Galina: The AIC, delta AIC, and AIC weights all reference an entire model and provide no information on how you should interpret the individual parameters within a model. If you believe based on your AIC weights that the model with ZONE + LABRADOR TEA + YEAR + ZONE x LABRADOR TEA is a reasonable candidate then your decision about interpreting the interaction should only be based on that term relative to other terms within this model. And, yes, if you were going to bother including an interaction in a model you better be willing to interpret it. And don't waste your time AIC model averaging the parameter estimates across these multiple models. There is no useful point for doing so other than deluding yourself into thinking you somehow have addressed model uncertainty. I'm including a previous post of mine to r-sig-ecolog list regarding the issues with AIC model averaging of individual regression parameter estimates, its ridiculous use for determining relative importance of predictors, etc.
Brian r-sig-ecology post: Joana and any others: You cannot obtain a valid or useful measure of relative importance of predictor variables across multiple models by applying relative AIC weights or using model averaged coefficients unless all your models included a single predictor (which, of course, is not what is usually done). And this applies to hurdle or any other models. AIC (and relative weights) apply to the log likelihood maximized for estimating a model that may be composed of 1 to many predictors. The log likelihood nor its associated AIC for a model has any ability to distinguish among the contributions of the individual predictors used in maximizing the log lilkelihood, and most useful definitions of relative importance of predictors within a model requires some ability to make that distinction. The best that AIC and relative AIC weights applied to individual predictor coefficients can tell you is the relative importance of the models in which the predictors occurred. And that is not the same as relative importance of predictors for most statisticians. It is quite possible to have a predictor with little relative importance within a model that has high relative importance, and, of course the opposite is true too. This AIC weights approach ignores that fact. Burnham and Anderson (2002, 2004) have done ecologists a great disservice by suggesting that AIC model weights can be used to address relative importance of individual predictors in models that included multiple predictors. AIC model weights can be used to assess the relative importance of models (that are combinations of one to many predictors) but are insufficient to address the relative importance of individual predictor variables because they donât recognize the differential contribution of individual predictors to the likelihood (or equivalently deviance, or sums of squares). Indeed, the use of AIC model weights, as employed by most people, acts as if for a given model that all predictors contributed equally to the likelihood and, thus, get the same weight for being in that model. That is a totally unreasonable assumption and never likely in practice. AIC weights are based on AIC that are computed from the log likelihood maximized by all predictors simultaneously. There is nothing in the theory behind AIC that suggests you can attribute the log likelihood equally to all predictor variables in the model. Iâm not sure why Burnham and Anderson (2002) propagated such a notion as it totally conflicts with and ignores a large body of statistical literature on methods for assigning relative importance of predictors within a given statistical model. Examples from some accessible statistical and ecological literature include: Bring, J. 1994. How to standardize regression coefficients. The American Statistician 48: 209-213. Chevan, A., and M. Sutherland. 1991. Hierarchical partitioning. The American Statistician 45: 90-96. Christensen, R. 1992. Comments on Chevan and Sutherland. The American Statistician 46: 74. Grömping, U. 2007. Estimators of relative importance in linear regression based on variance decomposition. The American Statistician 61: 139-147. Kruskal, W., and R. Majors. 1989. Concepts of relative importance in recent scientific literature. The American Statistician 43: 2-6. MacNally, R. 2000. Regression model-building in conservation biology, biogeography and ecology: The distinction between â and reconciliation of â âpredictiveâ and âexplanatoryâ models. Biodiversity and Conservation 9: 655-671. Murray, K., and M. M. Conner. 2009. Methods to quantify variable importance: implications for the analysis of noisy ecological data. Ecology 90:348-355. The paper by Murray and Conner (2009) is a simulation study that confirms and states what was obvious to most statisticians â AIC was not designed to differentiate among contributions of individual predictors within a model and, thus, is not appropriate for evaluating relative importance of individual predictors. Stick to using AIC weights to assess relative importance of models (note however that if all your models had single predictors then ranking models would be the same as ranking predictors). Relative importance is a slippery concept with many interpretations. But useful ones for predictors variables within a given regression model typically are related to contribution to reducing the objective function used in statistical estimation (minimizing negative log likelihood, sums of squares, deviance, etc.) and expected change in the response variable given a unit change in the predictor (these are well discussed in Bring 1994). In essence, the relative importance of individual predictors would have to involve the relative importance within a given model, i.e., how much it contributes to the likelihood that is maximized, or equivalently the minimization of residual sums of squares or minimization of deviances (â2´ log likelihood) relative to the other predictors in that model. A couple of possibilities for determining relative importance of predictors within a model exist. The ratio of *t-*statistics (parameter estimate divided by its standard error) corresponds directly to the ratio of appropriately standardized variables (Bring 1994). Note that this standardization of variables (by their partial standard deviations) is different than what is often done by default in most statistical software. The hierarchical partitioning approach of Chevan and Sutherland (1991) is based on similar ideas but considers more contrasts of models. It is available in the hier.part library in R. Murray and Connor (2009) simulation study found that it worked quite well. So I think what needs to be done is to conduct some reasonable assessment of relative