Abstract
Recall that the type of regression model you use is determined mostly by the properties of the response variable. Well what if you have more than one response variable?
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The mean is written m ij here, not μ ij, because in a hierarchical model, it is a random quantity, not a fixed parameter.
- 2.
Note that fix=1 does not mean “fix the variance to one”; it means “fix the variances for all random effects from the first one onwards”. If a model has three different types of random effects in it, fix=2 would fix the variance of the second and third random effects but not the first.
- 3.
Although if the purpose of the study is prediction, additional features that are known to often help are shrinking parameters (using a LASSO or assuming regression coefficients are drawn from a common distribution), or in large datasets, using flexible regression tools (like additive models) to handle non-linearity.
References
Anderson, E. (1935). The irises of the Gaspe Peninsula. Bulletin of the American Iris Society, 59, 2–5.
Anderson, T. W. (2003). An introduction to multivariate statistical analysis (3rd ed.). New York: Wiley.
Bates, D. M. (2010). lme4: Mixed-effects modeling with R. New York: Springer.
Brooks, M. E., Kristensen, K., van Benthem, K. J., Magnusson, A., Berg, C. W., Nielsen, A., Skaug, H. J., Machler, M., & Bolker, B. M. (2017). glmmTMB balances speed and flexibility among packages for zero-inflated generalized linear mixed modeling. The R journal, 9, 378–400.
Clark, J. S. (2007). Models for ecological data. Princeton, NJ: Princeton University Press.
Clark, J. S., Gelfand, A. E., Woodall, C. W., & Zhu, K. (2014). More than the sum of the parts: Forest climate response from joint species distribution models. Ecological Applications, 24, 990–999.
Cressie, N., Calder, C. A., Clark, J. S., Hoef, J. M. V., & Wikle, C. K. (2009). Accounting for uncertainty in ecological analysis: The strengths and limitations of hierarchical statistical modeling. Ecological Applications, 19, 553–570.
Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
Diggle, P. J., Heagerty, P., Liang, K.-Y &, Zeger, S. (2002). Analysis of longitudinal data (2nd ed.). Oxford University Press.
Evans, M., & Swartz, T. (2000). Approximating integrals via Monte Carlo and deterministic methods. Oxford statistical science series. OUP Oxford. ISBN: 9780191589874.
Gelman, A., Stern, H. S., Carlin, J. B., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis. Chapman and Hall/CRC.
Hadfield, J. D. et al. (2010). MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package. Journal of Statistical Software, 33, 1–22.
Hartig, F. (2020). DHARMa: Residual diagnostics for hierarchical (multi-level/mixed) regression models. R package version 0.3.2.0.
Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning (2nd ed.). New York: Springer.
Johnson, N., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distributions. Wiley series in probability and statistics. Wiley. ISBN: 9780471128441.
Letten, A. D., Keith, D. A., Tozer, M. G., & Hui, F. K. (2015). Fine-scale hydrological niche differentiation through the lens of multi-species co-occurrence models. Journal of Ecology, 103, 1264–1275.
Ovaskainen, O., & Abrego, N. (2020). Joint species distribution modelling: With applications in R. Cambridge: Cambridge University Press.
Robert, C., & Casella, G. (2013). Monte Carlo statistical methods. Springer.
Warton, D. I. (2008). Penalized normal likelihood and ridge regularization of correlation and covariance matrices. Journal of the American Statistical Association, 103, 340–349.
Weisbecker, V., & Warton, D. I. (2006). Evidence at hand: Diversity, functional implications, and locomotor prediction in intrinsic hand proportions of diprotodontian marsupials. Journal of Morphology, 267, 1469–1485.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Warton, D.I. (2022). More Than One Response Variable: Multivariate Analysis. In: Eco-Stats: Data Analysis in Ecology. Methods in Statistical Ecology. Springer, Cham. https://doi.org/10.1007/978-3-030-88443-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-88443-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-88442-0
Online ISBN: 978-3-030-88443-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)