Zero-inflated Poisson (ZIP) regression is a model for count data with excess zeros. It assumes that with probability p the only possible observation is 0, and with probability 1p, a Poisson(A) random variable is observed. For example, when manufacturing equipment is properly aligned, defects may be nearly impossible. But when it is misaligned, defects may occur according to a Poisson(A) distribution. Both the probability p of the perfect, zero defect state and the mean number of defects A in the imperfect state may depend on covariates. Sometimes p and A are unrelated; other times p is a simple function of A such as p = 1/(1 + AT) for an unknown constant r. In either case, ZIP regression models are easy to fit. The maximum likelihood estimates (MLE's) are approximately normal in large samples, and confidence intervals can be constructed by inverting likelihood ratio tests or using the approximate normality of the MLE's. Simulations suggest that the confidence intervals based on likelihood ratio tests are better, however. Finally, ZIP regression models are not only easy to interpret, but they can also lead to more refined data analyses. For example, in an experiment concerning soldering defects on printed wiring boards, two sets of conditions gave about the same mean number of defects, but the perfect state was more likely under one set of conditions and the mean number of defects in the imperfect state was smaller under the other set of conditions; that is, ZIP regression can show not only which conditions give lower mean number of defects but also why the means are lower.

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. CitationHamilton, V.E., et al., "Evidence for widespread hydrated minerals on asteroid (101955) Bennu." Nature astronomy 3, 4 (2019): p.

Christensen, P. R.; Drouet d'Aubigny, C. Y.; Hamilton, V. E.; Reuter, D. C.; Rizk, B.; Simon, A. A.; Asphaug, E.; Bandfield, J. L.; Barnouin, O. S.; Barucci, M. A.; Bierhaus, E. B.; Binzel, R. P.; Bottke, W. F.; Bowles, N. E.; Campins, H.; Clark, B. C.; Clark, B. E.; Connolly, H. C.; Daly, M. G.; Leon, J. de; Delbo', M.; Deshapriya, J. D. P.; Elder, C. M.; Fornasier, S.; Hergenrother, C. W.; Howell, E. S.; Jawin, E. R.; Kaplan, H. H.; Kareta, T. R.; Le Corre, L.; Li, J.-Y.; Licandro, J.; Lim, L. F.; Michel, P.; Molaro, J.; Nolan, M. C.; Pajola, M.; Popescu, M.; Garcia, J. L. Rizos; Ryan, A.; Schwartz, S. R.; Shultz, N.; Siegler, M. A.; Smith, P. H.; Tatsumi, E.; Thomas, C. A.; Walsh, K. J.; Wolner, C. W. V.; Zou, X.-D. and Lauretta, D. S. (2019). Properties of rubble-pile asteroid (101955) Bennu from OSIRIS-REx imaging and thermal analysis. Nature Astronomy, 3 pp. 341-351. For guidance on citations see FAQs.Length of main text: 2956 words Length of methods: 3605 words Length of legends: 565 words Number of references: 53 main text, 69 including methods and supplementary information (refs 67 to 69 are cited in the SI only) , we show that asteroid (101955) Bennu's surface is globally rough, dense with boulders and low in albedo. The number of boulders is surprising given Bennu's moderate thermal inertia, suggesting that simple models linking thermal inertia to particle size do not adequately capture the complexity relating these properties. At the same time, we find evidence for a wide range of particle sizes with distinct albedo characteristics. Our findings imply that ages of Bennu's surface particles span from the disruption of the asteroid's parent body (boulders) to recent in situ production (micron-scale particles).

The Open University's repository of research publications and other research outputs The dynamic geophysical environment of (101955) Bennu based on OSIRIS-REx measurements