Suppose we have an anova model with two factors: Populations and species. In our model, populations are the main factor and species are nested within populations. Sample sizes for each level of factors differ (unbalanced design). We used Type-I (sequential) sum of squares.
As you see in the table below, Species ‘a’ is found in both populations pop1 and pop3. Similarly, species ‘b’ is found in both pop1 and pop2. Here are my questions:
1- Can we use nested anova to test among species variation within populations?
If the answer is yes, then:
Does the (population mean square)/(species mean square) ratio still give the correct F-value for among population variation?
If the answer is no, what is the best way to test among species variation within populations?
Any suggestions? Thank you very much.
Factors are given as follows:
Not sure which softtware you are using. In GenStat, that I use, you should be able to get it right by defining the data structure as Population/Species, where '/' denotes a nesting symbol.
Thank you very much. I am using R. Yes, the R formula is similar: .~Population/Species. My major question is: Is it statistically acceptable to use nested anova with this data set? (Note that I am not interested in interaction effects)
I see no problem with that. A fundamental principle to get statistical analysis 'correct' is to faithfully account for the structure in data - in this case the data structure is Pop/Sp. With nested factors, you can't estimate interaction effects.
Species ‘a’ and species ‘b’ are found in combination with more than one population, so this is not fully nested structure but it should not be a problem using nested anova. Am I right? Thanks again.
Looking at the structure of your data as it appears in the table you provided, this should be OK.
Dear Caner, you appear to have a situation in between a nested design and a cross-classified design. This should imply that if you apply a nested design ANOVA analysis,
(1) You have a somewhat more precise comparison between populations than the nested ANOVA tells you (i.e. you are on the safe side).
(2) Your estimate of the within population variance is less precise than the nested ANOVA tells you.
However, I would also be concerned about what your within population samples represent. The question could be posed, whether they can be regarded as random samples of species, so the assumption that each of them represents a within population variance should be reconsidered. This also brings up another fundamental question: What is the question you want to answer? Your formulation "to test among species variation within populations" is not clear enough to me.
Best wishes, Rolf Sundberg
Mr Sundberg,
Maybe I should give more detail on what I am doing. I am developing an R function (actually modifying an existing R function, adonis in vegan package) that is suitable for hierarchical partitioning of genetic variation among and within multiple sources of variation. It is analogous to MANOVA and uses permutation procedure (correctly restricted for nested designs) to test the significance of the pseudo-F ratios. Conventionally, analysis of molecular variance (AMOVA) (Excoffier et al. 1992) was used for estimating population differentiation and hypotheses testing. However, it was not suitable for our study since it was designed for analysing variation in a single species with a strictly nested data structure.
In answer to your first question:
Yes, they are random samples of species.
To your second question:
My question is: How the total variation is partitioned into variation between-groups and variation within-groups? ’. Explicitly, in this case: ‘ Which contributes significantly to the total variation, the variation among species within populations or the variation among populations? ' I know that the sum of squares was correctly calculated, but I could not be sure about F ratios and hypotheses testing for this partially nested design.
Thank you very much,
Sincerely, Caner
Excoffier L, Smouse PE, Quattro JM (1992) Analysis of molecular variance inferred from metric distances among DNA haplotypes: application to human mitochondrial DNA. Genetics 131:479–491
Caner, your answer to my first question appears to imply that your situation is far from a conventional nested case. You appear to have a small group of species per population, and to take a small sample of individuals per population (i.e. not a sample of species, but a sample of individuals which are classified by species). If the total number of species is as limited as your data set indicates, I'm afraid you must think of your data as a severely unbalanced cross-classified set of data. The lack of balance is perhaps not only due to sampling but also to all species not existing in all populations?
If the interpretation above is adequate, your answer to my second is irrelevant.
Good luck!
Rolf
Mr Sundberg,
The total number of species and populations was not limited as this representative example. Original data set includes 64 populations and 17 species, however it is still very unbalanced, which is consistent with your interpretation. We collect 319 individuals and assign them to a species as you notice (I give the contingency table below).
In Excoffier’s amova, sums of squares are also analyzed in a nested analysis of variance framework. Excoffier’s original data set was also unbalanced. The only difference with our data is, samples (here populations) found in combination with only one region (I also give the contingency table below for this data set).
So, is my question still irrelevant?
What is the best way to analyze genetic variation with this data set?
Thank you,
Sincerely, Caner
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