My real fear about fourth order polynomial by categorical interaction is that you may just be fitting noise. ?The part about the plateau is what tipped me off to looking at a non-linear effect. ?If only one interaction showed this, it would seem to me that the others aren't showing the kind of biological process observed here.
Have you looked at the residual error with increasing order of the polynomials? ?If there is an inflation of this (or the other random effects), then it is likely an overfit equation.?Steve Denham Director, Biostatistics MPI Research, Inc.
From: Quentin Schorpp <quentin.schorpp at ti.bund.de>
To: Steve Denham <stevedrd at yahoo.com>; "r-sig-mixed-models at r-project.org" <r-sig-mixed-models at r-project.org>
Sent: Thursday, December 17, 2015 3:08 AM
Subject: Re: [R-sig-ME] Orthogonal Polynomial contrasts with ordered factors
Hello,
Thank you for your answer. If I understood you right, you mean significant effects? of higher order polynomials could point to non-linear relationships between response and explanatory variable (also if the explanator is a categorial factor), under the condition, that? lower order polynomials in between are "skipped" in the sense that theses are not significant? The significant effect of ("non successive") higher order polynomials could be either in interaction terms and without? However when thinking about fitting a non-linear model, i'm wondering if it would be justified in the case that significant ("non successive") higher order polynomials only occur in one interaction case (the discussed pattern is only observed in the interaction with one level of the second interaction factor).
p.S. Sorry for the worse formatting of the initial question, while writing the e-mail everything looked good, especially the factor by factor table to reveal the partly nested structure.
kind regards,
Quentin
Am 16.12.2015 um 12:38 schrieb Steve Denham:
I'll take a crack at the B questions. ?Keep in mind that this is an opinion only. ?A significant fourth order by other factor interaction, especially when the lower order polynomial by factor interactions are not significant means that you have probably fit a polynomial to a non-polynomial effect. ?Your mention of a plateau tends to support this, at least to me. ?Some sort of non-linear effect (sigmoidal, four or five factor logistic) may make more sense from a biological perspective. ? Steve Denham Director, Biostatistics MPI Research, Inc.
From: Quentin Schorpp <quentin.schorpp at ti.bund.de>
To: "r-sig-mixed-models at r-project.org" <r-sig-mixed-models at r-project.org>
Sent: Tuesday, December 15, 2015 8:14 AM
Subject: [R-sig-ME] Orthogonal Polynomial contrasts with ordered factors
Hello,
I would appreciate to get to know more about the use of polynomial
contrasts in lme4::glmer.
Does anybody could give me an advice for literature about that subject.
In particular
A:? I read, that if a second order polynomial is significant in the
summary output, then it is supposed to be significant AFTER the first
order polynomial was? ? ? ? ? ? ? ? taken into account. Is that right?
B1 : What happens if i use an ordered factor with another factor
(ordered or not) in an itneraction term? What does a signficant
interaction of the second? ? ? ? ? ? ? ? factor (any level) with the
fourth power polynomial of the first ordered factor tell me?
B2: And waht does it tell me when the lower order polynomials are not
significant in the interaction?
For more interested readers:
_The data_
My data is abundances of earthworms. I sampled 15 fields, three times
(samcam) during two years, with 4 pseudoreplicates per field (N=180).
The factor age_class describes the stage of development of the field, it
has 5 levels ((n = 3 replicates).
However, one of these levels A_Cm has n=6 since i had to switch the
fields in the second year.
Field.ID is my random factor, to control for the pseudoreplication per
field and the longitudinal character of the data.? For the sake of less
complexity samcam stayed non-ordered.
Here is the design
? field.ID\samcam1 2 3 1 4 4 4 2 4 4 4 3 4 4 4 4 4 4 4 5 4 4 4 6 4 4 4 7 4
4 4 8 4 4 4 9 4 4 4 10 4 4 4 11 4 4 4 _12 4 4 4_ 13 4 0 0 Fields had to
be switched in the second year 14 4 0 0 15 4 0 0 16 0 4 4 17 0 4 4 18 0 4 4
Other continuous predictor variables were scaled before analysis.
data structure:
? $ abundance? ? ? : num? 0 0 3 3 2 1 2 5 12 5 ...
