[R-sig-ME] explaining lme variance component results

David Afshartous dafshartous at med.miami.edu
Thu Sep 13 18:05:56 CEST 2007


All,
I don't have the initial e-mails from this thread and when I searched the
mail archives for "explaining lme variance component results" it doesn't
come up.  Do the SIGs need to be searched differently  then r-help?
Thanks,
David



On 9/6/07 5:07 AM, "John Maindonald" <john.maindonald at anu.edu.au> wrote:

> While I was out of contact with the list for a couple of days, I put the
> following together.  It is for the data that Mike gave initially, so
> that
> it assumes no replicates within POLE. The mean square for
> MONTH:TIME:TRANSECT:POLE is the Residual.
> 
> The variance component for TRANSECT is effectively zero.  For
> purposes of exposition, it can be set to zero.
> 
> The terms formula MONTH/TIME then suffices. lme() gives the numbers of
> groups as:
> 
> Number of Observations: 286
> Number of Groups:
>            MONTH TIME %in% MONTH
>                4              16
> 
> Observe that the design is then very nearly a complete balanced design:
> 
>> with(temp, table(MONTH,TIME))
>       TIME
> MONTH  1  2  3  4
>      4 18 17 18 18
>      5 18 18 18 17
>      6 18 18 18 18
>      7 18 18 18 18
> 
> Observe that
>    286/16=17.875
> less than 18 because two of the TIME:MONTH combinations have
> only 17 values.  The harmonic mean, which is 17.87, seems however
> preferable.
> 
> If the design were balanced, with replicates at
> MONTH/TIME/POLE level equal to n1=4/n2=4/n3=17.87
> then the mean squares would estimate, respectively
> 
> 4 x 17.87 s1^2 +  17.87 s2^2 + s3^2  = 4 x 17.87 (s1^2 +  s2^2 /4 +
> s3^2 / (4*17.87)
> 17.87 s2^2 + s3^2  = 17.87 ( s2^2 + s3^2 / 17.87)
> s3^2
> 
> In the first two cases, a second version of the formula is given
> that makes (to me, at least) better intuitive sense.
> 
> We have
>           Variance     Anova        DF     No. of repeats
>           Component    mean square         per 'parent'
> MONTH    0.75287      74.10751      3     4
> TIME     1.07395      20.52433     12     4
> Residual 1.20354      1.20356     270     17.87 (harmonic mean)
> 
> Observe that the residual mean square estimates the residual variance.
> The TIME variance component is calculated pretty much as
> (20.52433 - 1.20356)/17.87 = 1.0813, which does not quite agree
> with the ML estimate (nor should it; equating mean squares to expected
> mean squares is not the same as REML!)).
> 
> The statistical error is affected both by the statistical error in
> the TIME
> anova mean square, and by the statistical error in the Residual mean
> square.  The variance formula that is given below for MONTH can be
> readily adapted for this case also.
> 
> The MONTH variance component is calculated pretty much as: (74.10751 -
> 20.52433)/(4*17.87) = 0.7497 Again the agreement with the REML
> estimate is, quite properly, not perfect.
> 
> The estimate for s1^2 (MONTH} has a statistical error that is a
> compound of the errors in the ANOVA mean squares for both TIME and
> MONTH.  The variances of the two anova mean squares (both sample
> values of chi-squared statistics, under the usual assumptions) add,
> while the quantities themselves are subtracted.  The SE (sqrt of
> variance), for the estimate s1^2 of sig1^2, is
> 
> sqrt( (n2 * n3 * sig1^2 + n3 * sig2^2 + sig3^2)^2 / nu1
>       + (n3 * sig2^2 + sig3^2)^2 / nu2 ) / (n2*n3)
> 
> [The n's are the numbers of repeats.  The nu's are degrees of freedom.]
> 
> Variances are not, for quantities that are differences multiples of
> chi-squared statistics, a good basis for inference. (Here I am tempted
> to make rude comments about over-reliance on variances in much
> sample  survey work!). The variance calculations may however be useful
> in giving an idea of the relative contributions of the different
> sources of
> statistical noise.
