Summary: By means of Shannon’s relationship between information and probability, Youden’s index for rating diagnostic tests is shown to be a probability-scale analogue of the log-likelihood ratio of a positive test outcome.

Summary: A correspondence between Youden’s index for rating diagnostic tests and the log-likelihood ratio of a positive test outcome is illustrated by data calculated from two previously-published studies of binary tests.

For an ordinary least-squares regression model, the coefficient of determination (R²) describes the proportion (or percentage) of variance of the response variable explained by the model, and is a widely accepted summary measure of predictive power. A number of R²-analogues are available as summary measures of predictive power associated with logistic regression models, including models of disease risk. Tjur’s R² and McFadden’s R² are of particular interest in this context. Both of these metrics have transparent derivations, which reveal that they apply to different aspects of model evaluation. Tjur’s R² is a measure of separation between (known) actual states (e.g., gold standard determinations of “healthy” or “diseased” status) whereas McFadden’s R² is a measure of separation between predicted states (e.g., forecasts of disease status based on models of disease risk). This clarifies their interpretation in the context of evaluation of logistic regression models of disease risk. In addition, versions of both Tjur’s R² and McFadden’s R² may be obtained from analyses of disease risk that are not preceded by logistic regression analysis. Tjur’s R² and McFadden’s R² are shown to be useful, distinct summary measures of predictive power for epidemiological models of disease risk.

Spatial pattern, an important epidemiological property of plant diseases, can be quantified at different scales using a range of methods. The spatial heterogeneity (or overdispersion) of disease incidence among sampling units is an especially important measure of small-scale pattern. As an alternative to Taylor’s power law for the heterogeneity of counts with no upper bound, the binary power law (BPL) was proposed in 1992 as a model to represent the heterogeneity of disease incidence (number of plant units diseased out of n observed in each sampling unit, or the proportion diseased in each sampling unit). With the BPL, the log of the observed variance is a linear function of the log of the variance for a binomial (i.e., random) distribution. Over the last quarter century, the BPL has contributed to both theory and multiple applications in the study of heterogeneity of disease incidence. In this article, we discuss properties of the BPL and use it to develop a general conceptualization of the dynamics of spatial heterogeneity in epidemics; review the use of the BPL in empirical and theoretical studies; present a synthesis of parameter estimates from over 200 published BPL analyses from a wide range of diseases and crops; discuss model fitting methods, and applications in sampling, data analysis, and prediction; and make recommendations on reporting results to improve interpretation. In a review of the literature, the BPL provided a very good fit to heterogeneity data in most publications. Eighty percent of estimated slope (b) values from field studies were between 1.06 and 1.51, with b positively correlated with the BPL intercept parameter. Stochastic simulations show that the BPL is generally consistent with spatiotemporal epidemiological processes and holds whenever there is a positive correlation of disease status of individuals composing sampling units.

The evidential basis for disease management decision making is provided by data relating to risk factors. The decision process involves an assessment of the evidence leading to taking (or refraining from) action on the basis of a prediction. The primary objective of the decision process is to identify—at the time the decision is made—the control action that provides the best predicted end-of-season outcome, calculated in terms of revenue or another appropriate metric. Data relating to disease risk factors may take a variety of forms (e.g., continuous, discrete, categorical) on measurement scales in a variety of units. Log₁₀-likelihood ratios provide a principled basis for the accumulation of evidence based on such data and allow predictions to be made via Bayesian updating of prior probabilities.

Boundary line models for N₂O emissions from agricultural soils provide a means of estimating emissions within defined ranges. Boundary line models partition a two-dimensional region of parameter space into sub-regions by means of thresholds based on relationships between N₂O emissions and explanatory variables, typically using soil data available from laboratory or field studies. Such models are intermediate in complexity between the use of IPCC emission factors and complex process-based models. Model calibration involves characterizing the extent to which observed data are correctly forecast. Writing the numerical results from graphical two-threshold boundary line models as 3×3 prediction-realization tables facilitates calculation of expected mutual information, a measure of the amount of information about the observations contained in the forecasts. Whereas mutual information characterizes the performance of a forecaster averaged over all forecast categories, specific information and relative entropy both characterize aspects of the amount of information contained in particular forecasts. We calculate and interpret these information quantities for experimental N₂O emissions data.