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Binary spatial observations arise in environmental and ecological studies, where Markov random field (MRF) models are often applied. Despite the prevalence and the long history of MRF models for spatial binary data, appropriate model diagnostics have remained an unresolved issue in practice. A complicating factor is that such models involve neighborhood specifications, which are difficult to assess for binary data. To address this, we propose a formal goodness-of-fit (GOF) test for diagnosing an MRF model for spatial binary values. The test statistic involves a type of conditional Moran's I based on the fitted conditional probabilities, which can detect departures in model form, including neighborhood structure. Numerical studies show that the GOF test can perform well in detecting deviations from a null model, with a focus on neighborhoods as a difficult issue. We illustrate the spatial test with an application to Besag's historical endive data as well as the breeding pattern of grasshopper sparrows across Iowa., (© The Author(s) 2024. Published by Oxford University Press on behalf of The International Biometric Society.)

Sudden death syndrome (SDS) is an important soybean [Glycine max (L) Merrill] disease caused by the soilborne fungus Fusarium virguliforme. Currently, 14 quantitative trait loci (QTL) had been confirmed associated with resistance or tolerance to SDS. The objective of the study was to evaluate usefulness of 10 of these QTL in controlling disease expression. Six populations were developed providing a total of 321 F2-derived lines for the study. Recombinant inbred lines (RIL) used as parents were obtained from populations of 'Essex' × 'Forrest' (EF), 'Flyer' × 'Hartwig' (FH), and 'Pyramid' × 'Douglas' (PD). Disease resistance was evaluated in the greenhouse at three different planting times, each with four replications, using sorghum infested with F. virguliforme homogeneously mixed in the soil (Luckew et al., Crop Sci 52:2215-2223, 2012). Four disease assessment criteria-foliar disease incidence (DI), foliar leaf scorch disease severity (DS), area under the disease progress curve (AUDPC), and root rot severity-were used. QTL were identified in more than one of the disease assessment criteria, mainly associated with lines in the most resistant categories. Five QTL (qRfs4, qRfs5, qRfs7, qRfs12, and Rfs16) were associated with at least one of the disease assessments across multiple populations. Of the five, qRfs4 was associated with DI, AUDPC, and root rot severity, and Rfs16 with AUDPC and root rot severity. The findings suggest it may be possible for plant breeders to focus on stacking a subset of the previously identified QTL to improve resistance to SDS in soybean.