> 
> I'd expect, though I have not gone through the arithmetic in detail,
> that the estimate for SE[s1^2] will increase, if we allow for the
> possibility that the TRANSECT component of variance, even though
> estimated as zero, may actually be greater than zero.
> 
> John Maindonald             email: john.maindonald at anu.edu.au
> phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
> Centre for Mathematics & Its Applications, Room 1194,
> John Dedman Mathematical Sciences Building (Building 27)
> Australian National University, Canberra ACT 0200.
> 
> 
> On 5 Sep 2007, at 4:54 AM, Mike Dunbar wrote:
> 
>> Thanks to all, a couple more comments following up on Kevin's
>> comments below, and also ones sent to me directly.
>> 
>> I have plotted the data in many different ways, having spent
>> several years (yes!) trying to work out a suitable analysis for
>> these data. The aim of this particular analysis is to try to keep
>> things as simple as possible, I'm aware in particular that there
>> are differences between the behaviour of the factors across the
>> months (so one option is a month by month analysis - which I have
>> done but it was vetoed by co-workers for this paper so long as
>> there's a simple interpretation as well), and also that time is
>> generally the most important factor overall (this is already
>> documented by others - the data are of drifting macroinvertebrates
>> in rivers in case anyones interested).
>> 
>> The structure of the nesting is designed to mirror our expected
>> view of the correlations in the data based on spatial/temporal
>> proximity, a bit as Kevin describes below: so four times were
>> measured across a day and the experiment repeated across four
>> months, and for each of the 16 occasions, we have five transects,
>> within those four poles each, and not described previously, 1-3
>> measures at different heights on the poles.
>> 
>> Regarding the zero values: yes the normality is an assumption, I
>> hope to do better once this initial analysis is over. What I hoped
>> to show is despite this, and despite the assumptions of the
>> variance components analysis, there is evidence of an effect of
>> TRANSECT and / or POLE, once MONTH and TIME are accounted for.
>> 
>> What is very pertinent (thanks John) is the fact that in the data
>> as described, there is no replication within the lowest stratum,
>> POLE. There was one seeming replicate, but that must be an error.
>> This may well be the source of the problem that the POLE variance
>> component was large but not significant.
>> 
>> I had thought that despite the lack of replication within POLE that
>> it would still be possible to estimate a variance component for
>> POLE separately from the residual. The very wooly reasoning being
>> that the POLE component represents consistency in drift density
>> between POLEs across TRANSECT, TIME and MONTH, and residual
>> represents lack of consistency.
>> 
>> If my reasoning above is flawed, I really don't want to ditch the
>> POLE component, as its fairly central to the analysis, and I could
>> bring in HEIGHT to give replication within POLE (previous data is
>> for one height only). I'd prefer to do this as a fixed effect and
>> I've posted below some example data/code: can anyone comment if
>> this is valid?
>> 
>> Regarding the issue of magnitude of variance component/random
>> effect vs significance, I wonder if there is more too it than that,
>> certainly in this case we know that TIME is more important than
>> MONTH, despite being nested, but more critically, I can show some
>> data where the magnitude of the component doesn't seem to relate to
>> its significance. I'll post this in a separate mail to avoid
>> confusion, once again any comments are welcome. This gives me a
>> real headache explaining my results to my co-workers, let alone
>> reviewers. I ought to add that there could easily still be mistakes
>> where, as one regarding a non-replicate
>> has already been identified.
>> 
>> All the best again - hope this is interesting to others struggling
>> with similar issues??
>> 
>> Mike
>> 
>> 
>> 
>> 
>> varcor.2h.insects.hf <- lme(log(insectdens+1) ~ HEIGHT, random=~1|
>> MONTH/TIME/TRANSECT/POLE, data=temp2)
>> # introduce HEIGHT as a fixed effect, there are two heights per
>> pole for some poles: hence unbalanced
>> VarCorr(varcor.2h.insects.hf)
>> # variances: MONTH - 0.639, TIME: 1.248, TRANSECT: 0.013, POLE:
>> 0.160, Residual: 1.016
>> 
>> varcor.2h.insects.nospat.hf <- lme(log(insectdens+1) ~ HEIGHT,
>> random=~1|MONTH/TIME, data=temp2)
>> 
>> anova(varcor.2h.insects.hf,varcor.2h.insects.nospat.hf)
>> # two spatial factors together marginally signficant: p=0.06, but
>> test likely conservative
>> # simulation approach for null distribution (Faraway) probably too
>> difficult at this depth of nesting
>> intervals(varcor.2h.insects.hf)
>> # again some evidence for significance of TRANSECT, but POLE lower
>> bound close to 0.
