fixed and random factors in nested models

[Michele Fumi]

Hi all,

I already read the recent discussion on nested models in this blog. However, I still have some doubts mainly deriving from what I read in several texts where I noticed some discrepancies on how compute F-statistics. Using the function nested.update of Geomorph to correct my data having all fixed factors in one instance and all random factors (presentation and digitization error) in another, I noticed that pseudo F-statistics is always computed using the ratio MS_A/MS_B(A).

In “Geometric Morphometrics for Biologists (2^nd Ed.)” and in some texts dealing with statistics, when main factor is fixed, F-ratio is often computed as MS_A/MS_Residuals, but as I have already stated, some discrepancies with other texts exist. I did some trials using the old free program permanova (Anderson, 2005) allowing one to specify whether a factor is fixed or random.

I used the plethodon dataset and I did four trials:

1. considering both the factors, Species and Site, as fixed;

2. considering both the factors, Species and Site, as random;

3. considering Species as fixed and Site as random;

4. considering Species as random and Site as fixed.

In permanova, when main factor (Species) is fixed, F-ratio is computed as MS_A/MS_Residuals. When main factor is random, F-ratio is MS_A/MS_B(A) as well as in the remaining two simulations.

How pseudo F-statistics should be computed in nested models for main factors when they are, respectively, fixed or random? And, how these models should be managed in geomorph?

Thanks in advance,

Michele

[Mike Collyer]

Michele,

I do not understand what you might be experiencing with permanova.  I looked at the user guide and did not see documentation for how F values were calculated.  However, it seems it strange that mixed effects models are calculated as you suggest.  The nested.update function only applies to Y ~ A/B type mixed models where A is fixed (groups) and B is random, nested with A (subgroups).  I believe the rather generally accepted method of obtaining F ratios is MS(A)/MS(B|A); i.e., MS(groups)/MS(subgroups) for the fixed (groups) effect.  Also, MS(B)/MS(Residuals); i.e., MS(subgroups)/MS(within subgroups) for the random (subgroups) effect.

Here is a link that suggests this is true:

http://influentialpoints.com/Training/nested_anova-principles-properties-assumptions.htm

Here is another:

http://www.statsdirect.com/help/analysis_of_variance/nested.htm 

Here is another:

http://www.itl.nist.gov/div898/handbook/ppc/section2/ppc233.htm 

I also found this from Karl Broman's (University of Wisconsin) webpage, which I think shows nested ANOVA well via example:

https://www.biostat.wisc.edu/~kbroman/teaching/labstat/fourth/notes11.pdf

If B is not a factor that is “random” - meaning that the levels of B within A are not unique to A, sampled presumably and randomly from a distribution of possible B levels - then MS(B|A) is substituted with MS(Residuals) to calculate the F value for fixed effect, A.  For the species/site example, the sites are not unique to species.  Therefore, a fixed effects model is generally used.  An example where this would not be the case and a mixed effects model would be used is if one had species and populations nested within species.  The populations are subgroups and species comprise only their populations (populations are not shared between species).  One should evaluate if variation among species is meaningful compared to variation among the subgroups they comprise, not the individuals within populations (residuals).  That is what nested.update attempts to do.

However, nested.update is pretty limited and more complex F values can be considered.  That is difficult to accommodate for all kinds of research designs, but with procD.lm, for example, one can use all random SS outcomes to generate random F distributions, according to their own design.  This requires a bit of programming skill though.  The nested.update function just uses the SS values to calculate random F values in a prescribed way.

Hope that helps,

Mike

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