Fall 2008 Thursday, November 6 Speaker: Hail Liu Department of Statistics and Actuarial Science The University of Iowa "Constrained Generalized Additive Models for Zero-Inflated Data". Abstract: Zero-inflated data abound in ecological studies as well as in other scientific and quantitative fields. Nonparametric regression with zero-inflated response may be studied via the so-called Zero-Inflated Generalized Additive Model (ZIGAM). ZIGAM assumes that the conditional distribution of the response variable belongs to the zero-inflated 1-parameter exponential family which is a probabilistic mixture of the zero atom and the 1-parameter exponential family, where the zero atom accounts for an excess of zeroes in the data. The specification of a ZIGAM requires linking some smooth, additive function of the covariates to the probability of the zero atom (zero-inflation probability) and another (possibly distinct) smooth, additive function of the covariates to the mean of the (non-zero-inflated) 1-parameter exponential family. The smooth predictor function linked to the zero-inflation probability need not bear any relationship to the smooth predictor function linked to the mean of the exponential family, which is reasonable if the zero-inflation process is uncoupled from the process underlying the non-zero-inflated data. However, there are cases where the zero-inflation process and the non-zero-inflated data generating process are coupled in such a way that the two smooth functions bear a monotone relationship. For example, this may be the case in trawl surveys where zero inflation may arise from spatio-temporal aggregation of the study species owing to the fact that fish swim in schools. We propose the COnstrained Zero-Inflated Generalized Additive Model (COZIGAM) for analyzing zero-inflated data. The COZIGAM assumes that the response follows some mixture distribution from the zero-inflated 1-parameter exponential family, with further assumption that the probability of zero-inflation is some monotone function of the (non-zero-inflated) exponential family distribution mean. When the latter assumption obtains, the new approach provides a unified framework for modeling zero-inflated data. This bypasses the problems of two popular methods for analyzing zero-inflated data that either focus only on the non-zero data or model the presence-absence data and the non-zero data separately. We develop an iterative algorithm for penalized likelihood estimation of a COZIGAM, and derive formulas for constructing confidence intervals. The asymptotic properties including consistency and limiting distribution of the penalized likelihood estimator are derived. We also propose a Bayesian model selection criterion for choosing between the unconstrained and the constrained ZIGAMs. Some extensions will be discussed, including imposing additive-component-specific pro-portional constraints and a threshold model accounting for phase shift phenomena. The new approach is illustrated with both simulated data and real applications. 3:00 Refreshments in 241B Schaeffer Hall 3:30 Talk in 140 Schaeffer Hall

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