## Predictive model assessment for count data (2007)

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### BibTeX

@TECHREPORT{Czado07predictivemodel,

author = {Claudia Czado and Tilmann Gneiting and Leonhard Held},

title = {Predictive model assessment for count data},

institution = {},

year = {2007}

}

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### Abstract

Summary. We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for ordered discrete data. Our proposals include a non-randomized version of the probability integral transform, marginal calibration diagrams and proper scoring rules, such as the predictive deviance. In case studies, we critique count regression models for patent data, and assess the predictive performance of Bayesian age-period-cohort models for larynx cancer counts in Germany.

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Citation Context ...02, p. 587). If the predictive distribution is a member of a one-parameter exponential family, such as the binomial or Poisson, the standardizing term is routinely taken to be the saturated deviance (=-=McCullagh and Nelder, 1989-=-, pp. 33-34; Knorr-Held and Rainer, 2001, p. 114; Spiegelhalter et al., 2002, p. 606; Clements et al., 2005, p. 581). However, when the predictive distributions come from possibly distinct parametric ... |

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Citation Context ...nsform (PIT) that is tailored to count data, and the marginal calibration diagram. Section 3 discusses the use of scoring rules as omnibus performance measures. We stress the importance of propriety (=-=Gneiting and Raftery, 2007-=-), note examples, relate to classical measures of predictive performance, and 2sidentify the predictive deviance as a variant of the proper logarithmic score. Section 4 turns to a cross-validation stu... |

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Citation Context ...is regular if s(P, x) is finite, except possibly that s(P, x) = ∞ if px = 0. Let P denotes the class of probability measures on the set of the nonnegative integers. The Savage representation theorem (=-=Savage, 1971-=-; Gneiting and Raftery, 2007) states that a regular scoring rule S for count data is proper if and only if where h : P →ss(P, x) = h(P ) − ∞� k=0 h ′ k (P )pk + h ′ x (P ) is a concave function and h ... |

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Citation Context ...Gunther and Tay, 1998; Gneiting et al., 2007). The PIT histogram is typically used informally as a diagnostic tool; formal tests can also be employed though they require care in their interpretation (=-=Hamill, 2001-=-; Jolliffe, 2007). Deviations from uniformity hint at reasons for forecast failures and model deficiencies. U-shaped histograms indicate underdispersed predictive distributions, hump or inverse-U shap... |

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Citation Context ...y plotting the empirical CDF of a set of PIT values and comparing to the identity function, or by plotting the histogram of the PIT values and checking for uniformity (Diebold, Gunther and Tay, 1998; =-=Gneiting et al., 2007-=-). The PIT histogram is typically used informally as a diagnostic tool; formal tests can also be employed though they require care in their interpretation (Hamill, 2001; Jolliffe, 2007). Deviations fr... |

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Citation Context ...obabilistic forecasts, or predictive distributions, for count data, as they occur in a wide range of epidemiological, ecological, environmental, climatological, demographic and economic applications (=-=Christensen and Waagepetersen, 2002-=-; Gotway and Wolfinger, 2003; McCabe and Martin, 2005; Elsner and Jagger, 2006; Frühwirth-Schnatter and Wagner, 2006; Nelson and Leroux, 2006). Our focus is on the low count situation in which continu... |

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Citation Context ...or, is the same for all three forecasts. 4. Case study: Model critique for count regression Count data often show substantial extra variation or overdispersion relative to a Poisson regression model (=-=Dean and Lawless, 1989-=-; Winkelmann, 2005). Various alternatives have been suggested to accommodate this, such as negative binomial and mixed Poisson models (Lawless, 1987). In this section, we investigate whether the non-r... |

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Citation Context ...e range of epidemiological, ecological, environmental, climatological, demographic and economic applications (Christensen and Waagepetersen, 2002; Gotway and Wolfinger, 2003; McCabe and Martin, 2005; =-=Elsner and Jagger, 2006-=-; Frühwirth-Schnatter and Wagner, 2006; Nelson and Leroux, 2006). Our focus is on the low count situation in which continuum approximations fail; however, our results apply to high counts and rates as... |

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Citation Context ...l, ecological, environmental, climatological, demographic and economic applications (Christensen and Waagepetersen, 2002; Gotway and Wolfinger, 2003; McCabe and Martin, 2005; Elsner and Jagger, 2006; =-=Frühwirth-Schnatter and Wagner, 2006-=-; Nelson and Leroux, 2006). Our focus is on the low count situation in which continuum approximations fail; however, our results apply to high counts and rates as well, as they occur routinely in epid... |

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Citation Context ... be strictly proper. If s(Q, Q) ≤ s(P, Q) for all P and Q, the scoring rule is said to be proper. Propriety is an essential property of a scoring rule that encourages honest and coherent predictions (=-=Bröcker and Smith, 2007-=-; Gneiting and Raftery, 2007). Strict propriety ensures that both calibration and sharpness are being addressed. A scoring rule s for count data is regular if s(P, x) is finite, except possibly that s... |

