# Common Medical and Statistical Problems: The Dilemma of the Sample Size Calculation for Sensitivity and Specificity Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Interval Estimation Using Different Methods

#### 2.2. New Expressions for Coverage Probability and Expected Length of a Conditional Probability Interval

#### 2.3. Optimal and Approximate Sample Size Determination

## 3. Results

#### 3.1. New Expressions for Coverage Probability of Interval Estimation Methods

#### 3.2. Impact on Sample Sizes

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CI | Confidence interval |

CP | Coverage probability |

EL | Expected length |

Se | Sensitivity |

Sp | Specificity |

## Appendix A

## Appendix B

**Table A1.**Two-sided $100\times (1-\alpha )\%$ confidence intervals for a binomial proportion, $\left[L\right(X)\phantom{\rule{0.166667em}{0ex}},U(X\left)\right]$, where X is the number of successes, and R commands.

Method | R command $\left[L\right(X)\phantom{\rule{0.166667em}{0ex}},U(X\left)\right]$ |

Clopper–Pearson | prop.ci($X,n,\alpha ,\u201c\mathtt{ClopperP}\u201d$) |

$X=0$ | $\left[0\phantom{\rule{0.166667em}{0ex}},1-{\left(\frac{\alpha}{2}\right)}^{\frac{1}{n}}\right]$ |

$0<X<n$ | $\left[Bet{a}_{\frac{\alpha}{2}}(X,n-X+1)\phantom{\rule{0.166667em}{0ex}},Bet{a}_{1-\frac{\alpha}{2}}(X+1,n-X)\right]$ |

$X=n$ | $\left[{\left(\frac{\alpha}{2}\right)}^{\frac{1}{n}}\phantom{\rule{0.166667em}{0ex}},1\right]$ |

Bayesian-U | prop.ci($X,n,\alpha ,\u201c\mathtt{BayesianU}\u201d$) |

$X=0$ | $\left[0\phantom{\rule{0.166667em}{0ex}},1-{\alpha}^{\frac{1}{n+1}}\right]$ |

$0<X<n$ | $\left[Bet{a}_{\frac{\alpha}{2}}(X+1,n-X+1)\phantom{\rule{0.166667em}{0ex}},Bet{a}_{1-\frac{\alpha}{2}}(X+1,n-X+1)\right]$ |

$X=n$ | $\left[{\alpha}^{\frac{1}{n+1}}\phantom{\rule{0.166667em}{0ex}},1\right]$ |

Jeffreys | prop.ci($X,n,\alpha ,\u201c\mathtt{Jef}\u201d$) |

$X=0$ | $\left[0\phantom{\rule{0.166667em}{0ex}},1-{\left(\frac{\alpha}{2}\right)}^{\frac{1}{n}}\right]$ |

$X=1$ | $\left[0\phantom{\rule{0.166667em}{0ex}},Bet{a}_{1-\frac{\alpha}{2}}(2,n)\right]$ |

$1<X<n-1$ | $\left[Bet{a}_{\frac{\alpha}{2}}(X+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}},n-X+\frac{1}{2}),Bet{a}_{1-\frac{\alpha}{2}}(X+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}},n-X+\frac{1}{2})\right]$ |

$X=n-1$ | $\left[Bet{a}_{\frac{\alpha}{2}}(n,2)\phantom{\rule{0.166667em}{0ex}},1\right]$ |

$X=n$ | $\left[{\left(\frac{\alpha}{2}\right)}^{\frac{1}{n}}\phantom{\rule{0.166667em}{0ex}},1\right]$ |

Agresti–Coull | prop.ci($X,n,\alpha ,\u201c\mathtt{AgrestC}\u201d$) |

$\left[max\{\frac{X+2}{n+4}-{z}_{1-\frac{\alpha}{2}}\sqrt{\frac{X+2}{{(n+4)}^{2}}(1-\frac{X+2}{n+4})};0\},min\{\frac{X+2}{n+4}+{z}_{1-\frac{\alpha}{2}}\sqrt{\frac{X+2}{{(n+4)}^{2}}(1-\frac{X+2}{n+4})};1\}\right]$ | |

