this helps. Spencer
>>> >>
>>> >>
>>> >> Leigh Ann Starcevich wrote:
>>> >>> Dear guRus:
>>> >>>
>>> >>> I am using lmer for a mixed model that includes a random
>>> intercept
>>> >>> for a set of effects that have the same distribution, Normal(0,
>>> >>> sig2b). This set of effects is of variable size, so I am using an
>>> >>> as.formula statement to create the formula for lmer. For
>>> example, if
>>> >>> the set of random effects has dimension 8, then the lmer call is:
>>> >>>
>>> >>> Zs<- paste("Z",1:mb,sep="")
>>> >>> Trendformula <-as.formula(paste("LogY ~ WYear + (1+WYear|Site) +
>>> >>> (1|", paste(paste(Zs,collapse="+"), ")")))
>>> >>> fit2.4a<-lmer(Trendformula, data = testsamp)
>>> >>>
>>> >>> which, for mb=8, expands to:
>>> >>>
>>> >>> fit1<-lmer(LogY ~ WYear + (1 | Site) + (1 | Year) + (1 |
>>> Z1+ Z2 + Z3
>>> >>> + Z4 + Z5 + Z6 + Z7 + Z8), data = testsamp)
>>> >>>
>>> >>>
>>> >>> I have no problems with this. However, if the set of
>>> random effects
>>> >>> has a dimension of 30, then the lmer call is:
>>> >>>
>>> >>> fit2<-lmer(LogY ~ WYear + (1 | Site) + (1 | Year) + (1 |
>>> Z1+Z2 + Z3
>>> >>> + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 + Z10 + Z11 + Z12 + Z13 +
>>> Z14 + Z15 +
>>> >>> Z16 + Z17 + Z18 + Z19 + Z20 + Z21 + Z22 + Z23 + Z24 + Z25 + Z26 +
>>> >>> Z27 + Z28 + Z29+ Z30), data = testsamp)
>>> >>>
>>> >>> In this case, I get an error because the name "Z1+Z2 + Z3
>>> + Z4 + Z5
>>> >>> + Z6 + Z7 + Z8 + Z9 + Z10 + Z11 + Z12 + Z13 + Z14 + Z15 +
>>> Z16 + Z17
>>> >>> + Z18 + Z19 + Z20 + Z21 + Z22 + Z23 + Z24 + Z25 + Z26 +
>>> Z27 + Z28 +
>>> >>> Z29+ Z30" is too long to print in the output. Is there any way to
>>> >>> name the random effect in lmer so that the shorter (and more
>>> >>> descriptive) name may be used and the error avoided? Or
>>> is there a
>>> >>> way to combine these into a single variable prior to the lmer
>>> >>> function call? In SAS, I am able to parameterize these as
>>> a Toeplitz
>>> >>> structure with bandwidth 1.
>>> >>>
>>> >>> Thanks for any help.
>>> >>>
>>> >>> Leigh Ann Starcevich
>>> >>> Doctoral student
>>> >>> Oregon State University
>>> >>> Corvallis, Oregon
>>> >>> [[alternative HTML version deleted]]
>>> >>>
>>> >>> ______________________________________________
>>> >>> 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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R version 2.9.0 (2009-04-17)
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ISBN 3-900051-07-0
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> library(lme4)
Loading required package: Matrix
Loading required package: lattice
Attaching package: 'Matrix'
The following object(s) are masked from package:stats :
xtabs
The following object(s) are masked from package:base :
rcond
> sessionInfo()
R version 2.9.0 (2009-04-17)
i486-pc-linux-gnu
locale:
LC_CTYPE=en_US.UTF-8;LC_NUMERIC=C;LC_TIME=en_US.UTF-8;LC_COLLATE=en_US.UTF-8;LC_MONETARY=C;LC_MESSAGES=en_US.UTF-8;LC_PAPER=en_US.UTF-8;LC_NAME=C;LC_ADDRESS=C;LC_TELEPHONE=C;LC_MEASUREMENT=en_US.UTF-8;LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lme4_0.999375-31 Matrix_0.999375-26 lattice_0.17-25
loaded via a namespace (and not attached):
[1] grid_2.9.0
> load("testsamp.rda")
> testsamp <- within(testsamp, testsamp, Z29 <- Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9)
> str(testsamp)
'data.frame': 105 obs. of 20 variables:
$ Year : num 2008 2008 2008 2008 2008 ...
$ WYear : num 0 0 0 0 0 0 0 1 1 1 ...
$ Site : num 4 18 26 40 67 75 94 4 18 26 ...
$ LogY : num 0.849 0.809 0.734 1.467 0.716 ...
$ Y : num 2.34 2.25 2.08 4.34 2.05 ...
$ Z2 : num 0.472 0.472 0.472 0.472 0.472 ...
$ Z3 : num -0.456 -0.456 -0.456 -0.456 -0.456 ...
$ Z4 : num 0.394 0.394 0.394 0.394 0.394 ...
$ Z5 : num -0.308 -0.308 -0.308 -0.308 -0.308 ...
$ Z6 : num 0.219 0.219 0.219 0.219 0.219 ...
$ Z7 : num -0.142 -0.142 -0.142 -0.142 -0.142 ...
$ Z8 : num 0.0833 0.0833 0.0833 0.0833 0.0833 ...
$ Z9 : num -0.044 -0.044 -0.044 -0.044 -0.044 ...
$ Z10 : num 0.0207 0.0207 0.0207 0.0207 0.0207 ...
$ Z11 : num -0.0085 -0.0085 -0.0085 -0.0085 -0.0085 ...
$ Z12 : num 0.00295 0.00295 0.00295 0.00295 0.00295 ...
$ Z13 : num -0.00082 -0.00082 -0.00082 -0.00082 -0.00082 ...
$ Z14 : num 0.000158 0.000158 0.000158 0.000158 0.000158 ...
$ WYearCen: num -7 -7 -7 -7 -7 -7 -7 -6 -6 -6 ...
$ Z29 : num 0.218 0.218 0.218 0.218 0.218 ...
> xtabs(~ Z29, testsamp)
Z29
-1.63314896653237 -0.592612960445885 -0.394922672042026 -0.233838363771464
7 7 7 7
-0.214767257996307 -0.0925109179084174 -0.0376120692964267 0.0181504978689960
7 7 7 7
0.0208109028608321 0.129704969588931 0.197693713890356 0.21834978504526
7 7 7 7
0.227443614453709 0.269153880224658 2.11810584406015
7 7 7
> (fittest29.1 <- lmer(LogY ~ WYear + (1+WYear|Site) + (1|Z29), testsamp, subset = WYear<=10))
Linear mixed model fit by REML
Formula: LogY ~ WYear + (1 + WYear | Site) + (1 | Z29)
Data: testsamp
Subset: WYear <= 10
AIC BIC logLik deviance REMLdev
-147.5 -131.1 80.77 -167.7 -161.5
Random effects:
Groups Name Variance Std.Dev. Corr
Z29 (Intercept) 0.0095237 0.097589
Site (Intercept) 0.0765445 0.276667
WYear 0.0262909 0.162145 0.046
Residual 0.0011282 0.033589
Number of obs: 77, groups: Z29, 11; Site, 7
Fixed effects:
Estimate Std. Error t value
(Intercept) 0.9857992 0.1183912 8.327
WYear -0.0002676 0.0619989 -0.004
Correlation of Fixed Effects:
(Intr)
WYear -0.020
>
> proc.time()
user system elapsed
26.245 0.188 26.512
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