I prefer to think of ANOVA as a model comparison process. The process requires calculating SS for various parts, or course. But to me, how the process is carried out is important. While some prefer to have SS for model terms that remain consistent with respect to the other terms in the model (type 3 SS), my preference is that the sum of the term SS equals the model SS (type 1 SS), consistent with Dean’s explanation in this thread. This can probably be viewed as nothing more than a personal preference, but I will expand on this for the benefit of Morphmet readers who might not fully understand the logic.
It is perhaps unfortunate that ANOVA tables are as simple as they are typically presented. For example, with a factorial model, Y ~ A + B + AB, the sources of variation found in an ANOVA table are listed as A, B, and AB (or A:B for the interaction) and SS are provided. It might be more appropriate to describe the conditions. For type 3 SS the SS are SS(A|B,AB), SS(B|A,AB), and SS(AB|A,B); for type 1 SS the SS are SS(A|intercept), SS(B|A), and SS(AB|A,B); and for the sake of comprehensiveness, for type 2 SS the SS are SS(A|B), SS(B|A), and SS(AB|A,B). As Dr. Rohlf pointed out, for type 1 SS, the tests of effects can be viewed as arbitrary, because the model Y ~ B + A + AB will produce different SS for A and B because the sequence with which effects are introduced is different. However, I would argue that SS(B|A) and SS(A|B) are two different things, and type 2 SS is (in part) tantamount to doing type 1 SS with different combinations of term order. (I.e., type 2 SS tests the “main" effects given that other main effects are in the model, which for two-factor factorial models, is the second effect of type 1 SS for the two different orders of introduction of main effects.) If the interaction is “significant”, it becomes sort of a moot argument, as all three approaches have the same conditional format, and focusing on main effects is probably ill-advised. If the interaction is not significant, and is removed from the model, then types 2 and 3 SS are the same, and both are the same as type 1 SS run in both orders, focusing on the second effect in the process. A debate about which is better can be had for this situation. But if it comes to a default course of action for a program, here is why I am uncomfortable with type 3 SS. The SS for both of the main effects are SS(A|B, AB) and SS(B|A,AB). That is, e.g., SS(A|B, AB) (usually referred to simply as SSA) is found by calculating SSE(B,AB) - SSE(A,B,AB), where SSE is the summed squared error (residuals) for the models. This does not make much sense to me. To evaluate the effect, A, the model comparison uses a model that includes the interaction between A and B, but not one of the factors that is part of the interaction. This does not happen with type 1 or type 2 SS. I found a blog that does a pretty good job explaining this in more detail, plus demonstrates how to appropriately alter design matrices for type 3 SS in R, when using aov: https://mcfromnz.wordpress.com/2011/03/02/anova-type-iiiiii-ss-explained/ Again, in geomorph, one can perform type 2 and type 3 SS calculations (advanced.procD.lm accommodates this). I would like to add that advanced.procD.lm will also allow input of design matrices, which means one can use the model.matrix function with various contrasts to appropriately set up calculation of SS. The program is quite flexible this way. Understanding the process to calculate SS by model comparisons means one can choose the SS paradigm that is most appropriate for the research design and philosophical leaning. Cheers! -Mike Michael Collyer Associate Professor Biostatistics Department of Biology Western Kentucky University 1906 College Heights Blvd. #11080 Bowling Green, KY 42101-1080 Phone: 270-745-8765; Fax: 270-745-6856 Email: [email protected]<mailto:[email protected]> On Jul 10, 2015, at 10:31 PM, F. James Rohlf <[email protected]<mailto:[email protected]>> wrote: I don’t understand your objections to type 3 SS. The fact they do not add up for unbalanced designs is because the effects are correlated. Type 3 allows one to test an effect with the other correlated factors held constant. Each SS is adjusted for all other factors. It just tries to estimate what one would get if you had a proper balanced design. Being dependent on order makes the testing more arbitrary. It also adds the complexity that the SS are adjusted for the factors already tested and factors you will add later are ignored at that point. Testing in a different order my look like they give a different result (though not really because different things are being tested. Philosophically, one could decide that the whole idea of “significance testing” is not useful but that is a different issue. ---------------------- F. James Rohlf, Distinguished Professor Emeritus, Dept. of Ecol. & Evol. Research Professor, Dept. of Anthropology Stony Brook University The much revised 4th editions of Biometry and Statistical Tables are now available: http://www.whfreeman.com/Catalog/product/biometry-fourthedition-sokal http://www.whfreeman.com/Catalog/product/statisticaltables-fourthedition-rohlf P Please consider the environment before printing this email From: Collyer, Michael [mailto:[email protected]] Sent: Thursday, July 9, 2015 12:56 PM To: Waldir Miron; MORPHMET Subject: [MORPHMET] Re: [geomorph] New comment on ANOVAs and Geomorph. Dear Waldir, There are both philosophical and practical reasons we do not now or will not offer options for choice of sums of squares in procD.lm. First the practical reasons. It is not trivial to offer such an option. The various sums of squares (types 1, 2, 3, etc.) all involve a paradigm of model comparisons. (We have a couple of blog posts about this on www.geomorph.net<http://www.geomorph.net/>.). If one understands the paradigms, advanced.procD.lm allows one to calculate any type of sums of squares (SS). If you are determined to use type 3 SS, I recommend using advanced.procD.lm. As for the philosophical reason that type 3 SS is not something we offer, it is a paradigm we choose not to acknowledge as valid. The reason for this is simple. With type 3 SS the sum of the parts exceeds the total. Type 3 SS, as a paradigm, involves removing effects from the "full" model, one by one, and calculating the changes in summed squared residuals as the effect SS. For unbalanced designs, the sum of SS across effects will exceed the model SS. This does not make much sense. Furthermore, the process involves removing main effects but retaining any interaction with these effects for calculating SS. This is a bit silly. Type 2 SS avoids this problem, and again, advanced.procD.lm can accommodate the type 2 approach. By contrast, type 1 SS involves sequentially adding model effects and calculating the change in summed squared residuals as the effect SS. The sum of SS across the effects is the same as the full model SS. The order of effects is important, but we feel the biologist should have some cognizance of the appropriate order. Because geomorph is an R package, users are free to manipulate functions. Thus, it is possible to augment the R script to provide type 3 SS. But this not something we will do as a permanent fixture, for the reasons above. Good luck! Mike Collyer Sent from my iPhone On Jul 9, 2015, at 9:35 AM, Waldir Miron <[email protected]<mailto:[email protected]>> wrote: Waldir Miron<http://www.blogger.com/profile/10564967218097034020> has left a new comment on your post "ANOVAs and Geomorph<http://www.geomorph.net/2015/04/anovas-and-geomorph.html>": Dear (s), I am investigating fish shape variaton from different rivers basins and habitats within river basins. However, I have an unbalaced samples for both factors, it is recommended to do an Anova with SS error type III (marginal), to avoid some sampling bias. Unfortunately, I tried several times but I was not able to calculate SS type III using the Procrustes coordinates with the procD.lm function from geomorph package. It seems that the default type error of the function is the type I, and I would like to know if there is any argument to change that or a way to do it. I would appreciate any help. Regards. Posted by Waldir Miron to geomorph<http://www.geomorph.net/> at July 9, 2015 at 9:35 AM -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org<http://www.morphometrics.org/> To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]<mailto:[email protected]>. -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org To unsubscribe from this group and stop receiving emails from it, send an email to [email protected].
