Table 4.
Hypothetical data from screening experiment.
| Raw Data | In symbols |
Numerical illustration |
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|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Actual disease state |
Actual disease state |
|||||||||||
| Test Result | Yes | No | Subtotal | Test Result | Yes | No | Subtotal | |||||
| Positive | a | b | a + b | Positive | 4500 | 3500 | 8000 | |||||
| Negative | c | d | c + d | Negative | 500 | 1500 | 2000 | |||||
| Sub totals |
a + c |
b + d |
N |
Subtotal |
5000 |
5000 |
10 000 |
|||||
| Definitions |
Term |
Definition |
Formula |
Numerical Result |
Alternative Formula or Term |
|||||||
| Prevalence, Π | Fraction of test subjects with disease | (a + c)/N | 0.5000 | Assumed a priori probability of disease, relatively high in this illustration | ||||||||
| Sensitivity, S | Fraction of subjects with positive test given that test subject has disease; “true positive/disease” | a/(a + c) | 0.9000 | Hypothetical data show relatively high sensitivity | ||||||||
| False negative rate | Fraction of subjects with disease, but with negative test result | c/(a + c) | 0.1000 | (1 − S) | ||||||||
| Specificity, Sp | Fraction of test subjects with negative test given that the test subject does not have disease | d/(b + d) | 0.3000 | Hypothetical data show relatively low specificity | ||||||||
| False positive rate | Fraction of test subjects with no disease, but positive test result | b/(b + d) | 0.7000 | (1 − Sp) | ||||||||
| Probability of positive test | True positives + false positives divided by total tests | (a + b)/N | 0.8000 | P(T+) = ΠS + (1 − Π)(1 − Sp) | ||||||||
| Probability of negative test | True negatives + false negatives divided by total tests | (c + d)/N | 0.2000 | P(T−) = Π(1 − S) + (1 − Π)Sp | ||||||||
| Positive predictive value PPV | Post-test probability of disease given a positive result | a/(a + b) | 0.5625 | A posteriori probability of disease given positive test result | ||||||||
| Negative predictive value NPV | Post-test probability of no disease given a negative test result | d/(c + d) | 0.750 | A posteriori probability no disease given negative test result | ||||||||
| Accuracy | Proportion of correct test results | (a + d)/N | 0.6000 | ΠS + (1 − Π)Sp | ||||||||
| Likelihood ratio | The probability of a subject who has the disease testing positive divided by the probability of a subject who does not have the disease testing positive | S/(1 − Sp) | 1.2857 | |||||||||
| Regret given positive test |
Probability that disease free subject has positive test |
b/(a + b) | 0.4375 | (1 − Π)(1 − Sp)/(ΠS + (1 − Π)(1 − Sp)) | ||||||||
| Bayes Theorem | Positive test |
Negative test |
||||||||||
| True state |
P(Hi) A priori Probability Person tested is in this state |
P(T+/Hi) Probability of positive test in this state |
P(Hi)P(T+/Hi) Joint probability |
P(Hi/T+) A posteriori probability |
True state |
P(Hi) A priori Probability Person tested is in this state |
P(/T−Hi) Probability of negative test in this state |
P(Hi)P(T−/Hi) Joint probability |
P(Hi/T−) A posteriori probability |
|||
| Disease | 0.5000 | 0.90 | 0.4500 | 0.5625 | Disease | 0.5000 | 0.10 | 0.0500 | 0.2500 | |||
| No disease | 0.5000 | 0.70 | 0.3500 | 0.4375 | No disease | 0.5000 | 0.30 | 0.1500 | 0.7500 | |||
| Probability of positive test = P(T+) | 0.8000 | Probability of negative test = P(T−) | 0.2000 | |||||||||
Additional references providing useful background: Alberg et al. (2004), Eddy (1982), Goetzinger & Odibo (2011), Lalkhen & McClusky (2008).