? $ ID? ? ? ? ? ? : Factor w/ 180 levels "1","2","3","4",..: 1 2 3 4 5
6 7 8 9 10 ...
? $ field.ID? ? ? : Factor w/ 18 levels "1","2","3","4",..: 1 1 1 1 2 2
2 2 3 3 ...
? $ age_class? ? ? : Ord.factor w/ 5 levels "A_Cm"<"B_Sp_young"<..: 5 5
5 5 5 5 5 5 5 5 ...
? $ samcam? ? ? ? : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
? $ hole? ? ? ? ? : Factor w/ 4 levels "1","2","3","4": 1 2 3 4 1 2 3 4
1 2 ...
? $ scl.pH? ? ? ? : num? -1.553 -1.553 -1.553 -1.553 0.715 ...
? $ scl.mc? ? ? ? : num? -1.072 -1.072 -1.072 -1.072 -0.429 ...
? $ scl.cn? ? ? ? : num? -0.703 -0.703 -0.703 -0.703 -0.474 ...
? $ scl.sand? ? ? : num? -0.245 -0.245 -0.245 -0.245 -0.0127 ...
? $ scl.silt? ? ? : num? -0.897 -0.897 -0.897 -0.897 -1.529 ...
? $ scl.clay? ? ? : num? 1.19 1.19 1.19 1.19 1.66 ...
? $ scl.ata1? ? ? : num? 1.6471 1.6471 1.6471 1.6471 0.0894 ...
? $ scl.atb1? ? ? : num? 1.6658 1.6658 1.6658 1.6658 0.0659 ...
? $ scl.hum1? ? ? : num? -1.378 -1.378 -1.378 -1.378 0.429 ...
_my hyptheses are_
1. abundance increases with increasing age_class
2. If abundance increases over the age classes it will be observed by
increasing abundance during the period of sampling
(3. Abundance increases during the period of sampling)
_The Model was :_
best.mod <- glmer(abundance~ age_class*samcam + scl.prec1 +
scl.mc*scl.pH? + (1|field.ID)? ,data=data,family=poisson,
control=glmerControl(optimizer="bobyqa"))
here is The Model output of one of the best models revealed by
MuMIn::dredge:
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
? Family: poisson? ( log )
Formula: anc ~ age_class * samcam + I(scl.ats1^2) + scl.prec1 + (1 | field.ID)
? ? Data: data
Control: glmerControl(optimizer = "bobyqa")
? ? ? AIC? ? ? BIC? logLik deviance df.resid
? ? ? 731? ? ? 789? ? -348? ? ? 695? ? ? 162
Scaled residuals:
? ? Min? ? 1Q Median? ? 3Q? ? Max
-2.181 -0.696 -0.171? 0.644? 4.178
Random effects:
? Groups? Name? ? ? ? Variance Std.Dev.
? field.ID (Intercept) 0.263? ? 0.513
Number of obs: 180, groups:? field.ID, 18
Fixed effects:
? ? ? ? ? ? ? ? ? ? Estimate Std. Error z value Pr(>|z|)
(Intercept)? ? ? ? ? 0.6457? ? 0.2079? ? 3.11? 0.0019 **
age_class.L? ? ? ? ? 1.9924? ? 0.4416? ? 4.51? 6.4e-06 ***
age_class.Q? ? ? ? ? -0.8644? ? 0.4204? -2.06? 0.0398 *
age_class.C? ? ? ? ? -0.2373? ? 0.4007? -0.59? 0.5537
age_class^4? ? ? ? ? 0.7026? ? 0.3591? ? 1.96? 0.0504 .
samcam2? ? ? ? ? ? ? 0.8549? ? 0.2074? ? 4.12? 3.7e-05 ***
samcam3? ? ? ? ? ? ? 0.3074? ? 0.1852? ? 1.66? 0.0969 .