Wilson | prop.ci($X,n,\alpha ,\u201c\mathtt{Wils}\u201d$) |

$\left[\frac{2X+{z}_{1-\frac{\alpha}{2}}^{2}-{z}_{1-\frac{\alpha}{2}}\sqrt{{z}_{1-\frac{\alpha}{2}}^{2}+4X(1-\frac{X}{n})}}{2(n+{z}_{1-\frac{\alpha}{2}}^{2})}\phantom{\rule{0.166667em}{0ex}},\frac{2X+{z}_{1-\frac{\alpha}{2}}^{2}+{z}_{1-\frac{\alpha}{2}}\sqrt{{z}_{1-\frac{\alpha}{2}}^{2}+4X(1-\frac{X}{n})}}{2(n+{z}_{1-\frac{\alpha}{2}}^{2})}\right]$ | |

$X=0$ | $\left[0\phantom{\rule{0.166667em}{0ex}},{sin}^{2}(min\left\{arcsin\sqrt{\frac{\frac{3}{8}+\frac{1}{2}}{n+\frac{3}{4}}}+\frac{{z}_{1-\frac{\alpha}{2}}}{2\sqrt{n+\frac{1}{2}}};\frac{\pi}{2}\right\})\right]$ |

$0<X<n$ | $\left[{sin}^{2}(max\left\{arcsin\sqrt{\frac{\frac{3}{8}+X-\frac{1}{2}}{n+\frac{3}{4}}}-\frac{{z}_{1-\frac{\alpha}{2}}}{2\sqrt{n+\frac{1}{2}}};0\right\})\phantom{\rule{0.166667em}{0ex}},{sin}^{2}(min\left\{arcsin\sqrt{\frac{\frac{3}{8}+X+\frac{1}{2}}{n+\frac{3}{8}}}+\frac{{z}_{1-\frac{\alpha}{2}}}{2\sqrt{n+\frac{1}{2}}};\frac{\pi}{2}\right\})\right]$ |

$X=n$ | $\left[{sin}^{2}(max\left\{arcsin\sqrt{\frac{\frac{3}{8}+n-\frac{1}{2}}{n+\frac{3}{4}}}-\frac{{z}_{1-\frac{\alpha}{2}}}{2\sqrt{n+\frac{1}{2}}};0\right\})\phantom{\rule{0.166667em}{0ex}},1\right]$ |

Wald | prop.ci($X,n,\alpha ,\u201c\mathtt{Wald}\u201d$) |

$\left[max\{\frac{X}{n}-{z}_{1-\frac{\alpha}{2}}\sqrt{\frac{X}{{n}^{2}}(1-\frac{X}{n})};0\},min\{\frac{X}{n}+{z}_{1-\frac{\alpha}{2}}\sqrt{\frac{X}{{n}^{2}}(1-\frac{X}{n})};1\}\right]$ |

_{γ}and Beta

_{γ}(a,b) represent the γ—quantiles of the N(0,1) and the Beta

_{γ}(a,b) distributions, respectively.

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**Figure 1.**Coverage probability of sensitivity intervals varying with the sensitivity, for $\eta =0.25$ and $n=500$, obtained with the new formula (solid line) and the conventional formula (dashed line), for Clopper–Pearson (

**left**) and Wilson (

**right**). The horizontal spotted line marks the nominal confidence level, $95\%$.

**Figure 2.**Coverage probability of sensitivity intervals varying with the sensitivity, admitting $n=500$, obtained for $\eta =0.10$ (grey line) and $\eta =0.25$ (black line), for Clopper–Pearson (

**top left**), Agresti–Coull (

**top right**), Wilson (

**bottom left**), and Wald (

**bottom right**). The horizontal spotted line marks the nominal confidence level, $95\%$.

**Figure 3.**Coverage probability of sensitivity intervals varying with the prevalence, admitting $n=500$, obtained for $\mathrm{Se}=0.96$ (solid line) and $\mathrm{Se}=0.73$ (dashed line), for Clopper–Pearson (

**top left**), Agresti–Coull (

**top right**), Wilson (

**bottom left**), and Wald (

**bottom right**). The horizontal spotted line marks the nominal confidence level, $95\%$.

**Figure 4.**Expected length of sensitivity intervals varying with the sample size (n), for $\eta =0.05$; 0.10; 0.15; 0.25, and $\mathrm{Se}=0.70$; 0.80; 0.90; 0.95; 0.99, using Clopper–Pearson and Wilson methods with $95\%$ nominal confidence level.

**Figure 5.**Expected length of specificity intervals varying with the sample size (n), for $\eta =0.05$; 0.10; 0.15; 0.25, and $\mathrm{Sp}=0.70$; 0.80; 0.90; 0.95; 0.99, using Clopper–Pearson and Wilson methods with $95\%$ nominal confidence level.

