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Comparison Between Individual Testing and Sample Pooling for Whole Populations


infected / condition positive (P)
the number of real cases of infection
$$=$$
not infected / condition negative (N)
the number of actually uninfected people
$$=$$
5.00%
Individual Testing
Expected Values
Total Cost
at per test
$$=$$
Tests Used
each person tested separately
$$=$$

true positive (TP)
eqv. with hit
$$=$$
true negative (TN)
eqv. with correct rejection
$$=$$
false positive (FP)
eqv. with false alarm Type I error
$$=$$
false negative (FN)
eqv. with miss Type II error
$$=$$

sensitivity, recall, hit rate or true positive rate (TPR)
$$\mathrm {TPR} ={\frac {\mathrm {TP} }{\mathrm {P} }}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}=1-\mathrm {FNR} $$
$$=$$
specificity, selectivity, or true negative rate (TNR)
$$\mathrm {TNR} ={\frac {\mathrm {TN} }{\mathrm {N} }}={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FP} }}=1-\mathrm {FPR} $$
$$=$$
85.00%

100.00%

Sample Pooling
Expected Values
25 Optimal Pool Size:

Total Cost
at per test
$$=$$
Optimal Total Cost
cost at optimal pool size
$$=$$
Tests Used
pools and retests
$$=$$
Minimum Tests Possible
tests used at optimal pool size
$$=$$

true positive (TP)
eqv. with hit
$$=$$
true negative (TN)
eqv. with correct rejection
$$=$$
false positive (FP)
eqv. with false alarm Type I error
$$=$$
false negative (FN)
eqv. with miss Type II error
$$=$$

effective sensitivity / true positive rate (TPR)
$$\mathrm {TPR} ={\frac {\mathrm {TP} }{\mathrm {P} }}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}=1-\mathrm {FNR} $$
$$=$$
effective specificity / true negative rate (TNR)
$$\mathrm {TNR} ={\frac {\mathrm {TN} }{\mathrm {N} }}={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FP} }}=1-\mathrm {FPR} $$
$$=$$

precision or positive predictive value (PPV)
$$\mathrm {PPV} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FP} }}=1-\mathrm {FDR} $$
$$=$$
negative predictive value (NPV)
$$\mathrm {NPV} ={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FN} }}=1-\mathrm {FOR} $$
$$=$$
miss rate or false negative rate (FNR)
$$\mathrm {FNR} ={\frac {\mathrm {FN} }{\mathrm {P} }}={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TP} }}=1-\mathrm {TPR} $$
$$=$$
fall-out or false positive rate (FPR)
$$\mathrm {FPR} ={\frac {\mathrm {FP} }{\mathrm {N} }}={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TN} }}=1-\mathrm {TNR} $$
$$=$$
false discovery rate (FDR)
$$\mathrm {FDR} ={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TP} }}=1-\mathrm {PPV} $$
$$=$$
false omission rate (FOR)
$$\mathrm {FOR} ={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TN} }}=1-\mathrm {NPV} $$
$$=$$
Prevalence Threshold (PT)
$$PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}$$
$$=$$
Threat score (TS)
$$\mathrm {TS} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} +\mathrm {FP} }}$$
$$=$$
accuracy(ACC)
$$\mathrm {ACC} ={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {P} +\mathrm {N} }}={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {TP} +\mathrm {TN} +\mathrm {FP} +\mathrm {FN} }}$$
$$=$$
balanced accuracy (BA)
$$\mathrm {BA} ={\frac {TPR+TNR}{2}}$$
$$=$$
F1 score
is the harmonic mean of precision and sensitivity
$$\mathrm {F} _{1}=2\cdot {\frac {\mathrm {PPV} \cdot \mathrm {TPR} }{\mathrm {PPV} +\mathrm {TPR} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}$$
$$=$$
Matthews correlation coefficient (MCC)
$$\mathrm {MCC} ={\frac {\mathrm {TP} \times \mathrm {TN} -\mathrm {FP} \times \mathrm {FN} }{\sqrt {(\mathrm {TP} +\mathrm {FP} )(\mathrm {TP} +\mathrm {FN} )(\mathrm {TN} +\mathrm {FP} )(\mathrm {TN} +\mathrm {FN} )}}}$$
$$=$$
Fowlkes–Mallows index (FM)
$$\mathrm {FM} ={\sqrt {{\frac {TP}{TP+FP}}\cdot {\frac {TP}{TP+FN}}}}={\sqrt {PPV\cdot TPR}}$$
$$=$$
informedness or bookmaker informedness (BM)
$$\mathrm {BM} =\mathrm {TPR} +\mathrm {TNR} -1$$
$$=$$
markedness(MK) or deltaP
$$\mathrm {MK} =\mathrm {PPV} +\mathrm {NPV} -1$$
$$=$$