Hello,

I think you can use ANOVA test by putting the Collour as factor , you put your data as showed in picture attached, and run your ANOVA test.

You can also calculate The Mahalanobis distance between two points in a p-dimensional space, taking into account the variance-covariance structure on these p dimensions. The square of this distance is defined by:

dM² = (x1 - x2) Σ-1 (x1 - x2)

So we have to calcuate three Mahalanobis Dsitance: (1) between "Green vs Blue" (2) "Green vs Red" (3) " Red vs Blue".

Best wishes

What is the null hypothesis you want to test? If this is that the dropping location is independent of the type of samara and you can assume that all the individual datra are independent (except that they belong to the same group), you can bootstrap the pattern by re-assigning the type (red, blue, green) to the coordinates. For each bootstrap sample you can calculate the Eukidean distances between the "red" the "green" and the "blue" points. You can then find how often you get larger distances than the one of your actual sample (the proportion of such instances is the estimate of the p-value).

If the independence assumption is not justified (for example, because the starting positions of the falling samara were systematically different between the groups), then you will need a model of the distribution of the group centers. As an initial guess I would model the polar coordinates of the group center with the radius as gamma-distributed and the angle as uniform distributed. The shape and the rate parameter could be estimated from the current sample (shape = mean²/variance and rate = mean/variance). According to this, a series of random group centers can be generated and the distances can be calculated (as before).

Thanks Chellai Fatih I was thinking of a Mahalanobis distance matrix, I will try!

@Jochen Wilhelm this was conducted indoors under controlled conditions so the starting point was static. I will try using euklidian distances!

Hi, Eric,

The analyses described herein are very popular, and often work quite well. JIC you'd like to try a modern nonparametric method that by formulation identifies the most accurate solution possible for a given application, I offer this response.

You have a class ("dependent") variable with three categories, and two ordered attributes ("independent variables").

Your question implies that you wish to combine the red and blue class categories, and discriminate this pooled group from the green class category.

First I feel that I should mention Simpson's Paradox -- combining groups can result in incorrect solutions. Here are two open-access articles on this topic:

https://odajournal.com/2015/04/21/estimating-inter-rater-reliability-using-pooled-data-induces-paradoxical-confounding-an-example-involving-emergency-severity-index-triage-ratings

https://odajournal.com/2013/09/19/the-use-of-unconfounded-climatic-data-improves-atmospheric-prediction

Nevertheless, concerning the two-class-category analysis you asked about, you can use a modern analysis which: (a) requires no distributional assumptions, (b) is invariant over any monotonic transformation of the attributes, (c) provides an exact p-value, (d) provides effect strength indices which eliminate the effect of chance and correct for model complexity, (e) conducts jackknife analysis as well as many other cross-generalizability analyses, and (f) finds the most accurate model possible for your data. Here is a brief article which introduces this modern paradigm:

https://odajournal.com/2017/04/18/what-is-optimal-data-analysis

And an article that discusses understanding and application of a simple one-attribute optimal analysis:

https://odajournal.com/2018/11/26/visualizing-application-and-summarizing-accuracy-of-oda-models

Here are two articles describing and demonstrating a maximum-accuracy classification tree model (there are many more, search for CTA or classification tree analysis in my research article page):

Article Automated CTA Software: Fundamental Concepts and Control Commands

https://odajournal.com/2016/10/06/cta-vs-not-chi-square-fear-and-specific-recommendations-do-synergistically-affect-behavior

There are hundreds of other examples of such analyses, in the open-access ODA eJournal, and in pay-for-view journals. Several hundred such articles are presented in a sorted list on the publications tab of the eJournal. Using your web browser, go to the secure (https:) webpage at ODAJournal dot com.

Best wishes!

Eric Wahlsteen , The Euclidean distance corresponds to the distance of Mahalanobis in the case where the covariance matrix is the identity matrix, that is to say that the variables are centered reduced and independent.

In your case I don't think that is verified , so the Mahalanobis is recomended .

Paul Yarnold thanks for your answer and your references. A lot to read!

New ODA Method Discovered!

One beautiful feature of ODA is that it adapts to fit the analytic problem.

One troublesome feature of legacy methods is that data must often be adapted to meet the assumptions of the method.

Your question in this thread is a wonderful example of the former!

EXISTING ODA METHOD

In your (this particular) application, given the distribution of observations on the X and Y axes, it is obvious that the three class categories (red, blue, green) could simply be well-discriminated on the basis of observations’ scores on the X axis.

