[R-sig-eco] logistic regression and spatial autocorrelation
Aitor Gastón
aitor.gaston at upm.es
Thu Aug 25 23:22:34 CEST 2011
The limiting sample size in logistic regression is the minimum between the
number of positive and negative cases, in Tim's data 132 positive cases
(species occurrences). A minimum of 10 events per estimated parameter are
recommended based on external validation studies to avoid overfitting (see
Harrell, 2001. Regression Modeling Strategies. Springer). Therefore, with
Tim's data up to 13 parameters could be estimated (e.g., 13 variables
without nonlinear terms, or 6 variables in quadratic form, or 4 variables
using restricted cubic splines with 4 knots...).
Aitor
--------------------------------------------------
From: <cparker at pdx.edu>
Sent: Thursday, August 25, 2011 8:08 PM
To: "Pedro Lima Pequeno" <pacolipe at gmail.com>
Cc: <r-sig-ecology at r-project.org>; "Tim Seipel" <t.seipel at env.ethz.ch>
Subject: Re: [R-sig-eco] logistic regression and spatial autocorrelation
> Tim pointed put that he has only 132 samples out of 2800 with a species
> present and I am curious what people think about how well we can model
> that with logistic regression.
> -Chris
>
>
>
>
> On Aug 25, 2011, at 10:36 AM, Pedro Lima Pequeno <pacolipe at gmail.com>
> wrote:
>
>> Hi Tim,
>>
>> there are several ways of dealing with spatial autocorrelation in
>> ecological models (see e.g. Dormann 2007: Methods to account for
>> spatial autocorrelation in the analysis of species distributional
>> data: a review; and Beale et al. 2010: Regression analysis of spatial
>> data). As always, this is an area of active research, so the right or
>> wrong thing to do is not as clear as it may seem. Some have even
>> concluded that "changes in coefficients between spatial and
>> non-spatial methods depend on the method used and are largely
>> idiosyncratic", so that "researchers may have little choice but to be
>> more explicit about the uncertainty of models and more cautious in
>> their interpretation" (Bini et al. 2009: Coefficient shifts in
>> geographical ecology: an empirical evaluation of spatial and
>> non-spatial regression). Thus, new methods are emerging at a faster
>> rate than people studying and comparing their properties. Nonetheless,
>> I think some observations are useful:
>> 1) there are methods explicitly designed to detect spatial
>> autocorrelation such as Moran's autocorrelograms or variograms
>> (available in several R packages). As already pointed out, the
>> autocorrelation function is "well behaved" with linear, equally spaced
>> series in time or space;
>> 2) minimal adequate model selection with the AIC is sensitive to
>> residual autocorrelation; it tends to generate unstable and overfitted
>> models. Thus, when applying any model selection procedure, you should
>> account for the uncertainty in the process by averaging the model set
>> (or model predictions) with respect to the relative support of each
>> model (e.g. Akaike weights). Since you have a large sample, you could
>> account for residual spatial autocorrelation using eigenvector
>> filtering, which produces synthetic variables that capture spatial
>> patterns and can be included in linear models as explanatory variables
>> - a more intuitive approach if you don't want to mess with random
>> factors (see e.g. Diniz-Filho et al. 2008: Model selection and
>> information theory in geographical ecology). This can be implemented
>> with packages spacemakeR and vegan;
>> 3) autocorrelation in model residuals is not the only - nor most
>> important - problem in biological modeling; model misspecification is
>> the major issue. Residual autocorrelation often arises due to not
>> including relevant explanatory variables, interaction terms, assuming
>> an inappropriate response shape and/or an inadequate variance
>> structure, or any combination of these. All these things need to be
>> checked for proper model validation, for instance by partial
>> regression plots (or added-varialbe plots), which help you see the
>> shape of the response to each explanatory variable after account for
>> the variation in the remaining explanatory set.
>> At the same time, you could plot the residuals againts both model
>> predictions and explanatory variables. Since residuals are the
>> stochastic component of the model ("noise"), its relation with the
>> systematic components should be random; clear patterns in these plots
>> are indications of misspecification.
