[R-sig-ME] AIC Comparison for MLM with Different Distributions

[Ben Bolker]

  Only when the response variable is transformed.
  [please keep r-sig-mixed-models in the cc: list if possible when
following up on questions ... ]

  cheers
    Ben Bolker

On 2020-03-07 12:16 p.m., Kate R wrote:
> 
> Hi Ben,
> 
> Thank you for your reply! Would I also apply the Jacobian correction to
> the Gamma with log-link, or is it only used when the response variable
> is transformed?
> 
> Many thanks again!
> Katie
> 
> 
>     ------------------------------
> 
>     Message: 2
>     Date: Wed, 4 Mar 2020 09:55:55 -0500
>     From: Ben Bolker <bbolker using gmail.com <mailto:bbolker using gmail.com>>
>     To: r-sig-mixed-models using r-project.org
>     <mailto:r-sig-mixed-models using r-project.org>
>     Subject: Re: [R-sig-ME] AIC Comparison for MLM with Different
>             Distributions
>     Message-ID: <eae1791a-e56a-7c32-b872-f6fa93157857 using gmail.com
>     <mailto:eae1791a-e56a-7c32-b872-f6fa93157857 using gmail.com>>
>     Content-Type: text/plain; charset="utf-8"
> 
> 
>       I agree with Thierry's big-picture comment that you should generally
>     use broader/qualitative criteria to decide on a model rather than
>     testing all possibilities.  The only exception I can think of is if you
>     are *only* interested in predictive accuracy (not in inference
>     [confidence intervals/p-values etc.]), and you make sure to use
>     cross-validation or a testing set to evaluate out-of-sample predictive
>     error (although AIC *should* generally give a reasonable approximation
>     to relative out-of-sample error).
> 
>       Beyond that, if you still want to compute AIC (e.g. your supervisor or
>     a reviewer is forcing to do it, and you don't think you're in a position
>     to push back effectively):
> 
>       * as long as you include the Jacobian correction when you transform
>     the predictor variable (i.e. #2), these log-likelihoods (and AICs)
>     should in principle be comparable (FWIW the robustness of the derivation
>     of AIC is much weaker for non-nested models; Brian Ripley [of MASS fame]
>     holds a minority opinion that one should *not* use AICs to compare
>     non-nested models)
> 
>       * computing log-likelihoods/AICs by hand is in principle a good idea,
>     but is often difficulty for multi-level models, as various integrals or
>     approximations of integrals are involved.  The lmer and glmer
>     likelihoods (1-4) are definitely comparable. To compare across platforms
>     I often try to think of a simplified model that *can* be fitted in both
>     platforms (e.g. in this case I think a proportional-odds ordinal
>     regression where the response has only two levels should be equivalent
>     to a binomial model with cloglog link ...)
> 
>       cheers
>       Ben Bolker
> 
>     On 2020-03-03 5:29 p.m., Kate R wrote:
>     > Hi all,
>     >
>     > Thank you in advance for your time and consideration! I am a
>     > non-mathematically-inclined graduate student in communication just
>     learning
>     > multilevel modeling.
>     >
>     > I am trying to compare the AIC for 5 different models:
>     >
>     >
>     >    1. model.mn5 <- lmer(anxious ~ num.cm <http://num.cm> + num.pmc
>     + (1|userid), data = df,
>     >    REML = F)
>     >    2. model.mn5.log <- lmer(log(anxious) ~ num.cm <http://num.cm>
>     + num.pmc + (1|userid),
>     >    data = df, REML = F)
>     >    3. model.mn5.gamma.log <- glmer(anxious ~ num.cm
>     <http://num.cm> + num.pmc + (1|userid),
>     >    data = df, family = Gamma(link="log"))
>     >    4. model.mn5.gamma.id <http://model.mn5.gamma.id> <-
>     glmer(anxious ~ num.cm <http://num.cm> + num.pmc + (1|userid),
>     >    data = df, family = Gamma(link="identity"))
>     >    5. model.ord5 <- clmm(anxious ~ num.cm <http://num.cm> +
>     num.pmc + (1|userid), data =
>     >    df, na.action = na.omit)
>     >
>     > (num.cm <http://num.cm> is the group mean and num.pmc is the
>     group-mean-centered score of
>     > the predictor)
>     >
>     > Despite many posts on various help forums, I understand that it's
>     possible
>     > to compare non-nested models with different distributions as long
>     as all
>     > terms, including constants, are retained (i.e. see Burnham &
>     Anderson, Ch
>     > 6.7 <https://www.springer.com/gp/book/9780387953649>), but that
>     different R
>     > packages or model classes might handle constants differently or use
>     > different algorithms (see point 7
>     <https://robjhyndman.com/hyndsight/aic/>),
>     > thus making it difficult to directly compare AIC values. To avoid
>     > this non-comparability pitfall, it was suggested in one post to
>     calculate
>     > your own log-likelihood (though I'm having trouble finding this
>     post again).
>     >
> 
> 
> 
> 
>     > Please could you help with the following:
>     >
>     >    - What is the best practice for comparing the AICs for these 5
>     models?
>     >    - What is the R-code for manually calculating the
>     log-likelihood and/or
>     >    the AIC to retain all terms, including constants?
>     >    - Can you compare ordinal models (clmm) with the continuous models?
>     >    - Do you recommend any other methods and/or packages for comparing
>     >    models with different distributions and/or links?
>     >
>     > Many thanks in advance for your time and consideration! I greatly
>     > appreciate any suggestions.
>     >
>     > Kind regards,
>     > K
>     >
>     >       [[alternative HTML version deleted]]
>     >
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