importance of the predictors within each of the top candidate models. Then you can assess those results across all those top models. A formal process would compute the relative importance for a predictor (say variable X5) within each of the models in which it occurs, and the relative importance of each of the models within which the same predictor (e.g., X5) occurs (this latter could be done with AIC weights), and then combine that information somehow (probably with some product function). All these problems with AIC weights extend to problems with calculating model averaged regression coefficients. While it certainly may make sense to compute model averaged predictions (*y*-hat in linear regression) there is little reason to think that assigning weights based on the entire model to estimates for individual predictors and then averaging them provides an enlightened way to interpret regression parameter estimates (the coefficients). While it is true that algebraically one can obtain the model averaged predictions directly or by combining the model averaged coefficients, interpreting the coefficients implies that the units associated with these rates of change are interpretable. And since the rates of change implied by the estimate for a predictor are always conditional on what other predictors are in the model (they are partial effects), it is strange indeed to pretend like you can interpret averages of them across multiple models, each of which included many different predictors, in a comprehensible fashion (Candolo, Davison, and Demetrio. 2003. A note on model uncertainty in linear regression. The Statistician 52 (part 2): 165-177). Blums (et al. 2005. Individual quality, survival variation and patterns of phenotypic selection on body condition and timing of nesting in birds. Oecologica 143: 365-376) provide another good example for why not to model average individual parameter estimates. In essence this approach due to Burnham and Anderson (2002) is trying to provide an interpretation of an individual predictor as if it wasnât really conditional (they use the terminology âunconditionalâ) on the other predictors in the model, something that is unlikely to ever be true except in a controlled experimental design. Strange indeed! It is very much like trying to interpret main effects in the presence of an interaction in an ANOVA: while it occasionally may not be too misleading, in general it canât be a good thing to do. Averaging the parameter estimates across multiple models also assumes a continuity of the parameter space that may or may not be true in any given application. It is certainly possible, depending on the collinearity among the predictors, for a given predictor variable to have a strong positive rate of change when estimated with one set of other predictors and a strong negative rate of change when estimated with another set of predictors, i.e., a multimodal parameter space. And the average of those rates could then be close to zero, and not representative of any effect that was really estimated. For example, consider a weighted average estimate of â0.021 for variable A with a standard error of 0.018 (almost as large as the estimate) indicating an approximate 95% confidence interval of [â0.057, 0.015]. But we donât know whether this means there was a fairly dispersed unimodal distribution of parameter estimates near â0.021 and overlapping zero or whether there was a bimodal distribution of parameter estimates with one mode >0 and the other mode <0. If the latter case, which is to be expected often when considering models that include different combinations of predictors that are correlated with each other differently, then the averaged value is of low information content at best or completely misleading at worst. If you really want to try and understand the relationships due to effects of the individual predictor variables, then use of model averaged parameter estimates is a very shaky foundation for those interpretations. I realize it may come as a shock to many ecologists to learn that few statisticians actually support the idea of computing model averaged coefficients based on AIC weights. But such is the case. There is no statistical theory that Iâm aware of that suggests AIC weighted average parameter estimates are a reasonable way to obtain shrinkage estimates (what Burnham and Anderson claim their weighted coefficient estimates are like) to assess relationships across multiple models. Again, I reiterate, it may make some sense to use AIC weighted average predictions for predictive purposes, but not for interpreting individual model coefficients. There are other procedures, like the lasso (least absolute shrinkage and selection operator) that are more reasonable statistical approaches to obtain shrinkage estimates of parameters and eliminate needless parameters. Brian S. Cade, PhD U. S. Geological Survey Fort Collins Science Center 2150 Centre Ave., Bldg. C Fort Collins, CO 80526-8818 email: [email protected] <[email protected]> tel: 970 226-9326 On Thu, May 23, 2013 at 4:21 PM, Galina Kamorina < [email protected]> wrote: > Hi, > I would be very graitful if someone could help me to figure out my problem. > > I used mixed-effects models to analyse my data and AIC approach for model > selection. I am studying an effect on Labrador tea on basal diameter of > spruce in 2 different habitats (wet and dry zones) during 3 years. > This is one of example of my AIC table: > > > > > Candidate > models > > > K > > > AICc > > > Î AICc > > > AICc Wt > > > > > > Zone + Labrador tea + Year > > > 9 > > > -17.75 > > > 0.00 > > > 0.80 > > > > > Zone + Labrador tea + Year + Zone à Labrador tea > > > 10 > > > -14.69 > > > 3.06 > > > 0.17 > > > > > Zone + Labrador tea + Year + Year à Labrador tea > > > 12 > > > -11.21 > > > 6.53 > > > 0.03 > > > > > Zone + Labrador tea > > > 6 > > > 71.14 > > > 88.88 > > > 0.00 > > > > > Zone + Labrador tea + Zone à Labrador tea > > > 7 > > > 73.85 > > > 91.59 > > > 0.00 > > > > I interpreted the main effect of zone, Labrador tea and Year. My question > is should I interpret the interaction term Zone à Labrador tea also? > Normally I interpreted the effect of variables that have been in the models > with Î AICc < 4. > One professor said I should not interpred interaction term if the main > effect is stronger. But at the same time I saw articles where author > interpreted the interaction term where Akaike weight was still high. > > Thank you in advance. > Galina > [[alternative HTML version deleted]] > > ______________________________________________ > [email protected] 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. > [[alternative HTML version deleted]]
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