I have carried a survey to consumers. A question can be seen as a vector (x, y, z) with x+y+z=1. For example, for a consumer I have: x=0.5, y=0.3, z=0.2.
Is there any statistical test/procedure to measure relationships among (x, y, z), as explained variable, and multiple explanatory variables, including ratio, ordinal and nominal ones, such as age, studies, etc.?
Thanks!
You have compositional data. Use this keyword to find the statistical methods to use, and the seminal work from Aitchison.
If I interpret your saying right then you have a single dependent variable (DV) which is explained by three independent variables (IV). To analyze their influence on the DV you can use the Analysis of Variance (ANOVA).
Re-explain your question again since it's hard to understand your problem.
Hi, Alexander:
I have 3 dependent variables (DV) (x, y, z), which are related since they sum up to 1: x+y+z=1. This relationship is important to be taken into account.
I have multiple independent variables (IV): ratio, ordinal and nominal variables.
I want to analyse the influence of IV on DV.
Thanks for your comments
You have compositional data. Use this keyword to find the statistical methods to use, and the seminal work from Aitchison.
Emmanuel has probably given you the best advice. Also, try thinking in terms of degrees of information. X, Y, and Z are only two pieces of information, not three, because of the constraint that X+Y+Z=1. Said another way, if you know X and Y, you know Z, because Z = 1-X-Y.
Maybe a structural equation modeling can help (?). more specifically a Latent Variable Structural Model. So you will have a latent variable that is a composition of X, Y, and Z, which in turn will be explained by your independent variables (IVs).
You can break your model into three, with one DP per model. If you've got the same ID for each model then use the -2 log likelihood test (for maximum likelihood estimation) to compare model fit between the three. I believe a chi-square test can be used to compare OLS models. This approach will also save you some degrees-of-freedom which is good if you have a small dataframe.
Also, I'd agree with Hassan Jafari on the SEM. But SEMS can sometimes get overcomplicated with direct and indirect effects.
First a Cluster with the variables X, Y, Z
Then apply univariate and multivariate techniques for a dependent variable (The cluster obtained)
Dear Carlos,
The problem you state is one of the treated using compositional data analysis. One of the ways would be to think that (x,y,z) is a composition on which log-ratio analysis is appropriate. It require that you are interested in the ratios between x,y,z, say the odds x/y, x/z, y/z more than in their differences x-y, y-z, x-z. In that case, the isometric log-ratio transformation (ilr) allows to represent (x,y,z) in coordinates. These coordinates, called balances, can be
b1 = ln ( g(x,y) / z ) , g(x,y) = sqrt(x y)
b2 = ln ( x / y ) .
This is a generalization of the logit transform.
Now your problem reduces to two multiple regressions: b1 predicted by the explanatory covariates and factors; and b2 also predicted by the explanatory variables, in the standard way of multiple regression and the associated standard tests. Once these models are fitted, the ilr transformation can be inverted.
For elementary compositional regression I would recommend
Egozcue, J. J., J. Daunis-i Estadella, Pawlowsky-Glahn, V., Hron, K., and P. Filzmoser, 2012: Simplicial regression. The normal model. Journal of Applied Probability and Statistics (JAPS), 6 (1&2), pages = 87-108.
For a general view of compositional data
Pawlowsky-Glahn, V., J. J. Egozcue, and R. Tolosana-Delgado, 2015: Modeling
and analysis of compositional data. Statistics in practice. John Wiley
& Sons, Chichester UK. 272 pp.
Thank you very much for your answer, Juan Jose. I read about compositional data and your proposal matches with the idea I got from it. Your text clarifies it further to me. Thanks again for your useful help!
I agree with Juan Jose. Care must be taken when analyzing compositional data with a unit sum constraint to not end up with predictions that are not scientifically possible. I would also recommend the resources that Juan has suggested. Good luck.
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