>> 
>> # delete transect term and just compare models with and without
>> pole term
>> varcor.2h.insects.pole.hf <- lme(log(insectdens+1) ~ HEIGHT,
>> random=~1|MONTH/TIME/POLE, data=temp2)
>> # test pole factor on its own. This is possible as pole is coded as
>> a combination of transect and pole within transect
>> anova(varcor.2h.insects.pole.hf,varcor.2h.insects.nospat.hf)
>> # p=0.019. This would be great if analysis is valid
>> 
>> 
>> 
>> # read in data: this time with one or two heights per pole
>> 
>> temp2 <-
>> structure(list(MONTH = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>> 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L
>> ), .Label = c("4", "5", "6", "7"), class = "factor"), TRANSECT =
>> structure(c(1L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L,
>> 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L,
>> 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L,
>> 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L,
>> 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L,
>> 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L,
>> 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L,
>> 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L,
>> 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L,
>> 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L,
>> 5L, 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 5L, 5L, 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 5L, 5L, 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 5L, 5L, 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 5L, 5L, 5L, 5L, 5L, 5L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
>> 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>> 5L, 5L, 5L, 5L, 5L, 5L), .Label = c("1", "2", "3", "4", "5"), class
>> = "factor"),
>>     POLE = structure(c(1L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L,
>>     6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L,
>>     13L, 13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L, 1L, 2L,
>>     2L, 3L, 4L, 4L, 5L, 5L, 5L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L,
>>     10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L,
>>     17L, 18L, 18L, 1L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L,
>>     7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L,
>>     13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L, 1L, 1L, 2L,
>>     2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L,
>>     10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L,
>>     17L, 18L, 18L, 1L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L,
>>     7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L,
>>     13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L, 1L, 2L, 2L,
>>     3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L,
>>     10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L,
>>     17L, 18L, 18L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L,
>>     7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L, 13L,
>>     14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L, 1L, 1L, 2L, 2L, 3L,
>>     4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L,
>>     11L, 11L, 13L, 13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L,
>>     1L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L,
