Skip to main content

Advertisement

SpringerLink
Log in

Bayesian Estimation for Diagnostic Testing of Biosecurity Risk Material in the Absence of a Gold Standard when Test Data are Incomplete

  • Published:
Journal of Agricultural, Biological, and Environmental Statistics Aims and scope Submit manuscript

Abstract

Diagnostic testing is used by biosecurity officers for the detection and identification of plant and animal pathogens, often informing high-consequence decisions such as restricting the entry of trade goods. It is rare that such tests can be considered gold standards; however, uncertainty can be reduced by using the results of other tests, measuring performance on samples of known status and incorporating prior knowledge from expert judgement. This article presents an extension to the methods of Joseph et al. (Am J Epidemiol 141:263–272, 1995), and Dendukuri and Joseph (Biometrics 57:158–167, 2001) for Bayesian estimation in the absence of a gold standard test, which allows for the use of incomplete test data. This extension is demonstrated with a novel application: the case study of myrtle rust from Holliday et al. (Plant Dis 97:828–834, 2013), which involves samples from potential biosecurity risk material on importation pathways to Australia. The samples were tested at two laboratories, and prior estimates for pathway prevalence were obtained by expert elicitation. The Bayesian estimation was based on a model with and without covariances for the test results to assess the assumption of conditional independence. The results show that pathogen prevalence, diagnostic sensitivity and diagnostic specificity can be estimated using all available data even where some samples have been subject to only one of two available tests. The results also indicate the importance of consideration of the assumption of conditional independence. The findings enable diagnostic testing laboratories and decision makers to make use of all test results and to explicitly incorporate prior knowledge to estimate pathogen prevalence and test accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Andersen S (1997) Re: “Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard”. American Journal of Epidemiology 145:290–291

  • Aven T (2013) On the meaning and use of the risk appetite concept. Risk Analysis 33:462–468

  • Beavers DP, Stamey JD (2012) Bayesian sample size determination for binary regression with a misclassified covariate and no gold standard. Computation Statistics and Data Analysis 56:2574–2582

  • Black MA, Craig BA (2002) Estimating disease prevalence in the absence of a gold standard. Statistics in Medicine 21:2653–2669

  • Burgman MA, McBride M, Ashton R, Speirs-Bridge A, Flander L, Wintle B, Fidler F, Rumpff L, Twardy C (2011) Expert status and performance. PLoS ONE 6:e22,998

  • Carnegie AJ, Lidbetter JR, Walker J, Horwood MA, Tesoriero L, Glen M, Priest MJ (2010) Uredo rangelii, a taxon in the guava rust complex, newly recorded on Myrtaceae in Australia. Australasian Plant Pathology 39:463–466

  • Casella G, George EI (1992) Explaining the Gibbs Sampler. The American Statistician 46:167–174

  • Coutinho TA, Wingfield MJ, Alfenas AC, Crous PW (1998) Eucalyptus rust: a disease with the potential for serious international implications. Plant Disease 82:819–825

  • Demissie K, White N, Joseph L, Ernst P (1998) Bayesian estimation of asthma prevalence, and comparison of exercise and questionnaire diagnostics in the absence of a gold standard. Annals of Epidemiology 8:201–208

  • Dendukuri N, Joseph L (2001) Bayesian approaches to modeling the conditional dependence between multiple diagnostic tests. Biometrics 57:158–167

  • Dongen SV (2006) Prior specification in Bayesian statistics: three cautionary tales. Journal of Theoretical Biology 242:90–100

  • Dorny P, Phiri IK, Vercruysse J, Gabriel S, Willingham AL, Brandt J, Victor B, Speybroeck N, Berkvens D (2004) A Bayesian approach for estimating values for prevalence and diagnostic test characteristics of porcine cysticercosis. International Journal for Parasitology 34:569–576

  • Dunson DB (2001) Commentary: practical advantages of Bayesian analysis of epidemiologic data. American Journal of Epidemiology 153:1222–1226

  • Enoe C, Georgiadis MP, Johnson WO (2000) Estimation of sensitivity and specificity of diagnostic tests and disease prevalence when the true disease state is unknown. Preventive Veterinary Medicine 45:61–81

  • Epstein LD, Munoz A, He D (1996) Bayesian imputation of predictive values when covariate information is available and gold standard diagnosis is unavailable. Statistics in Medicine 15:463–476

  • Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliability Engineering & System Safety 54(2):133–144

  • Garber A, Solomon N (1999) Cost-effectiveness of alternative test strategies for the diagnosis of coronary artery disease. Annals of Internal Medicine 130:719–728

  • Geisser S, Johnson W (1992) Optimal administration of dual screening tests for detecting a characteristic with special reference to low prevalence diseases. Biometrics 48:839–852

  • Georgiadis MP, Johnson WO, Gardner IA, Singh R (2003) Correlation-adjusted estimation of sensitivity and specificity of two diagnostic tests. Journal of the Royal Statistical Society: Series C (Applied Statistics) 52(1):63–76

  • Glen M, Alfenas AC, Zauza EAV, Wingfield MJ, Mohammed C (2007) Puccinia psidii: a threat to the Australian environment and economy - a review. Australasian Plant Pathology 36:1–16

  • Grgurinovic CA, Walsh D, Macbeth F (2006) Eucalyptus rust caused by Puccinia psidii and the threat it poses to Australia. EPPO Bulletin 36:486–489

  • de Groot JAH, Dendukuri N, Janssen KJM, Reitsma JB, Brophy J, Joseph L, Bossuyt PMM, Moons KGM (2012) Adjusting for partial verification or workup bias in meta-analyses of diagnostic accuracy studies. American Journal of Epidemiology 174:847–853

  • Hanson T, Johnson WO, Gardner IA (2003) Hierarchical models for estimating herd prevalence and test accuracy in the absence of a gold standard. Journal of agricultural, biological, and environmental statistics 8(2):223–239

  • Holliday JL, Jones SA, Simpson JA, Glen M, Edwards J, Robinson A, Burgman MA (2013) A novel spore collection device for sampling exposure pathways: a case study of Puccinia psidii. Plant Disease 97:828–834

  • Johnson WO, Gastwirth JL, Pearson LM (2001) Screening without a “Gold Standard”: the Hui-Walter paradigm revisited. American Journal of Epidemiology 153:921–924

  • Joseph L (1997) Re: “Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard” The first author replies. American Journal of Epidemiology 145:291

  • Joseph L, Gyorkos TW, Coupal L (1995) Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard. American Journal of Epidemiology 141:263–272

  • Kawanishi T, Uematsu S, Kakishima M, Kagiwada S, Hamamoto H, Horie H, Namba S (2009) First report of rust disease on ohia and the causal fungus, Puccinia psidii, in Japan. Journal of General Plant Pathology 75:428–431

  • Keefer DL, Bodily SE (1983) Three-point approximations for continuous random variables. Management Science 29:595–609

  • Krull CR, Waipara NW, Choquenot D, Burns BR, Gormley AM, Stanley MC (2013) Absence of evidence is not evidence of absence: feral pigs as vectors of soil-borne pathogens. Austral Ecology 38:534–542

  • Langrell SRH, Glen M, Alfenas AC (2008) Molecular diagnosis of Puccinia psidii (guava rust) - a quarantine threat to Australian eucalypt and Myrtaceae biodiversity. Plant Pathology 57:687–701

  • Lew RA, Levy PS (1989) Estimation of prevalence on the basis of screening tests. Statistics in Medicine 8(10):1225–1230

  • Lewis F, Sanchez-Vanquez MJ, Torgerson PR (2012) Association between covariates and disease occurrence in the presence of diagnostic error. Epidemiology and Infection 140:1515–1524

  • Liu J, Chen F, Yu H, Zeng P, Liu L (2014) A two-stage Bayesian method for estimating accuracy and disease prevalence for two dependent dichotomous screening tests when the status of individuals who are negative on both tests is unverified. BMC medical research methodology 14(1):110

  • Lu Y, Dendukuri N, Schiller I, Joseph L (2010) A Bayesian approach to simultaneously adjusting for verification and reference standard bias in diagnostic test studies. Statistics in Medicine 29:2532–2543

  • Makowski D, Denis J-B, Ruck L, Penaud A (2008) A Bayesian approach to assess the accuracy of a diagnostic test based on plant disease measurement. Crop Protection 27:1187–1193

  • Martin TG, Kuhnert PM, Mengersen K, Possingham HP (2005) The power of expert opinion in ecological models using Bayesian methods: impact of grazing on birds. Journal of Applied Ecology 15:266–280

