Dear Eimear, I don't think I am going to be able to answer all your questions, but I think I can get the ball rolling and help refine for what you need to look; hopefully someone will be able to elaborate on my explanation and get you all the way there.
First, I need to point out two core ideas from multivariate statistics (and I apologize if you already know this). First, many techniques in multivariate statistics stem from the idea that you can take variables present in your data and combine them into "new" variables that have some desirable properties. Usually this involves taking a weighted sum of the variable values, so in more mathematical parlance we say that the new variables are linear combinations. A key point is that these linear combinations are still random variables and, as a result, we can visualize the linear combinations in the same way we visualize any multivariate data set. Second, although multidimensional, a random, multivariate data point originates, by definition, from a probability distribution, and we can make inferences about the distribution. For example, we can draw confidence regions, which will contain a population parameter, often the mean vector, with a known probability. In the graphic you mention, the statistical software JMP first transformed the sample space into a space of discriminate classifiers. Then it estimated where the population mean for each species is located in the discriminate space, along with regions composed of other probable locations. If an infinite number of these regions were drawn, 95% of them would contain the true population mean. What I do not know, however, is HOW it transformed the original sample space into the discriminate space. In LDA, this process is relatively straight-forward. It is geometrically equivalent to scaling each variable so that they are uncorrelated and have unit variance and then performing a PCA on the group means. This process is possible because every group has the same covariance structure. This is not the case in QDA, and the descriptions that I have read generally do not describe QDA in terms of finding linear combinations or applying some sort of transformation. In my (admittedly young) opinion, how you present the results of your QDA depends on what you want to say about the results. If you want to establish a rule for differentiating otoliths from different populations, then presenting the results of a leave-one-out cross validation (or even more ideally, classification accuracies from a test set) may be sufficient. In contrast, if you want to say that the populations are different from each other, it gets a little bit trickier because you have to address what you mean when you say different. Often discriminate analysis can be pulled into this question because one could argue that populations are different if samples in those populations can be classified to the correct group, and thus can be distinguished from each other. This is a legitimate argument, certainly, but it is important to remember, in my opinion, that this was not the goal of the individuals who created these methods, so you need to be cautious about how you interpret the results. Take a look at this paper for more thoughts on the process: Kovarovic, K., Aiello, L. C., Cardini, A., & Lockwood, C. A. (2011). Discriminant function analyses in archaeology: are classification rates too good to be true?. *Journal of Archaeological Science*, *38*(11), 3006-3018. Getting back to your issue, though, in this latter case, it may make more sense to provide an ordination as well, so long as you understand what the ordination is actually showing and whether what it is showing is relevant to your argument; I am afraid I cannot lend advice on that point without knowing how the sample space was transformed. Finally, in response to your question 3, if I understand what you looking at properly, these two plots are describing two different things. The plot of the discriminate axes is showing you a transformed sample space and where your data resides in this sample space. In contrast, the confidence ellipses estimates where the mean vector for each population is in the transformed sample space and regions where the mean vector is also likely to be located (see above). Hope this helps (sorry it was so long), James Soda On Sun, Jun 28, 2015 at 1:51 AM, Eimear Egan <eimearmariee...@gmail.com> wrote: > Hi all. > > I am examining otolith morphology using elliptical Fourier analysis and > shape indices. I am using the MASS package in R to run a quadratic > discriminant function analysis and I have gotten myself in a bit of a tizzy! > > I am a confused about how to present the results from the QDF. Most people > state the Jack-knife classifications in a table but I have seen one paper > that has plotted the centroid and 95% confidence ellipse (see link) > > http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0034481 > > 1. I don't understand what this plot represents. Is it the posterior > probabilities and if so is it correct to represent the data like this? > > 2. I have trawled through google and came across a link with code for the > iris dataset. The author also plots the centroid and 95% confidence ellipse > and this is of the posterior probabilities (but uses linear DA). > > 3. I have ran a linear DA on my data just to get to grips with > discriminant function in R. Ordination of the two discriminant axes (I have > three groups) versus plotting the posterior probabilities and confidence > ellipses produce very different results...... as expected I guess. Are both > methods correct but just different ways of visualizing the data? And how > does this apply in the context of QDF > > Any insights would be greatly appreciated. > > Thanks, > > Eimear > > > *Eimear Marie Ceileadh Egan* > *PhD Candidate* > Marine Ecology Research Group. > University of Canterbury, > School of Biological Sciences, > Private Bag 4800, Christchurch 8140, New Zealand > *Telephone* +64 3 364 2987 > > > > > -- > 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 morphmet+unsubscr...@morphometrics.org. > -- 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 morphmet+unsubscr...@morphometrics.org.
