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I have a dataset of discrete (ordinal, meristic, and nominal) variables describing morphological wing characters on several closely related species of insects. What I'm looking to do is conduct some kind of analysis that would give me a visual representation of the similarity of the different species based on the morphological characteristics. The first thing that popped into my head was PCA (this is the type of visualization I'm looking to create), but after looking into it (particularly other questions such as: Can principal component analysis be applied to datasets containing a mix of continuous and categorical variables? ), it seems PCA may be inappropriate for discrete data (PCA is used in these types of studies in the literature, but always with continuous data). Ignoring the statistical background of why this data is inappropriate, PCA does give me relatively perfect results with regard to my biological question (hybrid groups of interest fall right in the middle of their paternal groups).

I've also tried multiple correspondence analysis to appease the statistics (at least as far as my understanding goes), but I cannot seem to get a plot that is analogous to one I would get with PCA, where my observations (the biological individuals) are separated say by color to show the different groupings (different species, biologically speaking). It seems that this analysis is aimed at describing how the variables (here, my morphological characteristics) are related to each other, not the individual observations. And when I plot observations colored by group, I only get a single value (perhaps an average) describing the whole set of individuals. I've done the analysis in R, so perhaps I'm also just not R-savy enough to get my idea of the plot to work.

Am I correct in trying this kind of analysis with my data, or am I way off track? If you could not tell, my statistical expertise is limited, so the equations happening underneath these analyses are all way over my head. I'm trying to conduct this analysis completely descriptively (I don't need to do any more downstream number crunching), and I've read that if this is the case, PCA will suffice, but want to make sure I'm not violating too many statistical assumptions.

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    $\begingroup$ You should be able to get the kind of plot you want with multiple correspondence analysis. If you can give us a link to your data we could have a look. Multidimensional scaling is another possibility, but MCA can be seen as a kind of multidimensional scaling $\endgroup$
    – kjetil b halvorsen ♦
    Commented Apr 20, 2018 at 16:19
  • $\begingroup$ Latent class clustering is another methodological option. Basically, LCA creates a 'model' the heterogeneity in the residual from which is used to cluster. Historically there have been 2 broad research streams in the literature, both sociological. Original LCA dates back to Lazarsfeld at Columbia in the 50s, was unsupervised and used categorical data-R's poLCA is an example of this. More recently supervised finite mixture models for LCA have been developed. I'm not aware of R modules but there is inexpensive commercial software that does it (Latent Gold). LG website has good papers on LCA $\endgroup$
    – user78229
    Commented Apr 20, 2018 at 18:34

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It depends a little bit on your purpose, but if you're after a visualization tool there's a trick with applying multidimensional scaling to the output of random forest proximity which can produce pretty pictures and will work for a mixture of categorical and continuous data. Here you would classify the species according to your predictors. But - and it's a big caveat - I don't know if anyone really knows what the output to these visualizations mean.

Another alternative might be to apply multidimensional scaling to something like the Gower similarity.

There's a hanging question - what's your ultimate purpose? What question do you want to answer? I like these techniques as exploratory tools to perhaps lead you to asking more and better questions, but I'm not sure what they explain or tell you by themselves.

Maybe I'm reading too much into your question, but if you want to explore which predictor variables have the values for the hybrids sitting between the two pure species, you might be better building a model to estimate the values for the predictor variables which lead to the species and the hybrids directly. If you want to measure how the variables are related to each other, perhaps build a correlation matrix - and there are many neat visualizations for this.

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  • $\begingroup$ Thank you for the input. Ultimately, all I want from this analysis is to have some quantitative measure of the similarity of some species compared to others (I have two species which just based on gestalt appearance look like another closely related species, but genetically appear similar to a different species, suggesting ancient hybridization). The main point of this research question is to investigate the genetics of the group, and this morphological analysis will simply add to the whole biological story. Would this multidimensional scaling lead to visualization similar to PCA? $\endgroup$
    – J D
    Commented Nov 23, 2012 at 20:28
  • $\begingroup$ You get similar visualizations. The idea/intuition of MDS is to construct a mapping from a high dimensional space (for you the space of morphological characteristics) to some low dimensional space (like a 2D flat plane) such that distance in the high dimensional space is "pretty much the same" as the low diensional space. You can then plot the 2D flat plane. But it's contingent on getting a distance metric for the high dimensional space from somewhere. $\endgroup$
    – Patrick Caldon
    Commented Nov 25, 2012 at 23:47

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