>>     9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L,
>>     15L, 16L, 17L, 17L, 18L, 18L, 1L, 1L, 2L, 2L, 3L, 4L, 4L,
>>     5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L,
>>     12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L,
>>     1L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L,
>>     9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L,
>>     15L, 16L, 17L, 17L, 18L, 18L, 1L, 1L, 2L, 2L, 3L, 4L, 4L,
>>     5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L,
>>     12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L,
>>     1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L,
>>     9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L, 15L,
>>     16L, 17L, 17L, 18L, 18L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L,
>>     6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L,
>>     12L, 13L, 13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L, 1L,
>>     2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L,
>>     10L, 10L, 11L, 11L, 12L, 12L, 13L, 13L, 14L, 14L, 15L, 16L,
>>     17L, 17L, 18L, 18L, 1L, 2L, 2L, 3L, 4L, 4L, 5L, 5L, 6L, 6L,
>>     7L, 7L, 8L, 8L, 9L, 9L, 10L, 10L, 11L, 11L, 12L, 12L, 13L,
>>     13L, 14L, 14L, 15L, 16L, 17L, 17L, 18L, 18L), .Label = c("11",
>>     "12", "13", "14", "23", "24", "31", "32", "33", "34", "41",
>>     "42", "43", "44", "51", "52", "53", "54"), class = "factor"),
>>     TIME = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>>     1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
>>     2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
>>     3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
>>     4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L), .Label
>> = c("1",
>>     "2", "3", "4"), class = "factor"), HEIGHT = structure(c(1L,
>>     2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L,
>>     1L, 2L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L, 1L, 1L,
>>     2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L,
>>     1L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L,
>>     1L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 1L, 2L,
>>     1L, 2L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 1L,
>>     1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 1L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L,
>>     2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,
>>     1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L), .Label = c("1", "2", "3"
>>     ), class = "factor"), insectdens = c(0, 0, 63.64, 11.99,
>>     14.57, 22.5, 15.53, 0, 20.49, 107.6, 0, 87.16, 19.82, 22.24,
>>     18.28, 51.92, 33.87, 42.1, 59.01, 0, 47.32, 15.78, 12.46,
>>     43.02, 12.23, 9.98, 27.75, 7.47, 38.92, 11.78, 11.18, 0,
>>     0, 120.6, 44.61, 24.02, 45.9, 26.78, 14.56, 80.2, 62.34,
>>     37.4, 32.44, 17.58, 47.52, 8.94, 26.01, 54.7, 9.19, 141.89,
>>     29.36, 10.39, 48.88, 14.6, 20.46, 158.34, 20.5, 9.52, 18.82,
>>     14.36, 47.94, 12.26, 45.76, 31.44, 53.82, 104.37, 112, 74.4,
>>     59.88, 73.38, 94.36, 73.78, 120.26, 305, 48.12, 129.45, 264.87,
>>     53.88, 129.36, 87.9, 107.03, 57.33, 145.53, 90.48, 95.2,
>>     110, 116.55, 110.44, 492, 50.7, 140.4, 68.16, 111.28, 104.8,
>>     59.76, 75, 91.92, 68.4, 12.92, 19.94, 22.6, 17.38, 53.6,