  • McBride MF, Fidler F, Burgman MA (2012) Evaluating the accuracy and calibration of expert predictions under uncertainty: predicting the outcomes of ecological research. Diversity and Distributions 18:782–794

  • Mendoza-Blanco JR, Tu XM, Iyengar S (1996) A Bayesian inference on prevalence using a missing-data approach with simulation-based techniques: applications to HIV screening. Statistics in Medicine 15:2161–2176

  • Mila AL, Yang XB, Carriquiry AL (2003) Bayesian logistic regression of soybean Sclerotinia stem rot prevalence in the U.S. north central region: accounting for uncertainty in parameter estimation. Phytopathology 93:758–764

  • Mireku E, Simpson JA (2002) Fungal and nematode threats to Australian forests and amenity trees from importation of wood and wood products. Canadian Journal of Plant Pathology 24:117–124

  • Morris WK, Vesk PA, McCarthy MA (2013) Profiting from pilot studies: analysing mortality using Bayesian models with informative priors. Basic and Applied Ecology 14:81–89

  • Nauta MJ (2000) Separation of uncertainty and variability in quantitative microbial risk assessment models. International Journal of Food Microbiology 57(1):9–18

  • Pennello GA (2011) Bayesian analysis of diagnostic test accuracy when disease state is unverified for some subjects. Journal of Biopharmaceutical Statistics 21:954–970

  • Pollino CA, White AK, Hart BT (2007) Examination of conflicts and improved strategies for the management of an endangered Eucalypt species using Bayesian networks. Ecological Modelling 201:37–59

  • Poole WK, Flynn PM, Rao AV, Cooley PC (1996) Estimating the prevalence of drug use from self-reports in a cohort for which biologic data are available for a subsample. American Journal of Epidemiology 144:413–420

  • Rahme E, Joseph L (1998) Estimating the prevalence of a rare disease: adjusted maximum likelihood. Journal of the Royal Statistical Society: Series D (The Statistician) 47(1):149–158

  • Reitsma JB, Rutjes AW, Khan KS, Coomarasamy A, Bossuyt PM (2009) A review of solutions for diagnostic accuracy studies with an imperfect or missing reference standard. Journal of clinical epidemiology 62(8):797–806

  • Rubin DB (1988) Using the SIR algorithm to simulate posterior distributions. Bayesian statistics 3(1):395–402

  • Schiffman M, Wheeler CM, Dasgupta A, Solomon D, Castle PE (2005) A comparison of a prototype PCR assay and Hybrid Capture 2 for detection of carcinogenic human papillomavirus DNA in women with equivocal or mildly abnormal papanicolaou smears. American Journal of Clinical Pathology 124:722–732

  • Scott AN, Joseph L, Belisle P, Behr MA, Schwartzman K (2008) Bayesian modelling of tuberculosis clustering from DNA fingerprint data. Statistics in Medicine 27:140–156

  • Simpson JA, Thomas K, Grgurinovic CA (2006) Uredinales species pathogenic on species of Myrtaceae. Australasian Plant Patholology 35:549–562

  • Speirs-Bridge A, Fidler F, McBride M, Flander L, Cumming G, Burgman M (2010) Reducing overconfidence in the interval judgments of experts. Risk Analysis 30:512–523

  • Spiegelhalter DJ, Best NG (2003) Bayesian approaches to multiple sources of evidence and uncertainty in complex cost-effectiveness modelling. Statistics in Medicine 22(23):3687–3709

  • Stubner S (2002) Enumeration of 16S rDNA of Desulfotomaculum lineage 1 in rice field soil by real-time PCR with SybrGreen\(_{{\rm TM}}\) detection. Journal of Microbiological Methods 50:155–164

  • Toft N, Jørgensen E, (2005) Diagnosing diagnostic tests: evaluating the assumptions underlying the estimation of sensitivity and specificity in the absence of a gold standard. Preventive veterinary medicine 68(1):19–33

  • Tommerup IC, Alfenas AC, Old KM (2003) Guava rust in Brazil - a threat to eucalyptus and other Myrtaceae. New Zealand Journal of Science 33:420–428

  • Tu XM, Kowalski J, Jia G (1999) Bayesian analysis of prevalence with covariates using simulation-based techniques: applications to HIV screening. Statistics in Medicine 18:3059–3073

