Regression analysis of forest damage by marginal models for correlated ordinal responses

Abstract

Studies on forest damage can generally not be carried out by common regression models, mainly for two reasons: Firstly, the response variable, damage state of trees, is usually observed in ordered categories. Secondly, responses are often correlated, either serially, as in a longitudinal study, or spatially, as in the application of this paper, where neighborhood interactions exist between damage states of spruces determined from aerial pictures. Thus so{called marginal regression models for ordinal responses, taking into account dependence among observations, are appropriate for correct inference. To this end we extend the binary models of Liang and Zeger (1986) and develop an ordinal GEE1 model, based on parametrizing association by global cross{ratios. The methods are applied to data from a survey conducted in Southern Germany. Due to the survey design, responses must be assumed to be spatially correlated. The results show that the proposed ordinal marginal regression models provide appropriate tools for analyzing the in uence of covariates, that characterize the stand, on the damage state of spruce.

Signals of summer drought in crown condition data from the German Level I network

“…Mortality and discoloration should be separately studied by risk analysis or marginal models (e.g. Fahrmeir and Pritscher 1996). Table 2; coefficients in accordance with the drought respective fructification hypothesis are bold letters, those contrary to both hypotheses are in italics; length of time series: 6 years NS (Pr > |t|) > 0.05, * (Pr > |t|) £ 0.05, ** (Pr > |t|) £ 0.01, *** (Pr > |t|) < 0.0001 Crown transparency is an unspecific, however easily accessible parameter of tree condition and meets the requirements of a broad-scaled area-representative monitoring.…”

Modeling individual migraine severity with autoregressive ordered probit models

“…While GEE could also be used for ordinal valued time series (see for example Liang et al (1992), Heagerty and Zeger (1996) and Fahrmeir and Pritscher (1996)), we prefer a likelihood based regression time series approach to investigate the influence of weather conditions on migraine severity. One major reason for this preference is to have a complete statistical model specification, which allows the usage of standard model comparison techniques and forecasts in dynamic models.…”

A spatial model with ordinal responses for grazing impact data

“…In addition, Rue (2001) proposes an algorithm for speeding up mixing and convergence with Bayesian spatial models (see also Knorr-Held and Rue, 2002). There are other potential methods of analysis: for example, Wolfe and Firth (2002) consider maximum likelihood estimation of ordinal response models; and both Fahrmeir and Pritscher (1996) and Huang et al (2002) consider the use of generalised estimating equations. There are other potential methods of analysis: for example, Wolfe and Firth (2002) consider maximum likelihood estimation of ordinal response models; and both Fahrmeir and Pritscher (1996) and Huang et al (2002) consider the use of generalised estimating equations.…”