**Table 1.**Optimal sample sizes (${n}_{optimal}$) corresponding to several sensitivities, and differences between the optimal and approximate (${n}_{aprox}$) sample sizes, $\delta ={n}_{optimal}-{n}_{aprox}$, admitting $\eta =0.10$, $\omega =0.05,\phantom{\rule{0.277778em}{0ex}}\xi ={10}^{-4}$, and $95\%$ nominal confidence level.

Clopper- | Anscombe | Agresti- | Bayesian | Jeffreys | Wilson | Wald | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Pearson | Coull | Uniform | ||||||||||||

$\mathsf{Se}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ |

0.75 | 11848 | −2 | 11855 | 5 | 11459 | −1 | 11449 | −11 | 11452 | 2 | 11538 | 78 | 11472 | 2 |

0.80 | 10165 | −45 | 10172 | −8 | 9869 | 39 | 9771 | 1 | 9847 | 37 | 9847 | 77 | 9864 | 54 |

0.85 | 8176 | −34 | 8184 | 4 | 7829 | −1 | 7852 | 22 | 7781 | −29 | 7853 | 23 | 7855 | 45 |

0.90 | 5880 | −10 | 5890 | 0 | 5589 | −21 | 5557 | 27 | 5488 | −2 | 5546 | 26 | 5532 | 12 |

0.91 | 5385 | 5 | 5395 | 5 | 5112 | 2 | 5024 | −26 | 5032 | 22 | 5031 | −19 | 5029 | 29 |

0.92 | 4877 | 7 | 4887 | −23 | 4626 | 6 | 4559 | 39 | 4521 | 31 | 4532 | 2 | 4514 | 14 |

0.93 | 4357 | 7 | 4368 | −22 | 4165 | 35 | 4045 | 35 | 3998 | 8 | 4024 | 4 | 3984 | 14 |

0.94 | 3824 | −16 | 3836 | 6 | 3638 | −2 | 3495 | 5 | 3463 | 33 | 3535 | 25 | 3441 | 11 |

0.95 | 3280 | 0 | 3293 | 3 | 3142 | 2 | 2991 | 21 | 2916 | 26 | 3008 | 28 | 2882 | 12 |

0.96 | 2723 | 3 | 2737 | 7 | 2653 | −7 | 2440 | 0 | 2341 | 1 | 2461 | 1 | 2302 | 12 |

0.97 | 2170 | 20 | 2185 | 15 | 2178 | −2 | 1915 | 5 | 1796 | 16 | 1944 | 4 | 1613 | −7 |

0.98 | 1584 | 4 | 1595 | 5 | 1722 | 2 | 1406 | 6 | 1272 | 12 | 1462 | 12 | 489 | 9 |

0.99 | 1079 | 9 | 1079 | 9 | 1284 | 14 | 931 | 11 | 913 | 13 | 1040 | 10 | NA | NA |

**Table 2.**Optimal sample sizes (${n}_{optimal}$) corresponding to several specificities, and differences between the optimal and approximate (${n}_{aprox}$) sample sizes, $\delta ={n}_{optimal}-{n}_{aprox}$, admitting $\eta =0.10$, $\omega =0.05,\phantom{\rule{0.277778em}{0ex}}\xi ={10}^{-4}$, and $95\%$ nominal confidence level.

Clopper- | Anscombe | Agresti- | Bayesian | Jeffreys | Wilson | Wald | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Pearson | Coull | Uniform | ||||||||||||

$\mathsf{Sp}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{optimal}}$ | $\mathit{\delta}$ |

0.50 | 1741 | −2 | 1742 | −1 | 1696 | 0 | 1709 | 11 | 1697 | −1 | 1696 | 0 | 1700 | 0 |

0.55 | 1724 | −1 | 1738 | 12 | 1691 | 2 | 1685 | −4 | 1680 | 0 | 1685 | −4 | 1695 | 1 |

0.60 | 1673 | −1 | 1674 | −2 | 1628 | −1 | 1640 | 11 | 1629 | 0 | 1640 | 11 | 1632 | −2 |

0.65 | 1600 | 12 | 1601 | 12 | 1543 | −2 | 1543 | −2 | 1556 | 11 | 1543 | −2 | 1547 | 0 |

0.70 | 1469 | −1 | 1471 | 1 | 1428 | 2 | 1425 | −1 | 1436 | 10 | 1424 | −2 | 1433 | 1 |