>>     102.6, 10.45, 151.92, 30.3, 0, 0, 0, 39.18, 34.96, 16.38,
>>     21.38, 18.32, 60.4, 35.48, 16.9, 0, 24.96, 56.28, 263.76,
>>     38.24, 37.12, 9.26, 30.76, 26.24, 25.88, 46.48, 7.2, 21.09,
>>     48.87, 0, 28.1, 10.09, 44.28, 67.26, 0, 0, 29.72, 50.49,
>>     63.92, 0, 0, 0, 18.28, 10.82, 7.5, 27.06, 21.48, 9.09, 21.94,
>>     13.56, 10.4, 13.25, 46.6, 31.74, 8.57, 11.98, 12.08, 30.55,
>>     12.46, 31.16, 27.27, 16.35, 78.15, 100.8, 13.54, 80.44, 69.35,
>>     104.55, 83.6, 37.32, 0, 107.7, 91.55, 21.52, 50.76, 22.28,
>>     17, 55.6, 52.85, 40.72, 15.76, 15.12, 41.08, 25.44, 10.79,
>>     87.36, 19.58, 19.94, 78.32, 13.04, 39.54, 40.55, 74.08, 14.37,
>>     34.68, 31.68, 69.4, 62.28, 13.13, 117.96, 41.02, 18.27, 72.66,
>>     34.74, 30.2, 69.86, 17.4, 100.89, 16.72, 95.7, 43.92, 0,
>>     27.6, 129.6, 73.64, 147.4, 107.82, 92.16, 46.9, 76.1, 52.78,
>>     52.32, 60.57, 46.7, 48.65, 49.41, 0, 54.8, 30.18, 59.2, 0,
>>     12.52, 0, 0, 15.89, 90.39, 35.42, 26.64, 8.54, 17.46, 52.98,
>>     7.88, 48.81, 12.68, 49.85, 32.67, 64.6, 41.2, 20.2, 8.47,
>>     80.29, 38.52, 17.28, 35.94, 41.55, 9.4, 237.25, 0, 38.88,
>>     24.56, 25.69, 0, 15.42, 0, 0, 0, 0, 467.64, 25.82, 36, 11.64,
>>     112.05, 31.54, 42.08, 0, 26.86, 79.74, 0, 27.18, 17.48, 0,
>>     34.95, 14.45, 43.88, 33.76, 23.24, 32.2, 16.29, 72.84, 189.99,
>>     436.05, 365.6, 259.98, 329.29, 228, 158.4, 140.91, 448.95,
>>     433.84, 47.11, 228.9, 193.13, 130.3, 335.73, 609.9, 202.54,
>>     371.88, 332, 360.36, 219.56, 338.91, 329.94, 139.15, 262.34,
>>     285.9, 357.76, 253.68, 353.35, 839.16, 368, 717.42, 840.18,
>>     2081.2, 900.15, 1052.03, 705.12, 1276.65, 512.25, 838.88,
>>     614.46, 734.58, 479.52, 286.38, 3020.4, 750.6, 885.96, 796.8,
>>     932.49, 824.67, 1476.09, 716.76, 576.46, 528.58, 568.8, 568.8,
>>     712.53, 1168.86, 1864.56, 997.26, 792.05, 1807.52, 899.25,
>>     939.03, 1487.7, 1121.12, 166.5, 84.96, 78.7, 31.98, 169.2,
>>     99.35, 124.2, 176.85, 116.88, 104.6, 45.43, 0, 82.44, 193.05,
>>     53.5, 204.49, 135.72, 201.9, 129.76, 49.71, 50.5, 93.06,
>>     239.98, 75.72, 221.54, 207.79, 218.24, 73.26, 96.4, 227.63,
>>     155.4, 141.7, 280.63, 98.25, 58.4, 16.6, 30.84, 141.72, 0,
>>     277.16, 313.82, 534.19, 104.74, 508.04, 67.62, 68.44, 119.7,
>>     215.37, 26.92, 0, 63.24, 48.68, 11.62, 81.36, 142.5, 65.07,
>>     28.06, 133.5, 126.54, 70.28, 79.62, 107.73, 36.16, 30.14,
>>     31.76, 407.76, 422.24, 274.24, 317.7, 241.5, 190.3, 644.49,
>>     162.17, 1104.24, 324.78, 268.24, 214.2, 449.25, 363.22, 475.57,
>>     197.12, 311.63, 154.28, 461.3, 352.52, 247.69, 382.65, 395.25,
>>     270.63, 399.84, 338.4, 529.48, 440.82, 394.56, 270.48, 322,
>>     441.22, 353.5, 452.4, 414.96, 699.72, 89.04, 173.7, 347.6,
>>     10150.24, 563.67, 353.94, 456.88, 117.92, 513, 245.48, 440.37,
>>     372.36, 398.86, 334.35, 428, 410.13, 398.06, 674.87, 438.75,
>>     226.16, 367.9, 416.8, 501.48, 522.6, 616.11, 421.2, 309.96,
>>     423.09, 232.08, 198.06, 48.66, 109.59, 49.59, 58.05, 152.08,
>>     0, 617.83, 64.66, 372.75, 32.07, 66.81, 112.24, 68.28, 83.64,
>>     157.48, 145.2, 46.24, 143, 99.18, 117.5, 158.05, 61.1, 91.68,
>>     67.5, 112.62, 98.21, 117.54, 58.92, 77.3, 0)), .Names = c("MONTH",
>> "TRANSECT", "POLE", "TIME", "HEIGHT", "insectdens"), class =
>> "data.frame", row.names = c(1L,
