Critical appraisal of jointness concepts in Bayesian model averaging: evidence from life sciences, sociology, and other scientific fields

ABSTRACT

Jointness is a Bayesian approach to capturing dependence among regressors in multivariate data. It addresses the general issue of whether explanatory factors for a given empirical phenomenon are complements or substitutes. I ask a number of questions about existing jointness concepts: Are the patterns revealed stable across datasets? Are results robust to prior choice and do data characteristics affect results? And importantly: What do the answers imply from a practical vista? The present study takes an applied, interdisciplinary and comparative perspective, validating jointness concepts on datasets across scientific fields with focus on life sciences (Parkinson's disease) and sociology. Simulations complement the study of real-world data. My findings suggest that results depend on which jointness concept is used: Some concepts deliver jointness patterns remarkably uniform across datasets, while all concepts are fairly robust to the choice of prior structure. This can be interpreted as critique of jointness from a practical perspective, given that the patterns revealed are at times very different and no concept emerges as overall advantageous. The composite indicators approach to combining information across jointness concepts is also explored, suggesting an avenue to facilitate the application of the concepts in future research.

KEYWORDS:

• Bayesian model averaging
• ecology
• economic growth
• jointness concepts
• Parkinson's disease
• sociology

AMS 2010 SUBJECT CLASSIFICATION:

• 62F15
• 91B40
• 91B62
• 92C50
• 92D40

JEL CLASSIFICATIONS:

• C11
• I19
• J31
• O47
• Q54

Acknowledgments

I would like to thank Professors Herman K. van Dijk and George Michailidis for their suggestions. Also, I thank two anonymous reviewers for valuable comments, as well as Professor Robert G. Aykroyd and the editors Journal of Applied Statistics, for evaluating the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Part of the exposition here borrows from my other work on BMA.

2. An additional distinction is made between significant jointness ($|J|>1$) and strong jointness ($|J|>2$), see Doppelhofer and Weeks [10, p. 221)].

3. Dobra et al.  [8] also apply the jointness concept due to Ley and Steel [21] to the Fernández et al. [16] data, but use the GGM (Gaussian Graphical Model) framework. Their results suggest the presence of a considerable amount of jointness.

4. The jointness tables in Amini [2] are ordered according to the covariates' position in the dataset, not according to PIP rank, as is the case in Doppelhofer and Weeks [10] and in my jointness tables presented in Section 5.

5. For this framework, the sensitivity analysis extends to the choice of prior model size: The estimations are re-run for different values of the prior model size (see the parameter $\overline{k}$ in the Sala-i-Martin et al. [38] article); it turns out that the results are reasonably robust across prior model sizes. The reported results are for $\overline{k}=7$.

6. Eicher et al. [13] interpret PIPs above $50\mathrm{\%}$ such that a regressor does have an impact on growth, while PIPs exceeding $95\mathrm{\%}$ are taken to imply a strong effect on growth. These authors also provide a critical view on the BACE significance criterion due to Sala-i-Martin et al. [38] but note that results do not appear to differ dramatically across the two criteria.

7. The scale for UPDRS ranges from 0 (healthy state) to 176 (maximum impairment). Sakar et al. [37] find that dysphonia measures are reliable predictors for the UPDRS metric from a relatively low threshold level already, namely 15. This criterion is fulfilled here.

8. The detailed, that is, exact coloring (in the electronic version of this article) introduced by Doppelhofer and Weeks [10] also distinguishes between significant versus strong jointness. Generally, blue respectively boldface highlights significant positive jointness, while yellow and orange respectively italics is for significant negative jointness. The jointness surface is a concept utilized by Strachan [39]; these plots are 3-dimensional visualizations of the matrices with the $J_{il}$ entries. The vertical dimension shows the sign and magnitude of the entries in the jointness matrix.

9. It is again the case that the HG-RIC prior structure delivers results with less instances of significant jointness and less instances of substitutability.

10. For complements, the variables forming the jointness pairs belong less often to the same concept and typically have low individual PIPs, which can be rationalized either since complements are more informative together than in isolation.

11. The consideration of the CIs approach has been suggested by an anonymous reviewer.

12. While many variables in the dataset are strongly correlated with each other, the correlation pattern cannot be matched with the jointness results, which again provides some support for the finding by Doppelhofer and Weeks [10] that there is no simple link between jointness and bivariate correlations.

13. The inclusion of a simulation study has been proposed by an anonymous reviewer.