define: Definition of `truePrevMulti` and `truePrevMulti2` model

Description

These utility functions generate definitions for the test results and priors used by truePrevMulti and truePrevMulti2.

Usage

define_x(h)
define_prior(h)
define_prior2(h)

Arguments

Number of tests

Details

The vector of apparent tests results, x, must contain the number of samples corresponding to each combination of test results. The models assume that the first value corresponds to the number of samples that tested positive on all tests and that the last value corresponds to the number of samples that tested negative on all tests.

Function truePrevMulti estimates true prevalence from individual samples tested with h tests, using the approach of Berkvens et al. (2006). The prior in the multinomial model consists of a vector theta, which holds values for the true prevalence (TP), the sensitivity and specificity of the first test (SE1, SP1), and the conditional dependencies between the results of the subsequent tests and the preceding one(s). define_prior generates the definition of prior for h tests.

Function truePrevMulti2 implements and extends the approach described by Dendukuri and Joseph (2001), which uses a multinomial distribution to model observed test results, and in which conditional dependence between tests is modelled through covariances. Argument prior consists of prior distributions for:

True Prevalence: TP
SEnsitivity of each individual test: vector SE
SPecificity of each individual test: vector SP
Conditional covariance of all possible test combinations given a truly positive disease status: vector a
Conditional covariance of all possible test combinations given a truly negative disease status: vector b

define_prior2 generates the definition of prior for h tests.

References

Berkvens D, Speybroeck N, Praet N, Adel A, Lesaffre E (2006) Estimating disease prevalence in a Bayesian framework using probabilistic constraints. Epidemiology 17:145-153
Dendukuri N, Joseph L (2001) Bayesian approaches to modeling the conditional dependence between multiple diagnostic tests. Biometrics 57:158-167

Examples

Run this code

# NOT RUN {
## how is a 2-test model defined?

define_x(2)
# Definition of the apparent test results, 'x', for 2 tests:
# x[1] : T1-,T2- 
# x[2] : T1-,T2+ 
# x[3] : T1+,T2- 
# x[4] : T1+,T2+

define_prior(2)
# Conditional probability scheme
# Definition of the prior, 'theta', for 2 tests: 
# theta[1] : P(D+) = TP
# theta[2] : P(T1+|D+) = SE1
# theta[3] : P(T1-|D-) = SP1
# theta[4] : P(T2+|D+,T1+)
# theta[5] : P(T2+|D+,T1-)
# theta[6] : P(T2-|D-,T1-)
# theta[7] : P(T2-|D-,T1+)

define_prior2(2)
# Covariance scheme
# Definition of the prior for 2 tests: 
# TP :    True Prevalence 
# SE[1] : Sensitity T1 
# SE[2] : Sensitity T2 
# SP[1] : Specificity T1 
# SP[2] : Specificity T2 
# a[1] :  Covariance(T1,T2|D+) 
# b[1] :  Covariance(T1,T2|D-)

## how is a 3-test model defined?

define_x(3)
# Definition of the apparent test results, 'x', for 3 tests:
# x[1] : T1-,T2-,T3- 
# x[2] : T1-,T2-,T3+ 
# x[3] : T1-,T2+,T3- 
# x[4] : T1-,T2+,T3+ 
# x[5] : T1+,T2-,T3- 
# x[6] : T1+,T2-,T3+ 
# x[7] : T1+,T2+,T3- 
# x[8] : T1+,T2+,T3+

define_prior(3)
# Conditional probability scheme
# Definition of the prior, 'theta', for 3 tests: 
# theta[1] : P(D+) = TP
# theta[2] : P(T1+|D+) = SE1
# theta[3] : P(T1-|D-) = SP1
# theta[4] : P(T2+|D+,T1+)
# theta[5] : P(T2+|D+,T1-)
# theta[6] : P(T2-|D-,T1-)
# theta[7] : P(T2-|D-,T1+)
# theta[8] : P(T3+|D+,T1+,T2+)
# theta[9] : P(T3+|D+,T1+,T2-)
# theta[10] : P(T3+|D+,T1-,T2+)
# theta[11] : P(T3+|D+,T1-,T2-)
# theta[12] : P(T3-|D-,T1-,T2-)
# theta[13] : P(T3-|D-,T1-,T2+)
# theta[14] : P(T3-|D-,T1+,T2-)
# theta[15] : P(T3-|D-,T1+,T2+)

define_prior2(3)
# Covariance scheme
# Definition of the prior for 3 tests: 
# TP :    True Prevalence 
# SE[1] : Sensitity T1 
# SE[2] : Sensitity T2 
# SE[3] : Sensitity T3 
# SP[1] : Specificity T1 
# SP[2] : Specificity T2 
# SP[3] : Specificity T3 
# a[1] :  Covariance(T1,T2|D+) 
# a[2] :  Covariance(T1,T3|D+) 
# a[3] :  Covariance(T2,T3|D+) 
# a[4] :  Covariance(T1,T2,T3|D+) 
# b[1] :  Covariance(T1,T2|D-) 
# b[2] :  Covariance(T1,T3|D-) 
# b[3] :  Covariance(T2,T3|D-) 
# b[4] :  Covariance(T1,T2,T3|D-)
# }