I(scl.ats1^2)? ? ? ? -0.3328? ? 0.1038? -3.21? 0.0013 **
scl.prec1? ? ? ? ? ? 0.2241? ? 0.0851? ? 2.63? 0.0085 **
age_class.L:samcam2? -0.3474? ? 0.5463? -0.64? 0.5248
age_class.Q:samcam2? 0.1074? ? 0.4766? ? 0.23? 0.8216
age_class.C:samcam2? 0.8910? ? 0.3644? ? 2.44? 0.0145 *
_age_class^4:samcam2 -1.0352 0.2601 -3.98 6.9e-05 *** _? # HERE is the significant interaction of interest!
age_class.L:samcam3? 0.1274? ? 0.5125? ? 0.25? 0.8038
age_class.Q:samcam3? -0.1801? ? 0.4429? -0.41? 0.6842
age_class.C:samcam3? 0.5659? ? 0.3615? ? 1.57? 0.1174
_age_class^4:samcam3 -0.6489 0.2617 -2.48 0.0131 * ___# HERE is the significant interaction of interest!
_Interpretation:_
I understood, that the relationship was linear in general, as indicated
by the second line of the output, and this did not change between the
sampling campaigns. However, during the second and third sampling
campaign the relationship of abundances in the age_classes was
characterised by a stronger slope in younger classes and reached a
plateau afterwards, as indicated by the fourth power.
The missing of the interaction between age_class and samcam1 is very
hard for me to understand
I'm thankful for any advices!
Quentin
--
Quentin Schorpp, M.Sc.
Th?nen-Institut f?r Biodiversit?t
Bundesallee 50
38116 Braunschweig (Germany)
Tel:? +49 531 596-2524
Fax:? +49 531 596-2599
Mail: quentin.schorpp at ti.bund.de
Web:? http://www.ti.bund.de
Das Johann Heinrich von Th?nen-Institut, Bundesforschungsinstitut f?r L?ndliche R?ume, Wald und Fischerei ? kurz: Th?nen-Institut ?
besteht aus 15 Fachinstituten, die in den Bereichen ?konomie, ?kologie und Technologie forschen und die Politik beraten.
Quentin Schorpp, M.Sc.
Th?nen Institute of Biodiversity
Bundesallee 50
38116 Braunschweig (Germany)
Tel:? +49 531 596-2524
Fax:? +49 531 596-2599
Mail: quentin.schorpp at ti.bund.de
Web:? http://www.ti.bund.de
The Johann Heinrich von Th?nen Institute, Federal Research Institute for Rural Areas, Forestry and Fisheries ? Th?nen Institute in brief ?
consists of 15 specialized institutes that carry out research and provide policy advice in the fields of economy, ecology and technology.
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--
Quentin Schorpp, M.Sc.
Th?nen-Institut f?r Biodiversit?t
Bundesallee 50
38116 Braunschweig (Germany)
Tel: +49 531 596-2524
Fax: +49 531 596-2599
Mail: quentin.schorpp at ti.bund.de
Web: http://www.ti.bund.de
Das Johann Heinrich von Th?nen-Institut, Bundesforschungsinstitut f?r L?ndliche R?ume, Wald und Fischerei ? kurz: Th?nen-Institut ?
besteht aus 15 Fachinstituten, die in den Bereichen ?konomie, ?kologie und Technologie forschen und die Politik beraten.
Quentin Schorpp, M.Sc.
Th?nen Institute of Biodiversity
Bundesallee 50
38116 Braunschweig (Germany)
Tel: +49 531 596-2524
Fax: +49 531 596-2599
Mail: quentin.schorpp at ti.bund.de
Web: http://www.ti.bund.de
The Johann Heinrich von Th?nen Institute, Federal Research Institute for Rural Areas, Forestry and Fisheries ? Th?nen Institute in brief ?
consists of 15 specialized institutes that carry out research and provide policy advice in the fields of economy, ecology and technology.