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Citation Context ... score, where µP and σ 2 P nses(P, x) = � � x − µP 2 σP , (12) denote the mean and the variance of P , ought be approximately one when averaged over the predictions (Carroll and Cressie, 1997, p. 52; =-=Liesenfeld et al., 2006-=-, pp. 811, 818). Gotway and Wolfinger (2003, p. 1423) call the mean normalized squared error score the average empirical-to-model variability ratio, arguing also that it should be close to 11 issone. ... |

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Citation Context ...sidual or normalized squared error score, where µP and σ 2 P nses(P, x) = � � x − µP 2 σP , (12) denote the mean and the variance of P , ought be approximately one when averaged over the predictions (=-=Carroll and Cressie, 1997-=-, p. 52; Liesenfeld et al., 2006, pp. 811, 818). Gotway and Wolfinger (2003, p. 1423) call the mean normalized squared error score the average empirical-to-model variability ratio, arguing also that i... |

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Citation Context ...distributions, for count data, as they occur in a wide range of epidemiological, ecological, environmental, climatological, demographic and economic applications (Christensen and Waagepetersen, 2002; =-=Gotway and Wolfinger, 2003-=-; McCabe and Martin, 2005; Elsner and Jagger, 2006; Frühwirth-Schnatter and Wagner, 2006; Nelson and Leroux, 2006). Our focus is on the low count situation in which continuum approximations fail; howe... |

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Citation Context ...y, 1998; Gneiting et al., 2007). The PIT histogram is typically used informally as a diagnostic tool; formal tests can also be employed though they require care in their interpretation (Hamill, 2001; =-=Jolliffe, 2007-=-). Deviations from uniformity hint at reasons for forecast failures and model deficiencies. U-shaped histograms indicate underdispersed predictive distributions, hump or inverse-U shaped histograms po... |

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Citation Context ...ial or Poisson, the standardizing term is routinely taken to be the saturated deviance (McCullagh and Nelder, 1989, pp. 33-34; Knorr-Held and Rainer, 2001, p. 114; Spiegelhalter et al., 2002, p. 606; =-=Clements et al., 2005-=-, p. 581). However, when the predictive distributions come from possibly distinct parametric or non-parametric families, it is vital that the standardizing terms in the deviance are common (Spiegelhal... |

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Citation Context ...). Our focus is on the low count situation in which continuum approximations fail; however, our results apply to high counts and rates as well, as they occur routinely in epidemiological projections (=-=Knorr-Held and Rainer, 2001-=-; Clements, Armstrong and Moolgavkar, 2005). To this date, statistical methods for the assessment of predictive performance have been studied primarily from biomedical, meteorological and economic per... |

2 |
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Citation Context ...ect cancer incidence and mortality rates. Data from younger age groups (typically age < 30 years) for which rates are low are often excluded from the analysis. However, a recent empirical comparison (=-=Baker and Bray, 2005-=-) based on data from Hungary suggests that age-specific predictions based on full data are more accurate. A natural question arises here in how to quantify the quality of the predictive distributions.... |

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Citation Context ...ctive distribution, x ∼ P is a random count and v is standard uniform and independent of x, then u = Px−1 + v(Px − Px−1), x ≥ 1, (1) u = vP0, x = 0, (2) is standard uniform (Smith, 1985, pp. 286–287; =-=Frühwirth-Schnatter, 1996-=-, p. 297; Liesenfeld, Nolte and Pohlmeier, 2006, pp. 819–820). For time series data one typically considers onestep (or k-step) ahead predictions, based on a time series model fitted on past and curre... |

2 |
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Citation Context ...nd Moolgavkar, 2005). To this date, statistical methods for the assessment of predictive performance have been studied primarily from biomedical, meteorological and economic perspectives (Pepe, 2003; =-=Jolliffe and Stephenson, 2003-=-; Clements, 2005), focusing on predictions of dichotomous events or real-valued continuous variables. Here, we consider the hybrid case of count data, in which methods developed for either type of sit... |

2 |
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(Show Context)
Citation Context ...ogical, demographic and economic applications (Christensen and Waagepetersen, 2002; Gotway and Wolfinger, 2003; McCabe and Martin, 2005; Elsner and Jagger, 2006; Frühwirth-Schnatter and Wagner, 2006; =-=Nelson and Leroux, 2006-=-). Our focus is on the low count situation in which continuum approximations fail; however, our results apply to high counts and rates as well, as they occur routinely in epidemiological projections (... |

1 | Re: “Bayesian projections: What are the effects of excluding data from younger age groups - Clements, Hakulinen, et al. - 2006 |

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