Mostly About Screening

and Jordan Smoller2



(1)
Department of Epidemiology, Albert Einstein College of Medicine, Bronx, NY, USA

(2)
Department of Psychiatry and Center for Human Genetic Research, Massachusetts General Hospital, Boston, MA, USA

 



I had rather take my chance that some traitors will escape detection than spread abroad a spirit of general suspicion and distrust, which accepts rumor and gossip in place of undismayed and unintimidated inquiry.

Judge Learned Hand

October 1952



5.1 Sensitivity, Specificity, and Related Concepts


The issue in the use of screening or diagnostic tests is to strike the proper trade-off between the desire to detect the disease in people who really have it and the desire to avoid thinking you have detected it in people who really don’t have it.

An important way to view diagnostic and screening tests is through sensitivity analysis. The definitions of relevant terms and symbols are as follows:

T+ means positive test, T− means negative test, D+ means having disease, D− means not having disease. The symbol ∣ means, “given that,” so that P(T + ∣ D−) means positive test, given that there is no disease or D−.

Sensitivity: the proportion of diseased persons the test classifies as positive,



$$ \begin{array}{cc}\hfill =\frac{a}{a+c}=P\left(\mathrm{T}\kern-0.2em +\kern-0.12em \Big|\kern0.5em \mathrm{D}+\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{positive}\;\mathrm{test},\;\mathrm{given}\;\mathrm{disease}\right)\hfill \end{array} $$
Specificity: the proportion of nondiseased persons the test classifies as negative,



$$ \begin{array}{cc}\hfill =\frac{d}{b+d}=P\left(\mathrm{T}\kern-0.15em -\kern-0.12em \Big|\kern0.5em \mathrm{D}-\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{nagative}\;\mathrm{test},\;\mathrm{given}\;\mathrm{no}\;\mathrm{disease}\right)\hfill \end{array} $$
False-positive rate: the proportion of nondiseased persons the test classifies (incorrectly) as positive,



$$ \begin{array}{cc}\hfill =\frac{b}{b+d}=P\left(\mathrm{T}+\Big|\kern0.4em \mathrm{D}-\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{positive}\;\mathrm{test},\;\mathrm{given}\;\mathrm{no}\;\mathrm{disease}\right)\hfill \end{array} $$
False-negative rate: the proportion of diseased people the test classifies (incorrectly) as negative,



$$ \begin{array}{cc}\hfill =\frac{c}{a+c}=P\left(\mathrm{T}-\Big|\kern0.4em \mathrm{D}+\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{negative}\;\mathrm{test},\;\mathrm{given}\;\mathrm{disease}\right)\hfill \end{array} $$
Predictive value of a positive test: the proportion of positive tests that identify diseased persons,



$$ \begin{array}{cc}\hfill =\frac{a}{a+b}=P\left(\mathrm{D}+\Big|\kern0.4em \mathrm{T}+\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{disease}\;\mathrm{given}\;\mathrm{positive}\;\mathrm{test}\right)\hfill \end{array} $$
Predictive value of a negative test: the proportion of negative tests that correctly identifies nondiseased people,



$$ \begin{array}{cc}\hfill =\frac{d}{c+d}=P\left(\mathrm{D}-\Big|\kern0.4em \mathrm{T}-\right);\hfill & \hfill \left(\mathrm{probability}\;\mathrm{of}\;\mathrm{no}\;\mathrm{disease}\;\mathrm{given}\;\mathrm{negative}\;\mathrm{test}\right)\hfill \end{array} $$
Accuracy of the test: the proportion of all tests that are correct classifications,



$$ =\frac{a+d}{a+b+c+d} $$
Likelihood ratio of positive test: the ratio of probability of a positive test, given the disease, to the probability of a positive test, given no disease,



$$ \begin{array}{cc}\hfill =\frac{P\left(\mathrm{T}+|\mathrm{D}+\right)}{P\left(\mathrm{T}+|\mathrm{D}-\right)}\hfill & \hfill \kern-0.85em =\mathrm{positive}\;\mathrm{test},\;\mathrm{given}\;\mathrm{disease}\;\mathrm{versus}\;\mathrm{positive}\;\mathrm{test},\mathrm{given}\;\mathrm{no}\;\mathrm{disease}\hfill \end{array} $$




$$ \begin{array}{cc}\hfill =\frac{\mathrm{sensitivity}}{\mathrm{false}\;\mathrm{positive}\;\mathrm{rate}}\hfill & \hfill \kern-0.85em =\frac{\mathrm{sensitivity}}{1-\mathrm{specificity}}\hfill \end{array} $$

Likelihood ratio of a negative test:



$$ \begin{array}{cc}\hfill =\frac{P\left(\mathrm{T}-|\mathrm{D}+\right)}{P\left(\mathrm{T}-|\mathrm{D}-\right)}\hfill & \hfill =\mathrm{negative}\;\mathrm{test},\;\mathrm{given}\;\mathrm{disease}\;\mathrm{versus}\;\mathrm{negative}\;\mathrm{test},\mathrm{given}\;\mathrm{no}\;\mathrm{disease}\hfill \end{array} $$




$$ \frac{1-\mathrm{specificity}}{\mathrm{sensitivity}} $$

Note also the following relationships:
Nov 20, 2016 | Posted by in PUBLIC HEALTH AND EPIDEMIOLOGY | Comments Off on Mostly About Screening

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