>> 2L, 4L, 5L, 7L, 9L, 10L, 13L, 14L, 16L, 17L, 19L, 20L, 22L, 23L,
>> 25L, 26L, 28L, 29L, 31L, 32L, 34L, 35L, 37L, 38L, 40L, 41L, 43L,
>> 44L, 46L, 47L, 49L, 50L, 53L, 55L, 56L, 58L, 60L, 61L, 64L, 65L,
>> 67L, 68L, 70L, 71L, 73L, 74L, 76L, 77L, 79L, 80L, 82L, 83L, 85L,
>> 86L, 88L, 89L, 91L, 92L, 94L, 95L, 97L, 98L, 100L, 101L, 103L,
>> 104L, 106L, 107L, 109L, 111L, 112L, 115L, 116L, 118L, 119L, 121L,
>> 122L, 124L, 125L, 127L, 128L, 130L, 131L, 133L, 134L, 136L, 137L,
>> 139L, 140L, 142L, 143L, 145L, 146L, 148L, 149L, 151L, 152L, 154L,
>> 155L, 157L, 158L, 160L, 162L, 163L, 166L, 167L, 169L, 170L, 172L,
>> 173L, 175L, 176L, 178L, 179L, 181L, 182L, 184L, 185L, 187L, 188L,
>> 190L, 191L, 193L, 194L, 196L, 197L, 199L, 200L, 202L, 203L, 205L,
>> 206L, 208L, 209L, 211L, 213L, 214L, 217L, 218L, 220L, 221L, 223L,
>> 224L, 226L, 227L, 229L, 230L, 232L, 233L, 235L, 236L, 238L, 239L,
>> 241L, 242L, 244L, 245L, 247L, 248L, 250L, 251L, 253L, 254L, 256L,
>> 259L, 260L, 262L, 264L, 265L, 268L, 269L, 271L, 272L, 274L, 275L,
>> 277L, 278L, 280L, 281L, 283L, 284L, 286L, 287L, 289L, 290L, 292L,
>> 293L, 295L, 296L, 298L, 299L, 301L, 302L, 304L, 305L, 307L, 310L,
>> 311L, 313L, 315L, 316L, 319L, 320L, 322L, 323L, 325L, 326L, 328L,
>> 329L, 331L, 332L, 334L, 335L, 337L, 338L, 340L, 341L, 343L, 344L,
>> 346L, 347L, 349L, 350L, 352L, 353L, 355L, 356L, 358L, 359L, 361L,
>> 362L, 364L, 366L, 367L, 370L, 371L, 373L, 374L, 376L, 377L, 379L,
>> 380L, 382L, 383L, 385L, 386L, 388L, 389L, 394L, 395L, 397L, 398L,
>> 400L, 401L, 403L, 404L, 406L, 407L, 409L, 410L, 412L, 413L, 415L,
>> 417L, 418L, 421L, 422L, 424L, 425L, 427L, 428L, 430L, 431L, 433L,
>> 434L, 436L, 437L, 439L, 440L, 442L, 443L, 445L, 446L, 448L, 449L,
>> 451L, 452L, 454L, 455L, 457L, 458L, 460L, 461L, 463L, 464L, 466L,
>> 468L, 469L, 472L, 473L, 475L, 476L, 478L, 479L, 481L, 482L, 484L,
>> 485L, 487L, 488L, 490L, 491L, 493L, 494L, 496L, 497L, 499L, 500L,
>> 502L, 503L, 505L, 506L, 508L, 509L, 511L, 512L, 514L, 515L, 517L,
>> 519L, 520L, 523L, 524L, 526L, 527L, 529L, 530L, 532L, 533L, 535L,
>> 536L, 538L, 539L, 541L, 542L, 544L, 545L, 547L, 548L, 550L, 551L,
>> 553L, 554L, 556L, 557L, 559L, 560L, 562L, 563L, 565L, 566L, 568L,
>> 570L, 571L, 574L, 575L, 577L, 578L, 580L, 581L, 583L, 584L, 586L,
>> 587L, 589L, 590L, 592L, 593L, 595L, 596L, 598L, 599L, 601L, 602L,
>> 604L, 605L, 607L, 608L, 610L, 611L, 613L, 616L, 617L, 619L, 621L,
>> 622L, 625L, 626L, 628L, 629L, 631L, 632L, 634L, 635L, 637L, 638L,
>> 640L, 641L, 643L, 644L, 646L, 647L, 649L, 650L, 652L, 653L, 655L,
>> 656L, 658L, 659L, 661L, 662L, 664L, 667L, 668L, 670L, 672L, 673L,
>> 676L, 677L, 679L, 680L, 682L, 683L, 685L, 686L, 688L, 689L, 691L,
>> 692L, 694L, 695L, 697L, 698L, 700L, 701L, 703L, 704L, 706L, 707L,
>> 709L, 710L, 712L, 713L, 715L, 718L, 719L, 721L, 723L, 724L, 727L,
>> 728L, 730L, 731L, 733L, 734L, 736L, 737L, 739L, 740L, 742L, 743L,
>> 745L, 746L, 748L, 749L, 751L, 752L, 754L, 755L, 757L, 758L, 760L,
>> 761L, 763L, 764L, 766L, 769L, 770L, 772L, 774L, 775L, 778L, 779L,
>> 781L, 782L, 784L, 785L, 787L, 788L, 790L, 791L, 793L, 794L, 796L,
>> 797L, 799L, 800L, 802L, 803L, 805L, 806L, 808L, 809L, 811L, 812L,
>> 814L, 815L))
>> 
>> 
>> -- 
>> This message (and any attachments) is for the recipient on...{{dropped}}
> 
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