>> Finally, all this "model tinkering" is based on two fundamental
>> premises: you want to model a mean tendecy of response and the pattern
>> of variation around it. These are strictly statistical properties of
>> data - they have nothing to do with biology. If you don't really
>> believe the biological process you are studying implies a mean
>> response, but rather e.g. a maximum one (such as in population or
>> abundance limitation), than all these methods will actually induce you
>> to misspecify the model, but there are alternatives (see e.g. Cade et
>> al. 2005 - Quantile regression reveals hidden bias and uncertainty in
>> habitat models).
>>
>> 2011/8/25, Tim Seipel <t.seipel at env.ethz.ch>:
>>>
>>> Dear List,
>>> I am trying to determine the best environmental predictors of the
>>> presence of a species along an elevational gradient.
>>> Elevation ranges from 400 to 2050 m a.s.l. and the ratio of presences to
>>> absences is low (132 presences out 2800 samples)
>>>
>>> So to start I fit the full model of with the variable of interest.
>>>
>>> sc.m<-glm(PA~sp.max+su.mmin+su.max+fa.mmin+fa.max+Slope+Haupt4+Pop_density+Dist_G+Growi_sea+,data=sc.pa,'binomial')
>>>
>>> First, I performed univariate and backward selection using Akaike
>>> Information Criteria, and the fit was good and realistic given my
>>> knowledge of the environment though the D^2 was low 0.08. My final model
>>> was:
>>> ---------------------------------
>>> glm(formula = PA ~ Slope + sp.mmin + su.max + fa.mmin + Haupt4,
>>> family = "binomial", data = sc.pa)
>>>
>>> Deviance Residuals:
>>> Min 1Q Median 3Q Max
>>> -0.5415 -0.3506 -0.2608 -0.1762 3.0768
>>>
>>> Coefficients:
>>> Estimate Std. Error z value Pr(>|z|)
>>> (Intercept) -73.45212 23.13842 -3.174 0.00150 **
>>> Slope -0.03834 0.01174 -3.265 0.00109 **
>>> sp.mmin -15.34594 5.30360 -2.893 0.00381 **
>>> su.max 5.09712 1.70332 2.992 0.00277 **
>>> fa.mmin 13.52262 4.64021 2.914 0.00357 **
>>> Haupt42 -0.72237 0.27710 -2.607 0.00914 **
>>> Haupt43 -0.95730 0.37762 -2.535 0.01124 *
>>> Haupt44 -0.25357 0.24330 -1.042 0.29731
>>> ---
>>> Null deviance: 958.21 on 2784 degrees of freedom
>>> Residual deviance: 896.10 on 2777 degrees of freedom
>>> AIC: 912.1
>>>
>>> ----------------------
>>>
>>> I then realized that my residuals were all highly correlated (0.8-0.6)
>>> when I plotted them using acf() function.
>>>
>>> So to account for this I used glmmPQL to fit the full model:
>>>
>>> model.sc.c <- glmmPQL(PA ~
>>> sp.mmin+su.mmin+su.max+fa.mmin+Slope+Haupt4+Pop_density+Dist_G+Growi_sea,
>>> random=
>>> ~1|group.sc, data=sc.dat, family=binomial, correlation=corAR1())
>>>
>>> However, the algorithm failed to converge and all the p-vaules were
>>> either 0 or 1 and coefficient estimates approached infinity.
>>> Additionally the grouping factor of the random effect is slightly
>>> arbitrary and accounts a tiny amount of variation.
>>>
>>> ---
>>> So know I feel stuck between a rock and a hard place, on the one hand I
>>> know I have a lot of autocorrelation and on the other hand I don't have
>>> a clear way to include it in the model.
>>>
>>> I would appreciate any advice on the matter.
>>>
>>> Sincerely,
>>>
>>> Tim
>>>
>>> [[alternative HTML version deleted]]
>>>
>>> _______________________________________________
>>> R-sig-ecology mailing list
>>> R-sig-ecology at r-project.org
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-ecology
>>>
>>
>>
>> --
>> Pedro A. C. Lima Pequeno
>> Programa de Pós-graduação em Ecologia
>> Instituto Nacional de Pesquisas da Amazônia
>> Manaus, AM, Brasil
>>
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