From: Bias in trials comparing paired continuous tests can cause researchers to choose the wrong screening modality
Test 1
Observed Disease Status
Probability
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{ r [ 1 − Φ ( x − μ C 1 ) ] + r ( 1 − ψ ) × [ Φ ( x − μ C 1 , θ − μ C 2 , ρ C ) − Φ ( θ − μ C 1 , θ − μ C 2 , ρ C ) ] x < θ r [ 1 − Φ ( x − μ C 1 ) ] x ≥ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8C9B@
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{ ( 1 − r ) [ 1 − Φ ( x − μ N 1 ) ] − r ( 1 − ψ ) × [ Φ ( x − μ C 1 , θ − μ C 2 , ρ C ) − Φ ( θ − μ C 1 , θ − μ C 2 , ρ C ) ] x < θ ( 1 − r ) [ 1 − Φ ( x − μ N 1 ) ] x ≥ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@93F0@
{ r Φ ( x − μ C 1 ) − r ( 1 − ψ ) Φ ( x − μ C 1 , θ − μ C 2 , ρ C ) x < θ r Φ ( x − μ C 1 ) − r ( 1 − ψ ) Φ ( θ − μ C 1 , θ − μ C 2 , ρ C ) x ≥ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@87CD@
{ ( 1 − r ) Φ ( x − μ N 1 ) + r ( 1 − ψ ) Φ ( x − μ C 1 , θ − μ C 2 , ρ C ) x < θ ( 1 − r ) Φ ( x − μ N 1 ) + r ( 1 − ψ ) Φ ( θ − μ C 1 , θ − μ C 2 , ρ C ) x ≥ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8F01@