0.75 | 1325 | 8 | 1317 | 0 | 1273 | −1 | 1275 | 1 | 1282 | 9 | 1281 | 7 | 1274 | −1 |

0.80 | 1130 | −5 | 1138 | 6 | 1093 | 0 | 1093 | 7 | 1093 | 3 | 1093 | 7 | 1095 | 5 |

0.85 | 908 | −5 | 915 | 6 | 870 | 0 | 865 | −5 | 865 | −3 | 871 | 1 | 870 | 2 |

0.90 | 657 | 2 | 654 | −1 | 621 | −3 | 616 | 1 | 610 | 0 | 617 | 3 | 614 | 0 |

0.91 | 602 | 4 | 603 | 4 | 571 | 3 | 561 | −1 | 558 | 1 | 559 | −3 | 558 | 2 |

0.92 | 545 | 3 | 546 | 0 | 514 | 0 | 502 | −1 | 498 | −1 | 503 | −1 | 500 | 0 |

0.93 | 484 | 0 | 485 | −3 | 462 | 3 | 446 | 0 | 443 | −1 | 447 | 0 | 442 | 0 |

0.94 | 425 | −2 | 426 | 0 | 405 | 0 | 390 | 2 | 384 | 2 | 392 | 2 | 381 | −1 |

0.95 | 366 | 1 | 366 | 0 | 349 | 0 | 331 | 1 | 323 | 1 | 333 | 1 | 319 | 0 |

0.96 | 304 | 1 | 305 | 1 | 294 | −2 | 271 | −1 | 260 | 0 | 274 | 0 | 254 | −1 |

0.97 | 240 | 1 | 241 | −1 | 242 | −1 | 213 | 0 | 198 | 0 | 216 | 0 | 179 | −1 |

0.98 | 176 | 0 | 177 | 0 | 191 | −1 | 156 | 0 | 140 | 0 | 161 | −1 | 54 | 0 |

0.99 | 119 | 0 | 119 | 0 | 142 | 0 | 103 | 0 | 100 | 0 | 114 | −1 | NA | NA |

**Table 3.**Optimal sample sizes (${n}_{optimal}$) corresponding to several sensitivities, for the Clopper–Pearson and Wilson methods, with $\omega \phantom{\rule{0.166667em}{0ex}}$ varying between $0.05$ and $0.10$, admitting $\eta =0.10$, $\xi ={10}^{-4}$, and $95\%$ nominal confidence level.

Interval Width ($\mathit{\omega}$) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.05 | 0.06 | 0.07 | 0.08 | 0.09 | 0.10 | |||||||

$\mathbf{Se}$ | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson |

0.75 | 11,848 | 11,538 | 8280 | 7997 | 6119 | 5831 | 4711 | 4459 | 3742 | 3532 | 3046 | 2854 |

0.80 | 10,165 | 9847 | 7110 | 6826 | 5259 | 5006 | 4053 | 3825 | 3222 | 3003 | 2625 | 2428 |

0.85 | 8176 | 7853 | 5728 | 5445 | 4243 | 5445 | 3275 | 3995 | 2607 | 3054 | 2126 | 2399 |

0.90 | 5880 | 5546 | 4133 | 3838 | 3071 | 2839 | 2377 | 2165 | 1897 | 1719 | 1552 | 1389 |

0.91 | 5385 | 5031 | 3789 | 3523 | 2818 | 2578 | 2183 | 1987 | 1744 | 1567 | 1428 | 1272 |

0.92 | 4877 | 4532 | 3436 | 3158 | 2559 | 2340 | 1984 | 1797 | 1587 | 1419 | 1301 | 1158 |

0.93 | 4357 | 4024 | 3074 | 2808 | 2293 | 2074 | 1781 | 1604 | 1426 | 1270 | 1170 | 1036 |

0.94 | 3824 | 3535 | 2705 | 2454 | 2021 | 1827 | 1573 | 1404 | 1262 | 1121 | 1040 | 917 |

0.95 | 3280 | 3008 | 2326 | 2098 | 1743 | 1569 | 1360 | 1217 | 1094 | 976 | 905 | 801 |

0.96 | 2723 | 2461 | 1951 | 1742 | 1460 | 1307 | 1148 | 1028 | 925 | 829 | 768 | 691 |