  • Vettraino AM, Sukno S, Vannini A, Garbelotto M (2010) Diagnostic sensitivity and specificity of different methods used by two laboratories for the detection of Phytophthora ramorum on multiple natural hosts. Plant Pathology 59:289–300

  • Vidal E, Moreno A, Berolini E, Cambra M (2012) Estimation of the accuracy of two diagnostic methods for the detection of Plum pox virus in nursery blocks by latent class models. Plant Pathology 61:413–422

  • Vijan S, Hwang EW, Hofer TP, Hayward RA (2001) Which colon cancer screening test? a comparison of costs, effectiveness, and compliance. American Journal of Medicine 111:593–601

  • Vlems FA, Ladanyi A, Gertler R, Rosenberg R, Diepstra JHS, Rder C, Nekarda H, Molnar B, Tulassay Z, van Muijen GNP, Vogel I (2003) Reliability of quantitative reverse-transcriptase-PCR-based detection of tumour cells in the blood between different laboratories using a standardised protocol. European Journal of Cancer 39:388–396

Download references

Acknowledgments

This study was supported by the Australian Centre of Excellence for Risk Analysis (now named the Centre of Excellence for Biosecurity Risk Analysis). We sincerely thank the experts who participated in the elicitation of priors, the biosecurity officers and volunteers who facilitated the collection of samples, and the scientists who performed the laboratory testing. We are also very grateful to Andrew Robinson, Bonnie Wintle, Marissa McBride and Mark Burgman for their advice. Finally, we appreciate the time taken by two anonymous reviewers for providing detailed comments which have greatly improved the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Statistical Consulting Centre, University of Melbourne, Parkville, 3010, Australia

    Sandra Jane Clarke

  2. Centre of Excellence for Biosecurity Risk Analysis, School of Botany, University of Melbourne, Parkville, 3010, Australia

    Stuart Andrew Jones

Authors
  1. Sandra Jane Clarke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stuart Andrew Jones
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandra Jane Clarke.

Conditional Distributions for Gibbs Sampler

Conditional Distributions for Gibbs Sampler

1.1 Assuming Conditional Independence

Letting \(Y_{\cdot }=\sum Y_{i}\),

$$\begin{aligned} Y_{1}\mid n_1\pi ,S_1,C_1, S_2, C_2&\sim \hbox {Binomial}\Big (n_{1},\frac{\pi S_1S_2}{\pi S_1S_2+(1-\pi )(1-C_1)(1-C_2)}\Big )\\ Y_{2}\mid n_2\pi ,S_1,C_1,S_2, C_2&\sim \hbox {Binomial}\Big (n_{2},\frac{\pi S_1(1-S_2)}{\pi S_1(1-S_2)+(1-\pi )(1-C_1)C_2}\Big )\\ Y_{3}\mid n_3\pi ,S_1,C_1,S_2,C_2&\sim \hbox {Binomial}\Big (n_{3},\frac{\pi (1-S_1)S_2}{\pi (1-S_1)S_2+(1-\pi )C_1(1-C_2)}\Big )\\ Y_{4}\mid n_4\pi ,S_1,C_1,S_2,C_2&\sim \hbox {Binomial}\Big (n_{4},\frac{\pi (1-S_1)(1-S_2)}{\pi (1-S_1)(1-S_2)+(1-\pi )C_1C_2}\Big )\\ Y_{5}\mid n_5\pi ,S_1,C_1&\sim \hbox {Binomial}\Big (n_{5},\frac{\pi S_1}{\pi S_1+(1-\pi _i)(1-C_1)}\Big )\\ Y_{6}\mid n_6\pi ,S_1,C_1&\sim \hbox {Binomial}\Big (n_{6},\frac{\pi (1-S_1)}{\pi (1-S_1)+(1-\pi )C_1}\Big )\\ Y_{7}\mid n_7\pi ,S_2,C_2&\sim \hbox {Binomial}\Big (n_{7},\frac{\pi S_2}{\pi S_2+(1-\pi )(1-C_2)}\Big )\\ Y_{8}\mid n_8\pi ,S_2,C_2&\sim \hbox {Binomial}\Big (n_{8},\frac{\pi (1-S_2)}{\pi (1-S_2)+(1-\pi )C_2}\Big )\\ \pi \mid N, Y_{\cdot }&\sim \hbox {Beta}(Y_{\cdot }+\alpha _{\pi },N-Y_{\cdot }+\beta _{\pi })\\ S_1\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{5},Y_{6}, \alpha _{S1}, \beta _{S1}&\sim \hbox {Beta}(Y_{1}+Y_{2}+Y_{5}+\alpha _{S1},Y_{3}+Y_{4}+Y_{6}+\beta _{S1})\\ S_2\mid Y_1,Y_{2},Y_{3},Y_{4}, Y_{7},Y_{8}, \alpha _{S_2}, \beta _{S_2}&\sim \hbox {Beta}(Y_{1}+Y_{3}+Y_{7}+\alpha _{S_2},Y_{2}+Y_{4}+Y_{8}+\beta _{S_2})\\ C_1\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{5},Y_{6}, \alpha _{C_1}, \beta _{C_1}&\sim \hbox {Beta}(n_{3}+n_{4}+n_{6}-(Y_{3}+Y_{4}+Y_{6})+\alpha _{C_1},\\&n_{1}+n_{2}+n_{5}-(Y_{1}+Y_{2}+Y_{5})+\beta _{C_1})\\ C_2\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{7},Y_{8}, \alpha _{C_2}, \beta _{C_2}&\sim \hbox {Beta}(n_{2}+n_{4}+n_{8}-Y_{2}-Y_{4}-Y_{8}+\alpha _{C_2},\\&n_{1}+n_{3}+n_{7}-Y_{1}-Y_{3}-Y_{7}+\beta _{C_2}) \end{aligned}$$