For exposition, call the threshold (cutpoint) on the X axis which best discriminates the red and blue observations cutpoint RB, and call the threshold on X which best discriminates the blue and green observations cutpoint BG.

What does “best discriminates” mean? At the discretion of the analyst, ODA identifies thresholds which either discriminate groups with maximum overall accuracy (percent accurate classification or PAC), or with maximum accuracy after removing (accounting for) the effect of chance (effect strength for sensitivity or ESS).

In either case (i.e., regardless of the objective function which is optimized by the ODA model), every individual observation may also be weighted by its mass, or by its quality, or by the product of these two weights, etcetera. When weights are used ODA identifies thresholds which explicitly maximize weighted PAC or ESS.

The unweighted 3-class-category ODA model would be:

IF observation’s score on X cutpoint RB and cutpoint BG, THEN PREDICT class=green.

There are NO distributional assumptions for any ODA analysis, the EXACT p-value is obtained, and jackknife validity may be computed to estimate model CROSS-GENERALIZABILITY.

For clarity, here is an article demonstrating use of this methodology to assess inter-method (or inter-observer) reliability:

https://odajournal.com/2014/04/13/how-to-assess-inter-observer-reliability-of-ratings-made-on-ordinal-scales-evaluating-and-comparing-the-emergency-severity-index-version-3-and-canadian-triage-acuity-scale

NEW ODA METHOD

This is a stepwise procedure.

(Note: I was on the editorial board of Educational and Psychological Measurement, which IIRC was the first methodological journal which refused to accept articles using stepwise procedures. After my rotation on the board ended, I found a perfect stepwise regression application involving creating a short-form of a literacy assessment tool: Arozullah AM, Yarnold PR, Bennett CL, Soltysik RC, Wolf MS, Ferreira MR, Lee SYD, Costello S, Shakir A, Denwood C, Bryant FB, Davis T (2007). Development and validation of a short-form Rapid Estimate of Adult Literacy in Medicine. Medical Care, 45, 1026-1033.)

In the FIRST step discriminate the green versus the red and blue observations:

IF observation’s score on X > cutpoint BG, THEN PREDICT class=green. Otherwise PREDICT class=not green. Note that this model correctly classifies 100% of the observations, however the red and blue observations are co-mingled.

In the SECOND step omit the green observations from analysis: the red and blue groups must now be discriminated.

Ordinarily, in this application, the red and blue observations would be discriminated using a simple ODA model. For clarity, here is an article demonstrating this methodology:

https://odajournal.com/2018/11/26/visualizing-application-and-summarizing-accuracy-of-oda-models

However, in your application, it appears possible that a model involving scores on both the X and Y axes could yield a more accurate [(weighted) PAC or ESS] model. This would be accomplished using classification tree analysis (CTA), which chains multiple ODA models to obtain maximum-accuracy solutions. Here is an example of a CTA model:

https://odajournal.com/2016/10/06/cta-vs-not-chi-square-fear-and-specific-recommendations-do-synergistically-affect-behavior.

The ODA (Step 1) and CTA (Step 2) models would then be summarized as a single model:

If observation’s score on X > cutpoint BG, THEN PREDICT class=green;

Otherwise, if If observation’s score on (taken from CTA model) THEN PREDICT class=blue;

Otherwise, PREDICT class=red.

Fun!

Hi, Eric Wahlsteen

I created an artificial data set to illustrate the novometric stepwise CTA method which I discussed above--put flesh on the bones, so to speak. My simulated data were more noisy than data in your scatter plot. If interested, here is link to the 4-page article:

• https://odajournal.com/2019/04/06/novometric-stepwise-cta-analysis-discriminating-three-class-categories-using-two-ordered-attributes

I took a second look at your scatter plot--and in light of what I've learned, it appears to me that a simple linear model (red < blue < green on the X dimension) would do very well in training analysis (relatively strong model), with a bit of slippage in jackknife performance.

The attached PDF shows the simple linear ODA model which I mentioned (approximated by eye). As seen, sensitivity in training analysis is 100% correct for green observations, 94% for blue, and 68% for red. This level of accuracy corresponds to ESS = 80.10 -- indicating a strong effect in model development.

Reproducibility analysis is needed, I suggest a LOO. The PDF indicates the observations which may become incorrectly classified in LOO analysis. However, the effect would remain relatively strong in cross-generalizability analysis.

If interested. an annotated example of an ODA model, analysis, and interpretation is available here:

• https://odajournal.com/2018/11/26/visualizing-application-and-summarizing-accuracy-of-oda-models