0.97 | 2170 | 1944 | 1555 | 1403 | 1174 | 1064 | 930 | 849 | 763 | 697 | 639 | 586 |

0.98 | 1584 | 1462 | 1167 | 1080 | 903 | 845 | 732 | 686 | 613 | 576 | 525 | 494 |

0.99 | 1079 | 1040 | 833 | 807 | 679 | 659 | 571 | 554 | 491 | 476 | 431 | 417 |

**Table 4.**Optimal sample sizes (${n}_{optimal}$) corresponding to several specificities, for the Clopper–Pearson and Wilson methods, with $\omega \phantom{\rule{0.166667em}{0ex}}$ varying between $0.05$ and $0.10$, admitting $\eta =0.10$, $\xi ={10}^{-4}$, and $95\%$ nominal confidence level.

Interval Width ($\mathit{\omega}$) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.05 | 0.06 | 0.07 | 0.08 | 0.09 | 0.10 | |||||||

$\mathbf{Sp}$ | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson | Clopper-Pearson | Wilson |

0.50 | 1741 | 1696 | 1222 | 1184 | 901 | 864 | 692 | 663 | 547 | 521 | 445 | 421 |

0.55 | 1724 | 1685 | 1203 | 1172 | 888 | 855 | 683 | 656 | 542 | 517 | 440 | 418 |

0.60 | 1673 | 1640 | 1171 | 1130 | 862 | 833 | 665 | 636 | 527 | 500 | 428 | 405 |

0.65 | 1600 | 1543 | 1115 | 1077 | 822 | 786 | 630 | 601 | 501 | 474 | 406 | 384 |

0.70 | 1469 | 1424 | 1028 | 988 | 760 | 726 | 583 | 556 | 464 | 437 | 377 | 354 |

0.75 | 1325 | 1281 | 925 | 887 | 680 | 650 | 523 | 497 | 416 | 391 | 339 | 316 |

0.80 | 1130 | 1093 | 790 | 753 | 586 | 553 | 450 | 424 | 358 | 334 | 291 | 270 |

0.85 | 908 | 871 | 636 | 604 | 471 | 443 | 364 | 338 | 289 | 267 | 236 | 215 |

0.90 | 657 | 617 | 459 | 426 | 342 | 314 | 264 | 240 | 211 | 190 | 172 | 154 |

0.91 | 602 | 559 | 421 | 390 | 313 | 286 | 242 | 220 | 193 | 174 | 158 | 141 |

0.92 | 545 | 503 | 383 | 351 | 284 | 259 | 220 | 199 | 176 | 157 | 144 | 128 |

0.93 | 484 | 447 | 341 | 313 | 255 | 231 | 198 | 177 | 158 | 141 | 130 | 115 |

0.94 | 425 | 392 | 301 | 273 | 225 | 202 | 174 | 156 | 140 | 124 | 115 | 101 |

0.95 | 366 | 333 | 258 | 233 | 193 | 173 | 151 | 134 | 121 | 108 | 100 | 88 |

0.96 | 304 | 274 | 216 | 194 | 162 | 145 | 127 | 113 | 102 | 92 | 85 | 76 |

0.97 | 240 | 216 | 172 | 155 | 130 | 118 | 103 | 93 | 84 | 77 | 70 | 64 |

0.98 | 176 | 161 | 129 | 119 | 100 | 93 | 81 | 76 | 67 | 63 | 57 | 54 |

0.99 | 119 | 114 | 92 | 89 | 75 | 72 | 63 | 61 | 54 | 52 | 47 | 45 |

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**MDPI and ACS Style**

Oliveira, M.R.; Subtil, A.; Gonçalves, L. Common Medical and Statistical Problems: The Dilemma of the Sample Size Calculation for Sensitivity and Specificity Estimation. *Mathematics* **2020**, *8*, 1258.
https://doi.org/10.3390/math8081258

**AMA Style**

Oliveira MR, Subtil A, Gonçalves L. Common Medical and Statistical Problems: The Dilemma of the Sample Size Calculation for Sensitivity and Specificity Estimation. *Mathematics*. 2020; 8(8):1258.
https://doi.org/10.3390/math8081258

**Chicago/Turabian Style**

Oliveira, M. Rosário, Ana Subtil, and Luzia Gonçalves. 2020. "Common Medical and Statistical Problems: The Dilemma of the Sample Size Calculation for Sensitivity and Specificity Estimation" *Mathematics* 8, no. 8: 1258.
https://doi.org/10.3390/math8081258