1.2 Allowing for Conditional Independence

The following posterior distributions do not have a known form so sampling importance resampling (Rubin 1988) was used to sample the values at each iteration. Letting \(Y_{\cdot }=\sum Y_{i}, u_s=min(S_1,S_2)-S_1S_2\) and \(u_c=min(C_1,C_2)-C_1C_2\),

$$\begin{aligned} Y_{1}\mid n_1\pi ,S_1,C_1, S_2, C_2, covs, covc&\sim \hbox {Binomial}\Big (n_{1},\frac{\pi (S_1S_2+covs)}{\pi (S_1S_2+covs)+(1-\pi )((1-C_1)(1-C_2)+covc)}\Big )\\ Y_{2}\mid n_2\pi ,S_1,C_1,S_2, C_2, covs, covc&\sim \hbox {Binomial}\Big (n_{2},\frac{\pi (S_1(1-S_2)-covs)}{\pi (S_1(1-S_2)-covs)+(1-\pi )((1-C_1)C_2-covc)}\Big )\\ Y_{3}\mid n_3\pi ,S_1,C_1,S_2,C_2, covs, covc&\sim \hbox {Binomial}\Big (n_{3},\frac{\pi ((1-S_1)S_2-covs)}{\pi ((1-S_1)S_2-covs)+(1-\pi )(C_1(1-C_2)-covc)}\Big )\\ Y_{4}\mid n_4\pi ,S_1,C_1,S_2,C_2, covs, covc&\sim \hbox {Binomial}\Big (n_{4},\frac{\pi ((1-S_1)(1-S_2)+covs)}{\pi ((1-S_1)(1-S_2)+covs)+(1-\pi )(C_1C_2+covc)}\Big )\\ Y_{5}\mid n_5\pi ,S_1,C_1&\sim \hbox {Binomial}\Big (n_{5},\frac{\pi S_1}{\pi S_1+(1-\pi _i)(1-C_1)}\Big )\\ Y_{6}\mid n_6\pi ,S_1,C_1&\sim \hbox {Binomial}\Big (n_{6},\frac{\pi (1-S_1)}{\pi (1-S_1)+(1-\pi )C_1}\Big )\\ Y_{7}\mid n_7\pi ,S_2,C_2&\sim \hbox {Binomial}\Big (n_{7},\frac{\pi S_2}{\pi S_2+(1-\pi )(1-C_2)}\Big )\\ Y_{8}\mid n_8\pi ,S_2,C_2&\sim \hbox {Binomial}\Big (n_{8},\frac{\pi (1-S_2)}{\pi (1-S_2)+(1-\pi )C_2}\Big ) \end{aligned}$$
$$\begin{aligned} \pi \mid N, Y_{\cdot } \sim \hbox {Beta}(Y_{\cdot }+\alpha _{\pi },N-Y_{\cdot }+\beta _{\pi })&\\ p(S_1\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{5},Y_{6}, S_2,\alpha _{S_1},\beta _{S_1}, \beta _{covs})&\propto (S_1S_2+covs)^{Y_1}(S_1(1-S_2)\\&\quad -covs)^{Y_2}((1-S_1)S_2-covs)^{Y_3}\\&\times ((1-S_1)(1-S_2)+covs)^{Y_4}\\&\quad S_1^{Y_5}(1-S_1)^{Y_6}S_1^{\alpha _{S_1}-1}(1-S_1)^{\beta _{S_1}-1}\\&\quad (u_s-covs)^{\beta _{covs}-1}\\ p(S_2\mid Y_1,Y_{2},Y_{3},Y_{4}, Y_{7},Y_{8}, S_1,\alpha _{S_2},\beta _{S_2}, \beta _{covs})&\propto (S_1S_2+covs)^{Y_1}(S_1(1-S_2)\\&\quad -covs)^{Y_2}((1-S_1)S_2-covs)^{Y_3}\\&\times ((1-S_1)(1-S_2)+covs)^{Y_4}S_2^{Y_7}\\&\quad (1-S_2)^{Y_8}S_2^{\alpha _{S_2}-1}(1-S_2)^{\beta _{S_2}-1}\\&\quad (u_s-covs)^{\beta _{covs}-1}\\ p(C_1\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{5},Y_{6}, C_2,\alpha _{C_1},\beta _{C_1},\beta _{covc})&\propto ((1-C_1)(1-C_2)+covc)^{n_1-Y_1}\\&\quad ((1-C_1)C_2-covc)^{n_2-Y_2}\\&\quad (C_1(1-C_2)-covc)^{n_3-Y_3}\\&\times (C_1C_2+covc)^{n4-Y_4}(1-C_1)^{n_5-Y_5}\\&\quad C_1^{n_6-Y_6}C_1^{\alpha _{C_1}-1}(1-C_1)^{\beta _{C_1}-1}\\&\quad (u_c-covc)^{\beta _{covc}-1}\\ p(C_2\mid Y_{1},Y_{2},Y_{3},Y_{4},Y_{7},Y_{8}, C_1,\alpha _{C_2},\beta _{C_2}, \beta _{covc})&\propto ((1-C_1)(1-C_2)+covc)^{n_1-Y_1}\\&\quad ((1-C_1)C_2-covc)^{n_2-Y_2}\\&\quad (C_1(1-C_2)-covc)^{n_3-Y_3}\\&\times (C_1C_2+covc)^{n4-Y_4}(1-C_2)^{n_7-Y_7}\\&\quad C_2^{n_8-Y_8}C_2^{\alpha _{C_2}-1}(1-C_2)^{\beta _{C_2}-1}\\&\quad (u_c-covc)^{\beta _{covc}-1}\\ p(covs\mid S_1,S_2,Y_{1},Y_{2},Y_{3},Y_{4}, u_s,\alpha _{covs},\beta _{covs})&\propto (S_1S_2+covs)^{Y_1}(S_1(1-S_2)\\&\quad -covs)^{Y_2}((1-S_1)S_2-covs)^{Y_3}\\&\times ((1-S_1)(1-S_2)+covs)^{Y_4}\\&\quad covs^{\alpha _{covs-1}}(u_s-covs)^{\beta _{covs-1}}\\ p(covs\mid C_1,C_2,Y_{1},Y_{2},Y_{3},Y_{4}, u_c,\alpha _{covc},\beta _{covc})&\propto ((1-C_1)(1-C_2)+covc)^{n_1-Y_1}\\&\quad ((1-C_1)C_2-covc)^{n_2-Y_2}\\&\times (C_1(1-C_2)-covc)^{n_3-Y_3}\\&\quad (C_1C_2+covc)^{n4-Y_4} covc^{\alpha _{covc-1}}\\&\quad (u_c-covc)^{\beta _{covc-1}} \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clarke, S.J., Jones, S.A. Bayesian Estimation for Diagnostic Testing of Biosecurity Risk Material in the Absence of a Gold Standard when Test Data are Incomplete. JABES 20, 389–408 (2015). https://doi.org/10.1007/s13253-015-0214-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13253-015-0214-5

Keywords

Search

Navigation